The A-Level Physics Insert for January 2022 (Paper 4) is a datasheet of essential formulae provided during the exam. While you can use these equations directly, understanding their derivations deepens your grasp of physical principles and improves problem-solving skills. This article explores the step-by-step derivations of some of the most critical formulae from the insert.
Consider an object moving with constant acceleration a from initial velocity u to final velocity v over time t. Acceleration is defined as the rate of change of velocity:
考虑一个物体以恒定加速度 a 从初速度 u 加速到末速度 v，历时 t。加速度定义为速度的变化率：
a = (v − u) / t ⟹ v = u + a t
This gives the first SUVAT equation. For displacement s, we use the average velocity, which is (u+v)/2 for constant acceleration. Since displacement is average velocity multiplied by time:
Understanding the WJEC IGCSE Physics marking scheme is not just about knowing how many marks are awarded for a correct answer. It is a guide that reveals what examiners truly value, where the common pitfalls lie, and how to structure revision to maximise efficiency. In this article, we break down the assessment objectives, command words, mathematical requirements, and practical skill expectations that underpin the WJEC marking philosophy, empowering students to turn mark scheme insight into higher scores.
1. Overview of the WJEC IGCSE Physics Specification | WJEC IGCSE 物理课程大纲概览
The WJEC IGCSE Physics qualification (specification code 3120) is designed to provide a broad, coherent, and practical introduction to physics. It is assessed through two written papers and a practical assessment component, with all content drawn from a single-tier entry system that covers grades A* to G. The specification emphasises both knowledge with understanding and the application of scientific methods, ensuring that students develop critical thinking alongside factual recall.
WJEC IGCSE 物理资格证书（课程代码 3120）旨在提供广泛、连贯且注重实践的物理学入门。它通过两份书面试卷和一个实验评估部分进行评估，所有内容均来自统一的单层级考试体系，覆盖 A* 至 G 等级。该大纲强调知识与理解并存，以及科学方法的应用，确保学生在掌握事实性知识的同时发展批判性思维。
The syllabus is structured around core topics: forces and motion, electricity, waves, energy resources and energy transfer, solids, liquids and gases, magnetism and electromagnetism, radioactivity, and space physics. Each topic carries a specific weighting in the examination, which directly influences how students allocate their revision time. A detailed breakdown of topic weightings is not always published, but past papers show a consistent pattern: mechanics and electricity typically account for around 40% of the total marks, making them high-priority areas.
2. Assessment Objectives and Their Weightings | 评估目标及其权重
WJEC uses three Assessment Objectives (AOs) to evaluate student performance. AO1 focuses on demonstrating knowledge and understanding of scientific ideas, techniques, and procedures. This includes recalling facts, defining key terms, and describing phenomena. AO1 typically constitutes about 40% of the total marks across the qualification. AO2 requires students to apply knowledge and understanding in both familiar and unfamiliar contexts, including interpreting data, solving problems, and explaining patterns. This objective makes up roughly 40% as well. AO3, worth around 20%, assesses the ability to analyse, evaluate, and draw conclusions from scientific information, often involving practical scenarios or novel data sets.
Understanding this split is crucial because it shapes the nature of revision. Students who only memorise the textbook will struggle with AO2 and AO3 questions, which demand application and evaluation. The marking schemes consistently reward clear, logical steps in calculations and evidence-based reasoning in longer-answer questions. Therefore, effective preparation must go beyond recall and incorporate regular practice of past paper questions under timed conditions.
A common misconception is that AO3 only appears in practical-based questions. In reality, the analysis and evaluation skills are woven throughout the papers, including data interpretation in electricity graphs or evaluating the design of an experiment on motion. Mark schemes for AO3 questions usually carry specific indicative content, but also allow for alternative valid approaches, emphasising the importance of showing a line of reasoning.
3. Examination Structure and Component Breakdown | 考试结构与各部分分解
The WJEC IGCSE Physics assessment consists of two externally marked written papers and a practical assessment unit. Paper 1 covers the core content and lasts 1 hour 45 minutes, contributing 45% of the qualification. Paper 2 is also 1 hour 45 minutes long, covers additional content as well as synoptic elements, and contributes another 45%. The remaining 10% comes from the practical assessment, which is internally assessed and externally moderated, known as Unit 3: Practical Assessment.
Both written papers include a mix of question types: multiple choice, structured, and extended response. The mark allocation guides time management: a 6-mark extended question should receive proportionally more time than a 1-mark multiple choice. A typical paper contains around 80-100 marks, so students should aim to spend roughly 1 minute per mark on average. The practical assessment is conducted in the school laboratory, with students completing tasks that test planning, data collection, and analysis skills.
Table 1 summarises the component breakdown for the latest specification.
Component
Weighting
Duration
Paper 1: Core Physics
45%
1h 45m
Paper 2: Additional Physics
45%
1h 45m
Unit 3: Practical Assessment
10%
School-based
Knowing the structure allows targeted preparation: for instance, the synoptic nature of Paper 2 means students must be able to connect concepts from different topics, such as applying energy transfer ideas to electricity or forces to wave behaviour.
4. Types of Questions and Mark Allocation | 问题类型与分数分配
Multiple-choice questions typically test AO1 knowledge directly, requiring quick recall of definitions, units, or fundamental laws. Each correct option earns one mark; there is no negative marking. Structured questions break a larger problem into smaller parts, often starting with straightforward recall and progressing to application and analysis. The mark scheme for these questions uses a point-based system, where each mark corresponds to a specific piece of correct information or a step in a calculation.
Extended response questions, typically worth 4 to 6 marks, are assessed using a levels-of-response mark scheme. This approach judges the overall quality of the answer rather than simply tallying correct points. Examiners look for a logical structure, accurate use of scientific terminology, and clear explanations that link cause and effect. For example, a 6-mark question on the factors affecting the resistance of a wire would require not just listing factors but explaining the underlying mechanism with reference to electron flow and collisions with ions.
In calculation questions, marks are awarded for both the correct answer and the working out. Even if the final answer is wrong, students can earn method marks by showing the correct equation, substitution, and rearrangement. Mark schemes typically use the convention ‘M’ for method marks, ‘A’ for accuracy marks, and ‘B’ for independent marks. Thus, writing ‘F = m × a’ and substituting values can secure marks even if the arithmetic slips.
5. Command Words and What Examiners Expect | 指令词及考官的期望
Command words are the verbs that define the type of response required. In WJEC mark schemes, ‘State’ asks for a simple, brief answer, such as a definition or a name, with no explanation needed. ‘Describe’ requires a detailed account of a process or phenomenon, often using observations or characteristics. ‘Explain’ goes a step further: it demands giving reasons or causes, usually linking scientific principles to the observed effect. ‘Calculate’ expects numerical working leading to a final value, with units included.
Other command words include ‘Compare’, which needs similarities and differences, often using comparative language such as ‘higher than’ or ‘whereas’. ‘Evaluate’ demands a judgement based on evidence, weighing pros and cons. Crucially, if a question asks ‘Explain why’, a response that only describes what happens will not reach the higher marking bands. Examiners’ reports frequently note that students confuse ‘describe’ and ‘explain’, leading to lost marks.
For ‘Suggest’ questions, the mark scheme often accepts any scientifically plausible answer, awarding marks for creativity based on correct principles. This rewards students who can apply their knowledge flexibly rather than just regurgitating memorised facts. Therefore, when tackling ‘Suggest’ questions, candidates should propose ideas rooted in physics concepts, even if the exact scenario has not been taught.
WJEC IGCSE Physics uses a criterion-referenced grading system, but grade boundaries are set after each examination series to align with the difficulty of the papers. The raw mark needed for each grade boundary (A*, A, B, C, etc.) can fluctuate from session to session. Typically, to achieve an A*, a student must score around 80-85% of the total uniform marks, though this is an estimate. The practical assessment unit contributes to the overall grade, so strong performance in the practical can compensate for borderline written paper results.
WJEC IGCSE 物理采用标准参照的评分体系，但等级分数线在每次考试后根据试卷难度进行调整。每个等级（A*, A, B, C 等）所需的原始分阈值可能因考次而异。通常，要想获得 A*，学生需拿到统一标准分总分的大约 80-85%，不过这只是一个估计。实验评估单元对总成绩有贡献，因此实验部分的强劲表现可以弥补笔试纸卷的边缘成绩。
The raw marks from each component are scaled to uniform marks (UMS) to ensure consistency across different series. The maximum UMS for the qualification is typically 200 (90 for Paper 1, 90 for Paper 2, and 20 for the practical). Grade boundaries in UMS are fixed: A* at 180 UMS, A at 160, B at 140, C at 120, and so on. This stability means students can set target UMS scores early in their course.
Unit 3: Practical Assessment is worth 10% of the total IGCSE Physics marks, but its influence on the final grade should not be underestimated. The assessment is conducted in schools under controlled conditions, with tasks set by WJEC and marked by teachers using standardised marking criteria. The tasks typically cover planning an investigation, obtaining and recording evidence, analysing results, and evaluating the method and data.
The marking criteria for practical work align closely with AO3 skills. Students must demonstrate the ability to identify variables, select appropriate apparatus, take measurements with precision, and present data in tables with correct units. They are also marked on identifying anomalies, suggesting improvements, and drawing conclusions consistent with the evidence. One common weakness, as noted in examiner reports, is the failure to specify control variables clearly, leading to loss of marks even when the experiment is practically sound.
For instance, an investigation into how the length of a wire affects its resistance requires the current and the material to be kept constant. Simply stating ‘keep the same equipment’ is vague; the mark scheme expects ‘use the same wire material and maintain a constant current throughout’. Precision in language matters.
8. Mathematical Requirements and Marking | 数学要求与评分方式
Physics is inherently mathematical, and WJEC specifies the mathematical skills that can be assessed in the IGCSE exams. These include arithmetic, algebra, graph plotting and interpretation, geometry, and trigonometry. At least 20% of the marks across the qualification involve mathematical application. Mathematical marks are integrated within physics questions, not isolated. A typical calculation will involve substituting values into a formula, converting units, and stating the answer to an appropriate number of significant figures.
The mark scheme awards marks for the correct equation, substitution, rearrangement, and final answer with units. If a student writes the answer with an incorrect or missing unit, an accuracy mark (A mark) may be withheld. Significant figures also matter: as a rule of thumb, final answers should be given to the same number of significant figures as the least precise data provided in the question, or to 2 or 3 significant figures for general calculations. Pro tip: write the final answer to 3 significant figures when in doubt, unless the question specifies otherwise.
评分标准对正确的方程、代入、变形以及带单位的最终答案给予分数。如果学生写出答案时单位错误或遗漏，一个准确分 (A 分) 可能被扣掉。有效数字同样重要：根据经验法则，最终答案的有效数字位数应与题目提供的最不精确的数据一致，或一般计算取 2 到 3 位有效数字。专业建议：不确定时，最终答案写 3 位有效数字，除非题目另有规定。
Graph skills are a particular focus. Plotting points accurately, drawing a line of best fit, and calculating gradients and intercepts are routine tasks. Marks are awarded for choosing sensible scales, labelling axes with quantities and units, and deriving a correct relation from the gradient. Common mistakes include plotting points as crosses that are too large and forcing the best-fit line through the origin when it is not justified.
9. Common Pitfalls and How Mark Schemes Address Them | 常见失分点及评分标准如何处理
Examiner reports consistently highlight the same errors across different examination sessions. One frequent mistake is not reading the question carefully, such as providing a ‘description’ when ‘explanation’ is required, or missing the word ‘not’ in a prompt. Mark schemes will not reward mismatched responses. Another pitfall is omitting units in final answers of calculations; many schemes explicitly state ‘award the accuracy mark only if the correct unit is included’.
In extended writing, students often present bullet points instead of a coherent prose, which may limit the marks for structure and logical flow. The levels-of-response mark scheme expects a ‘well-structured account’ with ideas presented in a sequence. Furthermore, irrelevant information does not gain marks but may waste time, so sticking strictly to the question’s focus is key. Overgeneralisation, such as writing ‘the resistance increases’ without specifying what it depends on, can lead to vague answers that fail to satisfy the mark scheme requirements.
For numerical work, failure to show working means no method marks can be awarded if the final answer is wrong. The mark scheme is explicit: ‘Consequential errors are permitted but must follow through from a previously correct step’. This means if a student makes a slip in step 2 but uses that value correctly in step 3, they can still earn marks for step 3. However, if no working is shown, the examiner cannot award method marks, and an incorrect final answer yields zero.
10. Tips for Maximising Marks Using the Mark Scheme | 利用评分标准最大化得分的技巧
Active use of mark schemes during revision is one of the most effective strategies to improve exam performance. After attempting a past paper, students should mark their own work using the official mark scheme, paying close attention to the specific phrasing that earns marks. This exercise trains the student to understand exactly what the examiner is looking for, transforming mark schemes from a grading tool into a revision resource.
Another technique is to create ‘model answer’ flashcards for common extended response themes, such as ‘Explain how a transformer works’ or ‘Describe the process of radioactive decay and its applications’. On one side, write the question; on the other, write a full-mark answer using language directly from the mark scheme. This method embeds the expected terminology and logical structure into memory.
Time management in the exam can be optimised by understanding mark allocations. A 1-mark question should take no more than a minute, whereas a 6-mark question may require 7-8 minutes. Practising under timed conditions with a mark scheme in mind helps build the instinct to move on and not dwell on low-mark questions.
Finally, review the examiner reports alongside the mark schemes for the past papers you attempt. These reports provide commentary on where students commonly lost marks and what distinguished high-scoring scripts. They often include examples of both good and poor answers, giving a concrete reference for improvement.
📚 GCSE Physics: Work and Energy – Key Points | GCSE 物理：功与能量 考点精讲
In GCSE Physics, work and energy are core concepts that explain how forces cause movement and how energy is stored, transferred and conserved. Mastering these ideas helps you solve problems involving motion, machines and everyday energy changes. This article covers the essential definitions, equations and worked examples you need to ace your exams.
Work is done whenever a force makes something move. In everyday language, ‘work’ can mean any mental or physical effort, but in physics it has a strict meaning – it measures the amount of energy transferred by a force acting over a distance.
If you push against a wall and it does not move, no work is done on the wall because there is no displacement, even though you feel tired. Work requires both a force and movement in the direction of the force.
Work done is a scalar quantity measured in joules (J). One joule is equal to one newton-metre (1 J = 1 N m). The formula for work done is straightforward when force and displacement are in the same direction.
功是一个标量，单位是焦耳（J）。1焦耳等于1牛顿·米（1 J = 1 N m）。当力与位移同向时，功的计算公式非常简单。
Work done = force × distance moved in the direction of the force
W = F × d
W is work (J), F is force (N), and d is distance (m). If force is applied at an angle, you must use the component of force parallel to the movement. GCSE usually deals with forces directly along the line of motion.
W 为功（J），F 为力（N），d 为距离（m）。如果力与运动方向有夹角，则需使用沿运动方向的力分量。GCSE阶段通常处理力与运动共线的情况。
2. Calculating Work Done | 功的计算
In simple situations, you multiply the steady force by the distance the object moves while the force acts. For example, if a gardener pushes a wheelbarrow with a constant force of 80 N over a distance of 5 m, the work done is W = 80 N × 5 m = 400 J.
在简单情况下，可以用恒力的大小乘以物体在力作用期间移动的距离。例如，园丁用80 N的恒力推动独轮车移动5 m，则做功 W = 80 N × 5 m = 400 J。
If a force varies, work can be found from the area under a force-displacement graph, but GCSE papers focus on constant forces. Always check that the distance used is the distance moved in the direction of the force.
Be careful when forces are perpendicular to motion. If you carry a suitcase while walking horizontally, the upward force you apply does no work on the suitcase because there is no vertical movement. The suitcase does not gain gravitational potential energy horizontally.
Conversely, when you lift the suitcase vertically, work is done against gravity, and the work done equals the gravitational potential energy gained.
反之，当你竖直提起行李箱时，克服重力做了功，所做的功等于增加的重力势能。
3. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object because of its height above a reference point, usually the ground. The higher an object is lifted, the more GPE it stores.
重力势能是物体因其离地高度而储存的能量。物体被抬得越高，储存的重力势能越多。
The equation for GPE is:
重力势能的公式为：
GPE = m × g × h
where m is mass in kilograms (kg), g is gravitational field strength in newtons per kilogram (N/kg), and h is vertical height in metres (m). On Earth, g is about 9.8 N/kg, but exams often use 10 N/kg for simplicity.
其中 m 为质量（kg），g 为重力场强度（N/kg），h 为竖直高度（m）。在地球上 g 约为 9.8 N/kg，考试中常简化为 10 N/kg。
For example, a 2 kg book lifted onto a shelf 1.5 m high gains GPE = 2 kg × 10 N/kg × 1.5 m = 30 J. That energy originally came from the person lifting the book, who did 30 J of work.
例如，一本 2 kg 的书被抬到 1.5 m 高的书架上，获得的重力势能为 GPE = 2 kg × 10 N/kg × 1.5 m = 30 J。这份能量来自抬书的人，他做了 30 J 的功。
When an object falls freely, its GPE decreases as it converts into kinetic energy, assuming no air resistance. This conversion is a key example of the conservation of energy.
当物体自由下落时，其重力势能减少，转化为动能，假设没有空气阻力。这种转化是能量守恒的一个重要实例。
4. Kinetic Energy | 动能
Kinetic energy (KE) is the energy an object possesses due to its motion. Any moving object – a rolling ball, a flying arrow, a spinning wheel – has kinetic energy.
动能是物体由于运动而具有的能量。任何运动的物体——滚动的球、飞行的箭、旋转的车轮——都具有动能。
The kinetic energy equation is:
动能公式为：
KE = ½ × m × v²
m is mass (kg) and v is speed in metres per second (m/s). Because speed is squared, a small increase in speed causes a large increase in KE. Doubling the speed makes the kinetic energy four times greater.
m 为质量（kg），v 为速度（m/s）。由于速度是平方项，速度的小幅增加会导致动能大幅增加。速度加倍，动能变为原来的四倍。
This relationship explains why high-speed collisions are far more dangerous. A car travelling at 60 mph has four times the kinetic energy of the same car at 30 mph, not twice.
Kinetic energy is measured in joules (J). When a car brakes, friction converts its kinetic energy into thermal energy in the brakes and tyres, slowing it down.
动能以焦耳（J）为单位。汽车刹车时，摩擦力将其动能转化为刹车和轮胎中的热能，从而减速。
5. Elastic Potential Energy | 弹性势能
Elastic potential energy (EPE) is stored in objects that can be stretched or compressed, such as springs, elastic bands, and bungee cords. This energy is released when the object returns to its original shape.
弹性势能储存在可拉伸或压缩的物体中，如弹簧、橡皮筋和蹦极绳。当物体恢复原状时，这部分能量会被释放。
The formula for elastic potential energy, within the elastic limit, is:
This guide provides a concise summary of the most important concepts, formulas, and tips for the IGCSE CCEA Physics exam. Use it to quickly review key topics and common pitfalls.
1. Key Quantities, Units and Measurement | 关键物理量、单位和测量
A scalar quantity has only magnitude (e.g., mass, distance, speed). A vector quantity has both magnitude and direction (e.g., velocity, displacement, force). Always check if direction matters in an answer.
All base quantities are measured in SI units: length in metres (m), mass in kilograms (kg), time in seconds (s), electric current in amperes (A), temperature in kelvin (K). Prefixes like kilo (103), milli (10-3) and micro (10-6) must be known for conversions.
When using measuring instruments, parallax error is reduced by viewing the scale perpendicular to the pointer or meniscus. For micrometers and vernier calipers, remember to check for zero error before taking readings.
使用测量仪器时，视线应与指针或液面凹面垂直以减小视差。使用千分尺和游标卡尺前，务必检查零误差。
2. Motion | 运动
Distance is a scalar measuring the total path length; displacement is a vector measuring the shortest straight-line distance from start to finish. Speed is the rate of change of distance; velocity is the rate of change of displacement.
On a distance–time graph, the gradient gives speed. On a velocity–time graph, the gradient gives acceleration, and the area under the graph gives displacement. A horizontal line on a v–t graph means constant velocity (zero acceleration).
For uniform acceleration in a straight line, use the SUVAT equations. The key formulas are:
v = u + a t
s = u t + ½ a t2
v2 = u2 + 2 a s
s = ½ (u + v) t
These only apply when acceleration ‘a’ is constant. The acceleration of free fall near the Earth’s surface is approximately 9.8 m s-2 (often rounded to 10 m s-2 in CCEA problems).
只适用于加速度恒定的情况。地表附近自由落体加速度 g 约为 9.8 m s-2（CCEA 试题中常取 10 m s-2）。
3. Forces and Newton’s Laws | 力和牛顿定律
Newton’s First Law: An object remains at rest or in uniform motion in a straight line unless acted on by a resultant force. Mass is a measure of inertia.
牛顿第一定律：除非受到合力作用，物体将保持静止或匀速直线运动。质量是惯性的量度。
Newton’s Second Law: Resultant force = mass × acceleration, F = m a. This force must be in newtons (N) when mass is in kg and acceleration in m s-2.
牛顿第二定律：合力 = 质量 × 加速度，F = m a。当质量单位为 kg、加速度单位为 m s-2 时，力单位为牛顿(N)。
Newton’s Third Law: For every action force there is an equal and opposite reaction force. These forces act on different objects, so they do not cancel out for a single object.
Weight is the gravitational force on an object: W = m g. Always distinguish between mass (kg, scalar) and weight (N, vector). Friction and air resistance oppose motion; the terminal velocity is reached when weight equals air resistance, giving zero resultant force.
重量是物体所受的重力：W = m g。必须区分质量（kg，标量）与重量（N，矢量）。摩擦力和空气阻力阻碍运动；当重力与空气阻力平衡时，合力为零，物体达到终端速度。
4. Momentum | 动量
Momentum is defined as the product of mass and velocity: p = m v. Momentum is a vector quantity and has the unit kg m s-1.
动量定义为质量和速度的乘积：p = m v。动量是矢量，单位是 kg m s-1。
In a closed system, total momentum before a collision or explosion equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This principle of conservation of momentum is used to solve collision and recoil problems.
The change in momentum is called impulse: F t = Δp = m v – m u. Increasing the time of impact reduces the force, which explains how crumple zones, air bags and seat belts work to protect passengers.
动量的变化称为冲量：F t = Δp = m v – m u。延长碰撞时间可以减小作用力，这就是安全气囊、安全带和溃缩区的保护原理。
5. Energy, Work and Power | 能量、功和功率
Energy can exist in many forms (kinetic, gravitational potential, thermal, chemical, nuclear, etc.) and is always conserved. Work is done when a force moves an object in the direction of the force: W = F d (unit: joule, J).
能量有多种形式（动能、重力势能、热能、化学能、核能等），且总是守恒的。力使物体沿力的方向移动时做功：W = F d（单位：焦耳，J）。
Kinetic energy: KE = ½ m v2. Gravitational potential energy: GPE = m g h. For a falling object, the loss in GPE equals the gain in KE if air resistance is negligible.
动能：KE = ½ m v2。重力势能：GPE = m g h。忽略空气阻力时，物体下落减少的重力势能等于增加的动能。
Power is the rate of doing work or transferring energy: P = W / t (unit: watt, W). For an object moving at constant speed v against a force F, power can also be calculated as P = F v. Efficiency = useful energy output / total energy input (× 100%).
功率是做功或能量转移的速率：P = W / t（单位：瓦特，W）。物体以恒定速度 v 克服力 F 运动时，也可用 P = F v。效率 = 有用输出能量 / 总输入能量 (× 100%)。
6. Density and Pressure | 密度和压强
Density is mass per unit volume: ρ = m / V. The unit is kg m-3, though g cm-3 is also common. Recall that 1 g cm-3 = 1000 kg m-3.
密度是单位体积的质量：ρ = m / V。单位为 kg m-3，也常用 g cm-3。注意 1 g cm-3 = 1000 kg m-3。
Pressure on a surface is force per unit area: p = F / A (unit: pascal, Pa, where 1 Pa = 1 N m-2). A larger area reduces pressure for a given force, which is why wide tyres prevent sinking.
压强是单位面积上的力：p = F / A（单位：帕斯卡，Pa，1 Pa = 1 N m-2）。给定力时，增大面积可减小压强，因此宽轮胎不易下陷。
In a liquid, pressure increases with depth and density: p = ρ g h. This pressure acts equally in all directions. Atmospheric pressure at sea level is approximately 1.0 × 105 Pa.
液体压强随深度和密度增加：p = ρ g h。液体压强在各个方向上相等。海平面标准大气压约为 1.0 × 105 Pa。
7. Waves | 波动
Waves transfer energy without transferring matter. Transverse waves (e.g., light, water ripples) have oscillations perpendicular to the direction of energy transfer. Longitudinal waves (e.g., sound) have oscillations parallel to the direction of energy transfer.
The wave equation links speed (v), frequency (f) and wavelength (λ): v = f λ. Frequency is the number of complete waves per second, measured in hertz (Hz). The period T = 1 / f.
波速公式：v = f λ。频率是每秒完整波的个数，单位为赫兹(Hz)。周期 T = 1 / f。
Reflection, refraction and diffraction are key wave phenomena. Refraction occurs because the speed of the wave changes when it enters a different medium. Diffraction is the spreading of a wave as it passes through a gap or around an obstacle; it is most noticeable when the gap size is approximately equal to the wavelength.
The electromagnetic spectrum, in order of increasing frequency/decreasing wavelength, is: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays. All travel at the same speed in a vacuum (3.0 × 108 m s-1).
电磁波谱按频率递增/波长递减排列为：无线电波、微波、红外线、可见光、紫外线、X 射线、γ 射线。它们在真空中都以相同速度传播(3.0 × 108 m s-1)。
8. Light and Optics | 光和光学
The law of reflection: the angle of incidence (i) equals the angle of reflection (r), with both measured from the normal. Plane mirrors produce virtual, laterally inverted images.
反射定律：入射角(i)等于反射角(r)，均从法线量起。平面镜产生虚像，且左右颠倒。
Refractive index n describes how much light slows down in a medium: n = sin i / sin r, where i is the angle in a vacuum (or air) and r is the angle in the medium. Snell’s law: n₁ sin θ₁ = n₂ sin θ₂.
折射率 n 描述光在介质中的减速程度：n = sin i / sin r，其中 i 为真空（或空气）中的角度，r 为介质中的角度。斯涅尔定律：n₁ sin θ₁ = n₂ sin θ₂。
Total internal reflection occurs when light travels from a denser medium to a less dense medium at an angle of incidence greater than the critical angle c. The critical angle is given by sin c = 1 / n. This principle is used in optical fibres.
当光从光密介质射向光疏介质且入射角大于临界角 c 时，发生全内反射。临界角由 sin c = 1 / n 给出。光纤利用了这一原理。
Converging (convex) lenses can produce real or virtual images depending on object distance. Use the formula 1 / f = 1 / u + 1 / v (f = focal length, u = object distance, v = image distance). The magnification m = v / u = image height / object height.
会聚透镜（凸透镜）依据物距可成实像或虚像。使用公式 1 / f = 1 / u + 1 / v（f 焦距，u 物距，v 像距）。放大率 m = v / u = 像高 / 物高。
9. Electricity | 电学
Electric current is the rate of flow of charge: I = Q / t (unit: ampere, A). In metals, current is carried by free electrons. Conventional current flows from positive to negative.
Potential difference (voltage) is the energy transferred per unit charge: V = W / Q. Resistance is the opposition to current: R = V / I (unit: ohm, Ω). Ohm’s law states that V ∝ I for a resistor at constant temperature.
电势差（电压）是单位电荷转移的能量：V = W / Q。电阻是电流受到的阻碍：R = V / I（单位：欧姆，Ω）。欧姆定律指出，在温度不变时，电阻两端电压与电流成正比。
In series circuits, current is the same everywhere, and the total resistance is the sum of individual resistors: Rtotal = R₁ + R₂ + … In parallel circuits, the total current splits, and the combined resistance is given by 1 / Rtotal = 1 / R₁ + 1 / R₂ + …
Electrical power P = I V = I2 R = V2 / R. Energy consumed E = P t = I V t. The kilowatt-hour (kW h) is a unit of energy; 1 kW h = 3.6 × 106 J.
电功率 P = I V = I2 R = V2 / R。消耗的电能 E = P t = I V t。千瓦时(kW h)是能量单位；1 kW h = 3.6 × 106 J。
10. Electromagnetism and Radioactivity | 电磁学和放射性
A current-carrying wire in a magnetic field experiences a force (the motor effect) given by F = B I L when the field is perpendicular. Fleming’s left-hand rule predicts force direction. This principle is used in electric motors.
载流导线在磁场中会受到力（电动机效应），当磁场垂直时力为 F = B I L。弗莱明左手定则判定力方向。电动机利用此原理工作。
Electromagnetic induction: when a conductor cuts magnetic field lines, a potential difference is induced. Transformers change alternating voltage: Vp / Vs = np / ns. For an ideal transformer, input power equals output power: Vp Ip = Vs Is.
电磁感应：导体切割磁感线时产生感应电动势。变压器可改变交流电压：Vp / Vs = np / ns。理想变压器输入功率等于输出功率：Vp Ip = Vs Is。
Radioactive decay is a random process where unstable nuclei emit radiation. Alpha particles (α) are helium nuclei (2 protons + 2 neutrons), strongly ionising but weakly penetrating. Beta particles (β) are fast electrons, moderately ionising and penetrating. Gamma rays (γ) are electromagnetic waves, weakly ionising but highly penetrating.
The half-life is the time taken for half the radioactive nuclei in a sample to decay. It is constant for a given isotope. Atomic number (Z) is the proton number; mass number (A) is the total number of protons + neutrons. In nuclear equations, total A and total Z are conserved.
半衰期是指样品中一半放射性原子核发生衰变所需的时间，对特定核素是常数。原子序数 Z 为质子数；质量数 A 为质子加中子数。核反应方程中，总质量数和总原子序数守恒。
Published by TutorHao | Physics Revision Series | aleveler.com
Gravitation is one of the fundamental forces that govern the motion of planets, moons, satellites and everyday objects on Earth. In the IGCSE WJEC Physics syllabus, understanding gravitation is essential for explaining weight, free fall, orbital motion and the structure of the Solar System. This article distils the core points you need to master, from Newton’s law of universal gravitation to the behaviour of satellites in orbit.
Gravitation, or gravity, is a force of attraction that acts between any two masses. Unlike electric or magnetic forces, gravity is always attractive and never repulsive. It is the weakest of the four fundamental forces, yet it dominates on astronomical scales because it acts over infinite distances and all matter has mass.
In IGCSE Physics, we distinguish between two related concepts: mass (the amount of matter in an object, measured in kg) and weight (the gravitational force on that object, measured in N). The weight of an object depends on the gravitational field strength at its location.
2. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
Isaac Newton proposed that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres:
where F is the gravitational force (N), m₁ and m₂ are the two masses (kg), r is the distance between their centres (m), and G is the gravitational constant, approximately 6.67 × 10⁻¹¹ N m² kg⁻².
其中 F 是引力（N），m₁ 和 m₂ 是两个质量（kg），r 是它们中心之间的距离（m），G 是引力常数，大约为 6.67 × 10⁻¹¹ N m² kg⁻²。
The minuscule value of G explains why we do not notice gravitational attraction between everyday objects. However, when at least one mass is extremely large – like a planet – the force becomes significant.
G 的极小数值解释了为什么我们注意不到日常物体之间的引力。然而，当至少一个质量非常大——比如行星——时，这个力就变得显著。
This law is an inverse-square law: if the distance doubles, the force becomes one quarter; if the distance triples, the force becomes one ninth.
这是一个平方反比定律：如果距离加倍，力变为原来的四分之一；如果距离增加三倍，力变为九分之一。
3. Gravitational Field Strength (g) | 引力场强度 (g)
The gravitational field strength at a point is defined as the force per unit mass experienced by a small test mass placed at that point:
引力场强度定义为放置在该点的小检验质量所受到的每单位质量的力：
g = F / m
The unit of g is N/kg. This is equivalent to m/s², so g is also the acceleration of free fall in that field. On the Earth’s surface, g ≈ 9.8 N/kg (often rounded to 10 N/kg for IGCSE calculations).
g 的单位是 N/kg。这等同于 m/s²，因此 g 也是在该场中自由落体的加速度。在地球表面，g ≈ 9.8 N/kg（IGCSE 计算中常取 10 N/kg）。
Using Newton’s law of gravitation, the field strength at the surface of a planet of mass M and radius R is given by:
利用牛顿引力定律，质量为 M、半径为 R 的行星表面的场强为：
g = G M / R²
This equation shows that g depends on the planet’s mass and radius, not on the mass of the object experiencing the force. That is why all objects in free fall (ignoring air resistance) accelerate at the same rate.
这个公式表明 g 取决于行星的质量和半径，而与受力物体的质量无关。这就是为什么在自由落体中（忽略空气阻力）所有物体的加速度相同。
4. Weight and Mass – Clearing the Confusion | 重量与质量——消除混淆
A common IGCSE pitfall is confusing mass and weight. Remember the clear distinction:
IGCSE 常见的陷阱是混淆质量和重量。记住明确的区别：
Property 属性
Mass 质量
Weight 重量
Definition
Quantity of matter in an object
Gravitational force acting on the object
Unit
kilogram (kg)
newton (N)
Varies with location?
No, constant throughout the universe
Yes, depends on g
Measured by
Beam balance or electronic balance
Spring balance or newton meter
For example, an astronaut of mass 80 kg has a weight of about 800 N on Earth (g = 10 N/kg) but only about 130 N on the Moon (g ≈ 1.6 N/kg). Her mass remains 80 kg.
IGCSE questions often ask you to calculate weight using W = m g, and to explain why mass is a more fundamental property than weight.
IGCSE 题目常常要求你使用 W = m g 计算重量，并解释为什么质量是比重量更基本的属性。
5. Variation of g on Earth and Other Planets | 地球及其他行星上 g 的变化
The value of g is not perfectly constant over the Earth’s surface. It varies slightly due to:
g 的值在地球表面并非完全恒定。它会因以下因素而略有变化：
Altitude: g decreases with height above the Earth’s surface because the distance from the centre increases. This follows from g = G M / (R + h)².
Latitude: The Earth is not a perfect sphere – it bulges at the equator. Points at the poles are closer to the centre, so g is slightly larger there. Additionally, the Earth’s rotation causes a small centrifugal effect that reduces apparent weight at the equator.
Local geology: Dense rock formations can cause tiny local increases in g.
海拔高度：g 随着离地球表面的高度增加而减小，因为到地心的距离增大。这由 g = G M / (R + h)² 得出。
纬度：地球并非完美球体——它在赤道处隆起。两极的点更靠近地心，因此那里的 g 略大。此外，地球自转产生微小的离心效应，减小了赤道处的表观重量。
当地地质：致密的岩石构造可能导致 g 的微小局部增加。
On other planets, the surface gravitational field strength can be very different. For example, Jupiter’s g ≈ 25 N/kg because of its huge mass, while Mars’ g ≈ 3.7 N/kg. You may be asked to compare weights of the same object on different planets using the g values provided.
在其他行星上，表面引力场强度可能差异很大。例如，木星的 g ≈ 25 N/kg，因为它质量巨大；而火星的 g ≈ 3.7 N/kg。你可能需要利用给出的 g 值，比较同一物体在不同行星上的重量。
6. Orbital Motion – The Role of Gravity | 轨道运动——引力的作用
Gravity provides the centripetal force that keeps planets in orbit around the Sun and satellites around the Earth. For a body moving in a circular orbit of radius r at speed v, the required centripetal force is:
引力提供了使行星绕太阳运行、卫星绕地球运行的向心力。对于在半径为 r 的圆形轨道上以速度 v 运动的物体，所需的向心力为：
Fc = m v² / r
If this force is supplied entirely by gravity, then:
如果这个力完全由引力提供，那么：
G M m / r² = m v² / r
Here M is the mass of the central body (e.g., the Sun), and m is the mass of the orbiting body (e.g., a planet). The m cancels out, showing that the orbital speed of a satellite does not depend on its own mass:
其中 M 是中心天体（如太阳）的质量，m 是轨道天体（如行星）的质量。m 被抵消掉，表明卫星的轨道速度与它自身的质量无关：
v² = G M / r so v = √(G M / r)
This means that for a given central mass, the orbital speed decreases with increasing orbital radius. Planets farther from the Sun move more slowly.
这意味着对于给定的中心质量，轨道速度随着轨道半径的增大而减小。离太阳越远的行星运动得越慢。
7. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
Johannes Kepler formulated three empirical laws that describe planetary motion, and Newton later showed they are a consequence of his law of gravitation. IGCSE WJEC often focuses on the third law, but an overview of all three is useful:
First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus. (For IGCSE, orbits are often approximated as circles.)
Second Law (Law of Equal Areas): A line joining a planet to the Sun sweeps out equal areas in equal intervals of time, meaning planets move faster when closer to the Sun.
Third Law (Law of Harmonies): The square of the orbital period T is proportional to the cube of the semi‑major axis r (average distance from the Sun).
By combining Kepler’s third law with Newton’s law of gravitation, the constant for solar‑system objects is found to be 4π²/(G MSun). This relationship allows us to calculate the mass of the central body if we know T and r.
将开普勒第三定律与牛顿引力定律结合，可得太阳系物体的常数为 4π²/(G M太阳)。利用这一关系，如果我们知道 T 和 r，就可以计算中心天体的质量。
8. Types of Satellite Orbits | 卫星轨道的类型
Artificial satellites play a critical role in modern life, and their orbits are chosen according to their purpose. IGCSE WJEC expects you to be familiar with two main types:
Geostationary satellites: Orbit above the Earth’s equator at a height of about 36 000 km, with a period of exactly 24 hours. They appear fixed in the sky because their orbital period matches the Earth’s rotation. Used for telecommunications and weather monitoring.
Polar satellites: Orbit with a much lower altitude (typically 200–1000 km) and pass over the poles. Their orbital period is about 90–100 minutes, and as the Earth rotates beneath them, they scan the entire surface. Used for Earth observation, mapping, and spy satellites.
地球静止卫星：在地球赤道上方约 36 000 km 的高度运行，周期恰好为 24 小时。它们看似固定在天空中，因为轨道周期与地球自转一致。用于电信和天气监测。
For a geostationary orbit, there is only one specific altitude that gives the required period of 24 h. This can be calculated by equating the centripetal force and gravity, then substituting v = 2π r / T.
对于地球静止轨道，只有一个特定的高度能产生所需的 24 小时周期。这可以通过令向心力等于引力，然后代入 v = 2π r / T 来计算。
9. Gravitational Potential Energy (Simple Treatment) | 引力势能（简单处理）
In IGCSE, gravitational potential energy (GPE) near the Earth’s surface is given by:
在 IGCSE 中，地球表面附近的引力势能（GPE）由下式给出：
ΔEp = m g Δh
This formula is valid only when the change in height Δh is small enough that g can be considered constant. For large changes in altitude, g changes significantly, so the simple m g Δh does not apply, but this is beyond the IGCSE syllabus.
该公式仅在高度变化 Δh 足够小、g 可视为常数时有效。当高度变化很大时，g 变化显著，此时简单的 m g Δh 不适用，但超出了 IGCSE 大纲范围。
On a larger scale, the work done to move a mass against gravity changes the gravitational potential energy of the system. In an orbital context, a satellite in a higher orbit has more gravitational potential energy (less negative) than one in a lower orbit, but a lower kinetic energy because its speed is smaller. This energy trade‑off explains why satellites must fire rockets to move to a higher orbit.
10. Free Fall and Acceleration due to Gravity | 自由落体与重力加速度
An object is said to be in free fall when the only force acting on it is gravity. In the absence of air resistance, all objects near the Earth’s surface fall with the same acceleration, g ≈ 9.8 m/s². This is a direct consequence of the equivalence of gravitational and inertial mass, tested in IGCSE through the famous hammer‑and‑feather experiment on the Moon.
Because weight is W = m g, applying Newton’s second law (F = m a) gives m g = m a, so a = g. The mass cancels, proving that the acceleration of free fall is independent of the object’s mass.
因为重量 W = m g，应用牛顿第二定律 (F = m a) 得 m g = m a，所以 a = g。质量被抵消，证明自由落体加速度与物体质量无关。
In real life, air resistance opposes the motion, causing objects with larger surface area relative to mass to fall more slowly. IGCSE exam questions often require you to explain terminal velocity: when air resistance equals weight, the net force is zero and the object stops accelerating, falling at constant speed.
IGCSE WJEC practical assessments may include methods to determine g in the laboratory. One common method uses a simple pendulum. For small oscillations, the period T is given by:
IGCSE WJEC 实践考核可能包括在实验室测定 g 的方法。一种常见方法使用单摆。对于小幅度摆动，周期 T 由下式给出：
T = 2π √(L / g)
By measuring the period T for different lengths L, and plotting T² against L, a straight line through the origin is obtained. The gradient = 4π² / g, from which g can be calculated.
通过测量不同长度 L 对应的周期 T，并绘制 T² 对 L 的图，可以得到一条过原点的直线。斜率 = 4π² / g，由此可计算 g。
Another method is the free‑fall timing method: a ball bearing is dropped from a known height h and the time of fall t is measured using an electronic timer. Using s = ½ g t² (since u = 0), g can be found from g = 2h / t². A graph of h versus t² (or of 2h versus t²) gives a straight line whose gradient is g.
另一种方法是自由落体计时法：将钢球从已知高度 h 释放，用电子计时器测量下落时间 t。利用 s = ½ g t²（因 u = 0），可得 g = 2h / t²。绘制 h 对 t²（或 2h 对 t²）的图，得到一条直线，其斜率为 g。
Both methods require careful attention to uncertainties, such as reaction time when using a stopwatch, and ensuring small amplitudes for the pendulum.
两种方法都需要仔细关注不确定性，例如使用秒表时的反应时间，以及确保单摆摆幅较小。
12. Common Exam Pitfalls and Key Tips | 常见考试陷阱与关键提示
To score full marks on gravitation questions in IGCSE WJEC, keep these points in mind:
Always use the correct units: mass in kg, distance in m, force in N, g in N/kg (or m/s²).
When using F = G m₁ m₂ / r², remember that r is the distance between centres, not the altitude above the surface. If a satellite is at height h above Earth’s surface, r = REarth + h.
Do not confuse weight with mass. Weight is a force; it has both magnitude and direction (towards the centre of the planet).
In orbits, the gravitational force is the centripetal force; the orbiting object is continuously accelerating towards the centre, so its velocity changes direction, not speed (for circular orbits).
For Kepler’s third law, T must be in years (or seconds) and r in astronomical units (or metres) depending on the constant used. Be consistent with units.
In calculations of g on other planets, you may be given mass and radius in terms of Earth’s values; proportional reasoning often avoids tedious arithmetic.
要在 IGCSE WJEC 的引力问题中获得满分，请注意以下几点：
始终使用正确单位：质量用 kg，距离用 m，力用 N，g 用 N/kg（或 m/s²）。
使用 F = G m₁ m₂ / r² 时，记住 r 是中心之间的距离，而不是离地表的垂直高度。如果卫星在地球表面上方高度 h 处，r = R地球 + h。
Mastering gravitation not only secures marks in standalone questions but also supports topics like circular motion, energy, and astrophysics. Practise using the equations in varied contexts, and always draw a diagram showing forces and distances.
This article covers the essential concepts of ideal gases required for the AQA IGCSE Physics specification. You will explore the particle model, the meaning of pressure and temperature, the three experimental gas laws, the Kelvin scale, the ideal gas equation, and the kinetic theory that explains gas behaviour at a microscopic level.
Matter consists of tiny particles – atoms or molecules – that are in constant, random motion. In a gas, the particles are far apart compared with their size, move rapidly in all directions, and have negligible forces of attraction between them, except during collisions.
This model explains why gases are easily compressed and can expand to fill any container. The random motion of particles is often called Brownian motion when observed indirectly through larger visible particles, such as smoke particles in air.
Gas pressure is caused by the collisions of gas particles with the walls of their container. Each collision exerts a tiny force on the wall; the sum of countless collisions per second results in a steady, measurable pressure.
Pressure (p) is defined as force per unit area: p = F / A. In gases, if the number of collisions per second increases, or if the particles hit the walls with greater speed, the pressure rises.
压强（p）定义为单位面积上的力：p = F / A。在气体中，如果每秒碰撞次数增加，或者粒子以更大的速度撞击器壁，压强就会升高。
p = F / A
3. Temperature and the Kelvin Scale | 温度与开尔文温标
Temperature is a measure of the average kinetic energy of the particles in a substance. The higher the temperature, the faster the particles move on average.
温度是物质中粒子平均动能的量度。温度越高，粒子的平均运动速度越快。
For gas law calculations, we must use the absolute temperature scale, measured in kelvin (K). The kelvin scale starts at absolute zero (0 K), the theoretical temperature at which particles have minimum kinetic energy. To convert from degrees Celsius to kelvin:
For example, 25 °C is equivalent to 25 + 273 = 298 K. Note that a temperature change of 1 °C is exactly equal to a change of 1 K.
例如，25 °C 等于 25 + 273 = 298 K。注意，1 °C 的温度变化恰好等于 1 K 的变化。
4. Boyle’s Law: Pressure and Volume | 波义耳定律：压强与体积
Boyle’s law states that for a fixed mass of gas at constant temperature, the pressure p is inversely proportional to the volume V.
波义耳定律指出：对于一定质量的气体，在温度保持不变时，压强 p 与体积 V 成反比。
p ∝ 1/V or pV = constant
This means if you double the volume, the pressure halves, provided the temperature stays the same. A graph of p against V gives a curve, while a graph of p against 1/V (or pV against p) yields a straight line through the origin.
这意味着在温度不变的条件下，体积加倍则压强减半。p 对 V 的图线是一条曲线，而 p 对 1/V（或 pV 对 p）的图线是一条通过原点的直线。
Boyle’s law can be demonstrated experimentally by trapping air in a sealed syringe connected to a pressure gauge, and slowly changing the volume while recording pressure.
波义耳定律可通过实验演示：将空气封闭在连接压力计的注射器中，缓慢改变体积并记录压强。
5. Charles’s Law: Volume and Temperature | 查理定律：体积与温度
Charles’s law states that for a fixed mass of gas at constant pressure, the volume V is directly proportional to its absolute temperature T (in kelvin).
查理定律指出：对于一定质量的气体，在压强保持不变时，体积 V 与绝对温度 T（开尔文）成正比。
V ∝ T or V/T = constant
If the temperature doubles from 200 K to 400 K, the volume also doubles. A straight-line graph of volume against kelvin temperature passes through the origin. If you extrapolate the graph backwards, it crosses the temperature axis at absolute zero (−273 °C).
如果温度从 200 K 加倍到 400 K，体积也加倍。体积对开尔文温度的图线是一条通过原点的直线。若将图线向后外推，它与温度轴的交点即为绝对零度（−273 °C）。
A classic classroom experiment uses a capillary tube with a drop of sulfuric acid trapping a column of dry air, heated in a water bath, with temperature and volume measured.
经典的课堂实验使用一根毛细管，管内一滴硫酸封住一段干燥空气柱，在水浴中加热，并测量温度和体积。
6. The Pressure Law: Pressure and Temperature | 压强定律：压强与温度
The pressure law (sometimes called Gay-Lussac’s law) states that for a fixed mass of gas at constant volume, the pressure p is directly proportional to its absolute temperature T.
压强定律（有时称为盖-吕萨克定律）表明：对于一定质量的气体，在体积保持不变时，压强 p 与绝对温度 T 成正比。
p ∝ T or p/T = constant
This law can be investigated by heating a sealed metal sphere containing air, connected to a pressure gauge. As the temperature increases, the molecules move faster and hit the walls more frequently and with greater force, raising the pressure.
A graph of pressure against kelvin temperature is a straight line that, when extrapolated, passes through absolute zero.
压强对开尔文温度的图线是一条直线，外推后通过绝对零度。
7. The Combined Gas Law and the Ideal Gas Equation | 气体组合定律与理想气体方程
The three gas laws can be combined into a single relationship for a fixed mass of gas:
三条气体定律可以合并为针对一定质量气体的单一关系式：
(p₁V₁)/T₁ = (p₂V₂)/T₂
This is the combined gas law. It allows you to calculate how a gas’s pressure, volume or temperature changes when the other two variables change, as long as the amount of gas stays constant.
这就是气体组合定律。当气体质量不变，且其中两个变量变化时，可以用它计算另一个变量的变化。
To account for the amount of gas, we introduce the number of moles n and the molar gas constant R (8.31 J mol⁻¹ K⁻¹). This leads to the ideal gas equation:
为了考虑气体的量，我们引入摩尔数 n 和摩尔气体常数 R（8.31 J mol⁻¹ K⁻¹），从而得到理想气体方程：
pV = nRT
In IGCSE, you may not be required to perform detailed calculations with n and R, but understanding the relationship as pV / T = constant is essential.
在 IGCSE 阶段，可能不要求使用 n 和 R 进行详细计算，但理解 pV / T = 常数这一关系至关重要。
8. Absolute Zero and the Kelvin Temperature Scale | 绝对零度与开尔文温标
Absolute zero is the lowest possible temperature, where particles would theoretically possess zero kinetic energy. Its value is 0 K, equivalent to −273.15 °C (usually taken as −273 °C).
Charles’s law and the pressure law both predict that at absolute zero, the volume or pressure of an ideal gas would become zero. In reality, all gases liquefy or solidify before reaching this temperature, so the laws break down.
The kelvin scale is fundamental because it starts at absolute zero, making all gas law relationships simple proportions. Always convert temperatures to kelvin before using any gas law.
9. Kinetic Theory Explanation of the Gas Laws | 气体定律的分子运动论解释
The kinetic theory provides a microscopic explanation for the macroscopic gas laws. It is based on the assumptions that gas particles are tiny, far apart, in constant random motion, and that collisions are perfectly elastic (no kinetic energy is lost).
Boyle’s law: Reducing the volume of a gas at constant temperature means particles hit the walls more often because they have less distance to travel. Thus, pressure increases.
波义耳定律：温度不变时减小气体体积，粒子运动距离缩短，因此更频繁地撞击器壁，进而压强增大。
Charles’s law: Raising the temperature gives particles more kinetic energy; they move faster. To keep pressure constant, the volume must increase so that collisions with the walls occur at the same rate despite the higher speed.
The pressure law: At constant volume, heating increases particle speed, causing more frequent and harder collisions, so pressure rises.
压强定律：体积不变时加热，粒子速度加快，导致更频繁且更剧烈的碰撞，因此压强上升。
10. Assumptions of the Kinetic Theory Model | 分子运动论模型的假设
The kinetic theory model for an ideal gas makes several simplifying assumptions:
理想气体的分子运动论模型提出了几条简化假设：
1. The gas consists of a large number of identical, tiny particles moving in random directions. | 1. 气体由大量相同的、微小的粒子组成，它们向随机方向运动。
2. The volume of the particles themselves is negligible compared to the total volume of the gas. | 2. 粒子自身的体积与气体总体积相比可以忽略不计。
3. There are no intermolecular forces acting between particles except during elastic collisions. | 3. 除了弹性碰撞瞬间，粒子之间不存在分子间作用力。
4. Collisions between particles and with the container walls are perfectly elastic; kinetic energy is conserved. | 4. 粒子之间以及粒子与器壁之间的碰撞是完全弹性的，动能守恒。
5. The time of a collision is negligible compared with the time between collisions. | 5. 碰撞的时间与两次碰撞之间的时间相比可以忽略。
These assumptions hold well for real gases at low pressure and high temperature, but break down under extreme conditions.
这些假设在低压和高温下对实际气体成立得较好，但在极端条件下不再适用。
11. Ideal Gases vs Real Gases | 理想气体与实际气体
An ideal gas perfectly obeys the kinetic theory assumptions and the equation pV = nRT under all conditions. Real gases, however, deviate from ideal behaviour, particularly at high pressures and low temperatures.
At high pressure, particles are forced closer together, so their own volume becomes significant, and attractive intermolecular forces begin to play a role. At low temperatures, particles move slowly enough that these forces can cause the gas to condense into a liquid.
On a graph of pV against p, an ideal gas would give a horizontal straight line. Real gases show a curve that dips below the ideal line at moderate pressures, then rises steeply at very high pressures.
在 pV 对 p 的图线中，理想气体呈现一条水平直线。实际气体则在中等压力时下降至理想线以下，随后在极高压力时急剧上升。
12. Typical Exam Questions and Tips | 典型考题与答题技巧
IGCSE exam questions on ideal gases often require you to describe experiments, interpret graphs, perform simple calculations using the gas laws, and explain gas behaviour in terms of the kinetic theory.
Key tips: Always convert temperatures to kelvin before using any formula. State which gas law you are applying. When explaining with kinetic theory, link the motion of particles explicitly to the change in pressure, volume or temperature.
Common mistakes include forgetting to use kelvin, confusing direct and inverse proportionality, and stating that particles ‘expand’ when heated – they do not; the gas expands because the particles move faster and take up more space.
Practice plotting graphs of p against 1/V, V against T, and p against T. Be prepared to explain why extrapolated lines pass through the origin and what this reveals about absolute zero.
练习绘制 p 对 1/V、V 对 T 和 p 对 T 的图线。准备好解释为何外推直线通过原点，以及这说明了关于绝对零度的什么信息。
Published by TutorHao | Physics Revision Series | aleveler.com
Magnetism is a fundamental non-contact force that appears in permanent magnets, electromagnets and even in the Earth itself. In the IGCSE OCR Physics specification, you must learn how magnetic fields behave, how electric currents produce magnetic effects, and how these principles are applied in motors, generators and transformers. This article breaks down every essential concept, equation and practical application you need to master for your exam.
A magnet has two poles: a north-seeking (N) pole and a south-seeking (S) pole. Like poles repel each other, while unlike poles attract. The magnetic force is strongest at the poles.
磁铁有两个极：指北极（N）和指南极（S）。同名极相斥，异名极相吸。磁力在两极最强。
Only certain materials can be magnetised strongly. These are ferromagnetic materials such as iron, steel, cobalt and nickel. Steel retains magnetism well, so it is used for permanent magnets, while soft iron is easy to magnetise and demagnetise, making it ideal for temporary electromagnets.
Non-magnetic materials like wood, plastic and glass are not attracted to magnets at all. Paramagnetic materials, such as aluminium, experience a very weak attraction but are not covered in detail at IGCSE.
A magnetic field is the region around a magnet where a magnetic force can be detected. You can map the field using iron filings or a small plotting compass. Field lines always run from the north pole to the south pole outside the magnet, and continue through the magnet from south to north, forming closed loops.
The spacing of field lines indicates the strength of the field: the closer the lines, the stronger the field. At the poles, lines are most concentrated. Two unlike poles facing each other produce a uniform field pattern between them, with equally spaced parallel lines.
Remember that magnetic field lines never cross. The Earth itself behaves like a giant bar magnet, with its magnetic south pole near the geographic North Pole, which is why a compass needle’s north pole points northwards.
3. Magnetic Effect of a Current – Straight Wire | 电流的磁效应 – 直导线
Whenever an electric current flows through a conductor, it produces a magnetic field around it. For a long straight wire, the field lines form concentric circles centred on the wire. The direction of the magnetic field can be found using the right-hand grip rule: point your right thumb in the direction of conventional current (positive to negative), and your curled fingers show the direction of the magnetic field.
The strength of this magnetic field increases if you increase the current or bring the compass closer to the wire. Reversing the current reverses the direction of the field lines. This effect is the foundation of electromagnetism.
如果增大电流或让罗盘更靠近导线，磁场会增强。反转电流方向，磁感线方向也随之反转。这是电磁学的基础。
4. Solenoids and Electromagnets | 螺线管与电磁铁
A solenoid is a coil of wire. When current flows through it, the magnetic field pattern is similar to that of a bar magnet: one end acts as a north pole, the other as a south pole. The polarity can be determined by the right-hand grip rule applied to the solenoid: curl your right hand around the coil in the direction of current, and your thumb points towards the north pole.
You can increase the strength of the magnetic field of a solenoid by: increasing the current, increasing the number of turns, inserting a soft iron core inside the coil. The soft iron core becomes magnetised and greatly concentrates the field, producing a powerful electromagnet.
Electromagnets are used in relays, electric bells, circuit breakers and lifting magnets in scrap yards. Their advantage is that the magnetism can be switched on and off by controlling the current.
电磁铁常用于继电器、电铃、断路器和废料场的起重磁铁。其优点是可以通过控制电流来开关磁性。
5. The Motor Effect – Fleming’s Left-Hand Rule | 电动机效应 – 弗莱明左手定则
When a current-carrying conductor is placed in an external magnetic field, the two magnetic fields interact, producing a force on the wire. This is called the motor effect. The force is perpendicular to both the current direction and the magnetic field.
The direction of the force is given by Fleming’s left-hand rule: hold your left hand with the thumb, forefinger and second finger mutually at right angles. The First finger represents the magnetic Field direction (N to S), the seCond finger represents the Conventional Current (+ to -), and the ThuMb shows the direction of Motion (force). A common mnemonic is ‘F B C’ – Field, B-current, Motion – but remember the sequence: First finger Field, seCond finger Current, ThuMb Motion.
力的方向由弗莱明左手定则给出：伸出左手，使拇指、食指和中指两两垂直。食指（First finger）表示磁场方向（N 到 S），中指（seCond finger）表示常规电流方向（+ 到 -），拇指（ThuMb）表示运动（力）的方向。一个记忆窍门是 ‘F B C’，但记住顺序：食指磁场，中指电流，拇指运动。
The size of the force can be calculated using the equation:
F = B I L
力的大小可用公式计算：
F = B I L
where F is the force in newtons (N), B is the magnetic flux density in tesla (T), I is the current in amperes (A), and L is the length of the conductor in the magnetic field in metres (m). This equation assumes the wire is perpendicular to the field. If the wire is at an angle θ to the field, the effective component is B I L sinθ.
其中 F 是力（牛顿，N），B 是磁通量密度（特斯拉，T），I 是电流（安培，A），L 是导线在磁场中的长度（米，m）。该公式假设导线与磁场垂直。若导线与磁场成角度 θ，则有效分量为 B I L sinθ。
6. The DC Motor | 直流电动机
A simple DC motor consists of a rectangular coil of wire placed in a uniform magnetic field, with a current passed through it via a split-ring commutator and carbon brushes. The forces on the two sides of the coil produce a couple (a pair of opposite forces acting on different lines of action) that causes the coil to rotate.
The split-ring commutator is a clever device that reverses the direction of current in the coil every half turn. This ensures that the forces always push the coil in the same rotational direction, maintaining continuous rotation. Without a commutator, the coil would merely oscillate.
To increase the turning effect of a motor, you can: increase the current, use a stronger magnetic field, increase the number of turns on the coil, or place a soft iron core inside the coil to concentrate the field. The turning effect (torque) is maximised when the plane of the coil is parallel to the magnetic field lines.
Electromagnetic induction occurs when a conductor cuts through magnetic field lines, or when the magnetic field around a conductor changes. This induces a potential difference (voltage) across the ends of the conductor. If the conductor is part of a complete circuit, a current flows.
The size of the induced p.d. depends on the rate at which the magnetic field changes or the conductor cuts the field lines. A faster movement, a stronger magnetic field, or a larger number of turns in a coil all increase the induced voltage. Lenz’s law states that the direction of the induced current is such that it opposes the change causing it, which is a direct consequence of the conservation of energy.
Lenz’s law explains why you must do work to push a magnet into a coil; the induced current creates a magnetic field that repels the magnet. If the induced current aided the motion, you would get energy for free, which is impossible.
The generator effect is the practical application of electromagnetic induction. When a coil is rotated in a magnetic field, or a magnet is moved near a coil, a changing magnetic flux linkage induces an alternating p.d. and, if connected, an alternating current (a.c.).
For a moving conductor, you can use Fleming’s right-hand rule to find the direction of the induced conventional current. Arrange your right hand with thumb, forefinger and second finger mutually at right angles: First finger Field (N to S), ThuMb Motion of the conductor, and seCond finger gives the direction of induced Conventional Current. Notice the switch from the left hand for motors.
In a simple alternator (a.c. generator), slip rings and brushes maintain continuous electrical connection without reversing the current, so the output is a.c. In a dynamo, a split-ring commutator is used to produce direct current (d.c.) output. The frequency of the a.c. output equals the number of rotations per second of the coil.
A transformer is a device that changes the size of an alternating voltage. It consists of two coils wound on a complete soft iron core. The primary coil is connected to the input a.c. supply, and the secondary coil provides the output voltage. Transformers only work with alternating current because a changing magnetic field is required for induction.
The relationship between the voltages and the numbers of turns is given by the transformer equation:
Vp / Vs = Np / Ns
电压与匝数之间的关系由变压器方程给出：
Vp / Vs = Np / Ns
where Vp and Vs are the primary and secondary voltages, and Np and Ns are the numbers of turns on the primary and secondary coils. A step-up transformer has more turns on the secondary coil (Ns > Np) and increases voltage; a step-down transformer has fewer secondary turns and decreases voltage.
其中 Vp 和 Vs 是初、次级电压，Np 和 Ns 是初、次级线圈的匝数。升压变压器次级匝数更多（Ns > Np），电压增大；降压变压器次级匝数较少，电压减小。
If the transformer is 100% efficient (ideal transformer), the input power equals the output power: Vp × Ip = Vs × Is. In practice, some energy is lost as heat due to resistance in the coils and eddy currents in the core. Laminated cores reduce eddy current losses.
若变压器效率为 100%（理想变压器），输入功率等于输出功率：Vp × Ip = Vs × Is。实际中，由于线圈电阻和铁芯中的涡流，会有部分能量以热的形式损失。分层铁芯可减少涡流损失。
Transformers are essential in the National Grid: step-up transformers raise the voltage to hundreds of kilovolts for transmission, reducing current and thus I²R power loss in the cables; step-down transformers then lower the voltage to safe levels for homes and industries.
10. Magnetising and Demagnetising Methods | 磁化与退磁方法
A ferromagnetic material like steel can be magnetised by: stroking it repeatedly in one direction with a strong permanent magnet, placing it inside a solenoid connected to a direct current (d.c.) supply, or aligning it in the Earth’s magnetic field and hammering it. The stroking and d.c. solenoid methods produce a permanent magnet.
To demagnetise a magnet, you can: heat it strongly and let it cool in an east-west direction to avoid the Earth’s field, hammer it vigorously while pointing east-west, or place it inside a solenoid connected to an alternating current (a.c.) and slowly withdraw the magnet. The a.c. method repeatedly reverses and reduces the magnetic domains’ alignment, leaving the material unmagnetised.
11. Magnetic Materials and Domain Theory | 磁性材料与磁畴理论
Domain theory explains how ferromagnetic materials become magnetised. Inside a ferromagnetic material, there are tiny regions called domains, each acting as a tiny magnet with a north and south pole. In an unmagnetised material, the domains point in random directions, so their magnetic fields cancel out.
When the material is placed in a strong external magnetic field, the domains align in the direction of the field. The more domains that align, the stronger the overall magnetism. In a permanent magnet, most domains remain aligned even after the external field is removed. In a temporary magnet like soft iron, the domains easily fall back into random orientations once the field is gone.
Memorise the following equations exactly and know when to apply them: F = B I L for the motor effect when the conductor is perpendicular to the field. Vp / Vs = Np / Ns for ideal transformers. Power input = Power output for ideal transformers, so Vp Ip = Vs Is. Pay attention to units: B in tesla (T), I in amperes (A), L in metres (m), F in newtons (N).
准确记忆以下公式并知道何时使用：F = B I L 适用于导线垂直于磁场时的电动机效应。Vp / Vs = Np / Ns 适用于理想变压器。理想变压器输入功率 = 输出功率，即 Vp Ip = Vs Is。注意单位：B 为特斯拉（T），I 为安培（A），L 为米（m），F 为牛顿（N）。
In exams, always state the rule you are using – for example, ‘by Fleming’s left-hand rule, the force on the wire is upwards’. Clearly explain the direction of field, current and force/motion. When describing electromagnetic induction, use the phrase ‘cutting magnetic field lines’ and mention the rate of change. For transformers, stress that they only work with a.c. because a changing flux is needed for induction. Practise drawing the field patterns and labelling poles.
In GCSE AQA Physics, the topic of materials brings together ideas about density, states of matter, internal energy, specific latent heat, and the elastic behaviour of objects like springs. Understanding these concepts helps explain everything from why ice floats to how a spring stores energy. This revision guide walks you through each key point with clear explanations and practical examples.
Density is a measure of how much mass is packed into a given volume. The formula is ρ = m / V, where ρ is density (kg/m³), m is mass (kg), and V is volume (m³).
密度用来衡量单位体积内所含物质的质量。公式为 ρ = m / V，其中 ρ 代表密度（千克每立方米），m 代表质量（千克），V 代表体积（立方米）。
The particle model links density to the arrangement of particles. In solids, particles are tightly packed in a regular pattern, resulting in a high density. Liquids have particles that are close but can move past each other, so their density is usually slightly lower than that of solids. Gases have particles that are widely spaced and move randomly, giving them very low densities.
A denser material has more mass in the same volume. For example, a steel block is much denser than a block of expanded polystyrene of the same size.
密度较大的材料在相同体积下拥有更大的质量。例如，体积相同的一块钢就要比一块发泡聚苯乙烯重得多。
2. Measuring Density | 测量密度
To find the density of a solid object, first measure its mass using a balance. Then determine its volume. For a regular solid, volume is calculated from linear dimensions, e.g. V = length × width × height for a cuboid.
要测量固体的密度，首先用天平测出其质量。然后确定其体积。对于形状规则的固体，体积可通过尺寸计算得出，例如长方体体积 V = 长 × 宽 × 高。
For irregular solids, the volume is often measured by displacement. Submerge the object in a measuring cylinder partially filled with water; the rise in water level equals the volume of the object. If it floats, a sinker can be used to push it under water.
To measure the density of a liquid, place an empty container on a balance, zero it, pour the liquid into the container to record the mass, and then pour that liquid into a measuring cylinder to read the volume. Density is then calculated as ρ = m / V.
测量液体密度时，先将空容器放在天平上并回零，倒入液体后记录质量，再将液体倒入量筒读取体积。然后用 ρ = m / V 计算密度。
3. States of Matter | 物质的状态
Matter exists in three main states: solid, liquid, and gas. In a solid, particles vibrate about fixed positions in a regular lattice. In a liquid, particles are in close contact but can move past each other, so the liquid takes the shape of its container. In a gas, particles move randomly at high speed and fill the whole container.
Changes of state, such as melting, boiling, condensing, and freezing, occur when energy is transferred to or from a substance. The temperature remains constant during a change of state, even though heating or cooling continues.
The mass of a substance does not change during a change of state. That is because the number of particles stays the same, only the arrangement and energy of the particles change.
物质在状态变化过程中质量不变。这是因为粒子数目保持不变，变化的仅仅是粒子的排列方式和能量。
4. Internal Energy and Changes of State | 内能与状态变化
Internal energy is the total kinetic and potential energy of the particles inside a substance. Heating increases the internal energy; cooling decreases it.
内能是物质内部粒子的总动能和总势能之和。加热使内能增加，冷却使内能减少。
During a change of state, the temperature does not rise because the energy supplied breaks the bonds between particles rather than increasing their kinetic energy. This hidden energy is called latent heat.
A heating or cooling graph will show flat sections at the melting point and boiling point. The length of these flat sections depends on the amount of substance and the specific latent heat.
加热或冷却曲线会在熔点和沸点处出现水平段。水平段的长度取决于物质的数量和比潜热。
5. Specific Latent Heat | 比潜热
Specific latent heat is the amount of energy needed to change the state of 1 kg of a substance without changing its temperature. It is measured in J/kg.
比潜热是指使 1 千克物质在不改变温度的情况下改变状态所需的能量，单位是焦耳每千克。
There are two types: specific latent heat of fusion (for melting or freezing) and specific latent heat of vaporisation (for boiling or condensing). The latent heat of vaporisation is usually larger because all remaining particle bonds must be broken.
The energy needed for a change of state can be calculated using E = m L, where E is the thermal energy transferred (J), m is the mass (kg), and L is the specific latent heat (J/kg).
状态变化所需的能量可以用 E = m L 计算，其中 E 是传递的热能（焦耳），m 是质量（千克），L 是比潜热（焦耳每千克）。
6. Elastic Deformation and Hooke’s Law | 弹性变形与胡克定律
When a force is applied to an object, it can change shape. Elastic deformation means the object returns to its original shape when the force is removed. Inelastic (plastic) deformation means a permanent change of shape.
Hooke’s Law describes the elastic behaviour of many materials, particularly springs. It states that the extension of a spring is directly proportional to the force applied, as long as the elastic limit is not exceeded. The relationship is written as F = k x, where F is force (N), k is the spring constant (N/m), and x is extension (m).
胡克定律描述了许多材料（尤其是弹簧）的弹性行为。该定律指出，只要不超过弹性限度，弹簧的伸长量与所施加的力成正比。关系式写作 F = k x，其中 F 是力（牛顿），k 是弹簧常数（牛每米），x 是伸长量（米）。
The spring constant k is a measure of stiffness: a stiffer spring has a larger spring constant and produces a smaller extension for the same force.
弹簧常数 k 是刚度的量度：较硬的弹簧具有较大的弹簧常数，在相同的力作用下伸长量更小。
7. Spring Constant and Force-Extension Graphs | 弹簧常数与力-伸长图
A force-extension graph plots force on the y-axis against extension on the x-axis. For an elastic material obeying Hooke’s Law, the graph is a straight line passing through the origin.
力-伸长图以力为纵轴，以伸长量为横轴。对于遵循胡克定律的弹性材料，图像是一条通过原点的直线。
The gradient of the straight line equals the spring constant k. A steeper gradient means a larger spring constant, which indicates a stiffer spring.
直线的斜率等于弹簧常数 k。斜率越陡，弹簧常数越大，表明弹簧越硬。
If the line starts to curve, the material is no longer obeying Hooke’s Law and may have reached its limit of proportionality. Beyond this point, extension is no longer proportional to force.
8. Limit of Proportionality and Elastic Limit | 比例极限与弹性极限
The limit of proportionality is the point beyond which the relationship between force and extension is no longer linear. Hooke’s Law no longer applies.
比例极限是指力和伸长量之间的直线关系不再成立的那个点。从此点开始胡克定律不再适用。
The elastic limit is the maximum force that can be applied and still have the material return to its original length when the load is removed. For some materials, the elastic limit and limit of proportionality are very close; beyond the elastic limit, permanent deformation occurs.
Understanding these limits is essential for engineers who must ensure that structures and components do not deform permanently in use.
理解这些极限对于工程师至关重要，他们必须确保结构和部件在使用中不会发生永久变形。
9. Work Done in Stretching | 拉伸做功
When a spring is stretched or compressed elastically, work is done. This work is stored as elastic potential energy in the spring.
当弹簧被弹性拉伸或压缩时，会有做功。这些功以弹性势能的形式储存在弹簧中。
The energy stored can be calculated using the area under the force-extension graph. For a linear relationship, the energy is equal to Eₑ = ½ F x. Since F = k x, this can also be written as Eₑ = ½ k x².
储存的能量可以利用力-伸长图下的面积来计算。对于线性关系，能量等于 Eₑ = ½ F x。由于 F = k x，这也可以写成 Eₑ = ½ k x²。
In these equations, Eₑ is the elastic potential energy in joules (J), k is the spring constant in N/m, and x is the extension in metres. This relationship holds only if the spring is not stretched beyond its limit of proportionality.
Knowledge of density helps us choose materials for buoyancy, such as using low-density foam for life jackets. The particle model explains why gases are easily compressed and liquids are used in hydraulic systems.
Specific latent heat is crucial in designing cooling systems, like refrigerators, and understanding natural phenomena such as why sweating cools the skin. Elastic materials are used everywhere, from car suspensions to sports equipment; engineers must select spring constants appropriate for the load.
By connecting microscopic particle behaviour with macroscopic properties, you can explain a wide range of everyday observations and solve practical problems.
通过将微观粒子行为与宏观性质联系起来，你可以解释大量日常观察到的现象，并解决实际问题。
Published by TutorHao | GCSE Physics Revision Series | aleveler.com
Newton’s laws of motion form the bedrock of classical mechanics and are essential for any Edexcel A‑Level Physics student. This revision guide walks you through each law, common applications, and the key equations that you must master for the exams. We will also explore momentum, impulse, connected particles, circular motion, and gravitation — all rooted in Newton’s powerful principles.
Newton’s First Law states that an object will remain at rest or move with a constant velocity unless acted upon by a resultant external force. This property of matter is called inertia. The law implies that if the vector sum of forces on a body is zero, the body is in equilibrium: it may be stationary or moving at a steady speed in a straight line.
A common misconception is that a constant force is needed to maintain motion. In fact, when there is no net force (e.g., an ice hockey puck sliding on frictionless ice), the object continues moving indefinitely at constant velocity. Recognising when a body is in equilibrium is crucial for solving statics problems.
2. Mass, Inertial Mass and Gravitational Mass | 质量、惯性质量与引力质量
Mass is a measure of an object’s inertia — its resistance to change in velocity. The inertial mass m of a body is defined by the ratio of the net force applied to the resulting acceleration: m = Fₙₑₜ / a. In Edexcel Physics, you are also expected to distinguish between inertial mass and gravitational mass. Gravitational mass determines the strength of the gravitational force on an object, and experiments show that the two are equivalent; this equivalence underpins Einstein’s general relativity.
质量是物体惯性的量度——即物体对速度变化的抗拒程度。物体的惯性质量 m 由合外力与产生的加速度之比定义：m = Fₙₑₜ / a。在 Edexcel 物理中，你还需要区分惯性质量与引力质量。引力质量决定物体所受引力的大小，实验表明两者是等价的；这一等价性正是爱因斯坦广义相对论的基石。
In most A‑Level problems, we simply use ‘mass’ and apply Newton’s Second Law without distinguishing the two. However, a common exam question asks students to explain how mass can be measured using an inertial balance or how to compare masses via their accelerations under the same force.
The Second Law quantifies the effect of a resultant force: the acceleration a of an object is directly proportional to the net force Fₙₑₜ acting on it and inversely proportional to its mass m. In its simplest form, it is written as:
第二定律定量描述了合外力的作用效果：物体的加速度 a 与所受合外力 Fₙₑₜ 成正比，与质量 m 成反比。最简单的形式表示为：
Fₙₑₜ = m a
When using this equation, always ensure that you are working with the resultant force in a given direction. In Edexcel exams, you will often need to resolve forces into perpendicular components and apply F = ma along each axis separately, especially on inclined planes.
使用该方程时，务必确保你使用的是某一方向上的合力。在 Edexcel 考试中，常需要将力沿垂直方向分解，并分别沿每个轴应用 F = ma，特别是在斜面问题中。
For a body on a smooth inclined plane at angle θ, the component of weight down the slope is mg sin θ. The acceleration down the plane, in the absence of friction, is a = g sin θ.
对于放在光滑斜面上、倾角为 θ 的物体，重力沿斜面的分量是 mg sin θ。无摩擦时，物体沿斜面的加速度为 a = g sin θ。
4. Momentum and Newton’s Second Law in Terms of Momentum | 动量与动量形式的第二定律
Momentum p is defined as the product of mass and velocity: p = m v. It is a vector quantity with units kg m s⁻¹ or N s. Newton’s Second Law can be expressed in a more general form: the rate of change of momentum of an object is directly proportional to the resultant force acting on it, and takes place in the direction of that force.
动量 p 定义为质量与速度的乘积：p = m v。它是矢量，单位为 kg m s⁻¹ 或 N s。牛顿第二定律可以用更一般的形式表述：物体动量的变化率与其所受合外力成正比，且变化发生在该力的方向上。
Fₙₑₜ = Δp / Δt
For a system of constant mass, this reduces to F = m a. However, when mass changes (e.g., rocket propulsion), you must use the momentum form. Edexcel often tests your ability to link force and momentum in collision problems.
对于质量不变的系统，该式可简化为 F = m a。然而，当质量变化时（如火箭推进），必须使用动量形式。Edexcel 常在碰撞问题中考查你对力与动量关系的理解。
5. Impulse and Change in Momentum | 冲量与动量变化
Impulse J is defined as the product of the average force acting on an object and the time interval for which it acts: J = Fₐₗ₉ Δt. From the momentum form of the Second Law, impulse is equal to the change in momentum:
The impulse–momentum relationship is especially useful when analysing collisions, safety features like airbags, and sporting actions. The area under a force–time graph also gives the impulse experienced by an object.
In Edexcel papers, you are often asked to use this theorem to find the average force during a collision, such as a ball bouncing off a wall, or to explain how crumple zones reduce injury by increasing the impact time and thus reducing the force.
6. Newton’s Third Law: Action and Reaction | 牛顿第三定律：作用力与反作用力
Newton’s Third Law states that if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These forces are of the same type, act on different bodies, and are equal in magnitude but opposite in direction:
牛顿第三定律指出，若物体 A 对物体 B 施加一个力，则物体 B 同时对物体 A 施加一个大小相等、方向相反的力。这两个力类型相同、作用在不同物体上，大小相等、方向相反：
Fₐₚₒₙₙ Fₐₚₒₙₙ = – Fₐₚₒₙₙ (i.e., Fₐ on B = – F_B on A)
Common exam mistakes include pairing forces that act on the same body (e.g., weight and normal reaction on a book resting on a table) as a Third Law pair. The correct pair for the book’s weight is the gravitational pull of the book on the Earth, not the normal force from the table.
Understanding the Third Law is vital for analysing the motion of connected objects, rockets, and forces on inclined planes with pulleys. Always ask: “What is the other body?” to verify a Third Law pair.
7. Free-body Diagrams and Resolving Forces | 受力分析与力的分解
Drawing a clear free-body diagram is the first step to solving any force problem. Isolate the body, draw all forces acting on it as arrows originating from its centre of mass, and label them clearly: weight (mg), normal reaction (R or N), tension (T), friction (f or F_f), and any applied forces.
Forces that act at an angle, such as a pull at 30° to the horizontal, must be resolved into perpendicular components. Using trigonometry, the horizontal component is F cos θ and the vertical component is F sin θ. Apply Newton’s Second Law independently along each axis.
与水平方向成角度的力（如 30° 的拉力）必须分解为垂直分量。利用三角函数，水平分量为 F cos θ，竖直分量为 F sin θ。分别在每个轴上独立应用牛顿第二定律。
Equilibrium conditions are often tested: for a body at rest or moving with constant velocity, the net force in any direction is zero. This allows you to set up simultaneous equations to find unknown forces.
8. Connected Particles: Pulleys and Tension | 连接体问题：滑轮与张力
When two objects are connected by a light inextensible string passing over a smooth pulley, they share the same magnitude of tension and acceleration. Treat each mass separately: draw free-body diagrams, write down F = ma for each, and solve the resulting equations simultaneously.
当两个物体由一根轻质且不可伸长的绳子相连，并跨过一个光滑滑轮时，它们具有大小相同的拉力和加速度。分别处理每个物体：画受力图，对每个物体列出 F = ma，然后联立求解方程。
T − m₁g = m₁a (for the lighter mass being lifted) m₂g − T = m₂a (for the heavier mass falling)
Adding the two equations eliminates T and gives the acceleration: a = (m₂ − m₁)g / (m₂ + m₁). Such problems, known as Atwood machine problems, are classic Edexcel A‑Level questions and may include friction or an inclined table.
9. Circular Motion and Centripetal Force | 圆周运动与向心力
When an object moves in a circle at constant speed, its velocity is changing direction, so it experiences an acceleration towards the centre. This centripetal acceleration a_c = v² / r = ω² r. By Newton’s Second Law, there must be a resultant centripetal force:
当物体以恒定速率沿圆周运动时，其速度方向不断改变，因此它具有指向圆心的加速度。该向心加速度 a_c = v² / r = ω² r。根据牛顿第二定律，必然存在指向圆心的合力：
F_c = m v² / r = m ω² r
The centripetal force is not a new type of force; it is provided by tension, gravity, friction, or the normal reaction. For a car rounding a bend, friction provides the centripetal force; for a satellite, it is gravity.
Exam questions often require you to analyse situations like a bucket swinging in a vertical circle or a car on a banked track. Here you must resolve forces and apply F = ma in the radial direction.
考题常要求分析水桶在竖直面内摇转或汽车在倾斜轨道上行驶的情形。此时需要分解力并在径向应用 F = ma。
10. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
Newton’s Law of Universal Gravitation states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation:
where G is the gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻². Although this is not one of the three laws of motion, it is an extension of Newtonian mechanics and a key part of the Edexcel specification, especially for satellite motion, Kepler’s laws, and gravitational field strength g = GM / r².
其中 G 是引力常量，6.67 × 10⁻¹¹ N m² kg⁻²。虽然万有引力定律不属于三大运动定律，但它是牛顿力学的延伸，也是 Edexcel 考纲的重要组成部分，尤其在卫星运动、开普勒定律和引力场强度 g = GM / r² 方面。
Combining this law with centripetal force allows you to derive the orbital period of a satellite, showing that T² ∝ r³. Such derivations are common in A‑Level exams.
📚 Mastering Key Concepts from the AQA AS Physics PH02 June 2022 Examination Report | 掌握2022年6月AQA AS物理PH02考试报告关键概念
The June 2022 PH02 examination report for AQA AS Physics offers invaluable insight into common student misconceptions and areas where deeper conceptual understanding is required. This paper covers mechanics, materials and waves, and the examiners’ observations reveal patterns of error that can be corrected by revisiting fundamental principles. In this article, we unpack the key concepts highlighted in the report, providing clear explanations and practical tips to help you avoid similar mistakes and strengthen your grasp of the subject.
1. Phase Difference and Path Difference in Wave Superposition | 波叠加中的相位差与路径差
Many candidates confused phase difference with path difference or applied the conversion formula incorrectly. Phase difference Δφ (in radians) relates to path difference Δx by Δφ = 2π × Δx / λ, where λ is the wavelength. When two waves are exactly in phase, Δφ = 0, 2π, 4π … and constructive interference occurs. A path difference of one whole wavelength corresponds to a phase difference of 2π rad. Misidentifying the wavelength from a diagram or forgetting to express the answer in radians rather than degrees were common errors.
When analysing two-source interference patterns, remember that the nth-order bright fringe occurs when path difference equals nλ. For destructive interference, the path difference is (n + ½)λ, giving a phase difference of (2n+1)π. Always convert distances to the same unit and read any scale carefully.
分析双源干涉图样时，记住第 n 级明纹出现的条件是路径差为 nλ；暗纹对应路径差为 (n + ½)λ，相位差为 (2n+1)π。务必统一长度单位并仔细读标尺。
2. Standing Waves and Energy Transfer | 驻波与能量传递
A standing (stationary) wave forms when two progressive waves of equal frequency and amplitude travel in opposite directions and superpose. A fundamental misunderstanding persists that standing waves transfer energy along the medium. In reality, the net energy flux is zero – energy is confined between nodes, oscillating between kinetic and potential forms. The nodes are points of permanently zero displacement, while antinodes undergo maximum amplitude oscillation.
Examiners noted that students often drew sinusoidal standing waves with the same wavelength as the progressive wave but labelled nodes incorrectly. Remember that adjacent nodes (or adjacent antinodes) are separated by half a wavelength λ/2. For a string fixed at both ends, the fundamental frequency corresponds to one half‑wavelength between the supports. Use v = f λ, where v is the wave speed on the string determined by tension T and mass per unit length μ: v = √(T/μ).
考官指出，学生常画出与行波波长相同的正弦驻波但错误标记波节位置。请记住相邻波节（或相邻波腹）相距半个波长 λ/2。对于两端固定的弦，基频对应两个固定端之间恰好为半个波长。使用 v = f λ，其中弦上波速由张力 T 和线密度 μ 决定：
v = √(T/μ)
This relationship is frequently tested, and the derived frequency formula f = (1/2L)√(T/μ) for the fundamental mode must be used with consistent SI units.
这个关系经常被考查，基频公式 f = (1/2L)√(T/μ) 中的各量需使用一致的国际单位。
3. Interpreting Stress–Strain Graphs and Material Properties | 解读应力–应变曲线与材料特性
The stress–strain graph is a cornerstone of the materials topic, yet many answers in the PH02 report revealed poor understanding of key features. The linear portion obeys Hooke’s Law, and the gradient of this straight line gives the Young modulus E = stress/strain. Beyond the limit of proportionality, the graph curves. The elastic limit marks the point beyond which the material no longer returns to its original length; plastic deformation begins. Ultimate tensile strength (UTS) is the maximum stress on the graph. The distinction between ‘stiff’, ‘strong’, and ‘tough’ materials relies on interpreting gradients, maximum stress, and area under the graph.
应力–应变图是材料部分的核心，但PH02报告显示许多学生对关键特征缺乏清晰认识。线性部分符合胡克定律，该段直线的梯度即为杨氏模量 E = 应力/应变。越过比例极限后曲线弯曲；弹性极限是材料不再能恢复原长的临界点，塑性变形随之开始。抗拉强度 (UTS) 是曲线上的最大应力。区分“刚性强”“强度高”和“韧性好”的材料需分别依据曲线梯度、最大应力和曲线下面积。
Common errors included stating that the elastic limit and limit of proportionality are always the same, or misreading units on the axes. Remember that stress is force per unit area (Pa = N m⁻²) and strain is dimensionless (or ratio of extension to original length). When calculating Young modulus from a graph, always use the linear region: E = (F/A) / (ΔL/L₀). Provide the answer in pascals, or GPa where appropriate.
常见错误包括宣称弹性极限与比例极限始终重合，或读错坐标轴单位。请记住应力是单位面积受力 (Pa = N m⁻²)，应变无量纲；由曲线图计算杨氏模量时务必使用线性区域：E = (F/A) / (ΔL/L₀)，答案以帕斯卡或吉帕给出。
4. Young Modulus Calculations and Common Unit Errors | 杨氏模量计算与典型单位错误
The PH02 examiners highlighted persistent mistakes when students substituted values into the Young modulus formula without converting units correctly. The expression E = (F L₀) / (A ΔL) requires force in newtons, original length L₀ and extension ΔL in the same length unit (usually metres), and cross-sectional area A in m². A wire of diameter 0.5 mm has radius 0.25 mm = 2.5 × 10⁻⁴ m, giving area πr² ≈ 1.96 × 10⁻⁷ m². Many lost marks by using mm² or cm² without conversion.
PH02考官强调，学生代入杨氏模量公式时频繁出现单位换算错误。表达式 E = (F L₀) / (A ΔL) 中力用牛顿，原长 L₀ 和伸长量 ΔL 须用同一长度单位（通常为米），截面积 A 须用 m²。直径为 0.5 mm 的导线半径为 0.25 mm = 2.5 × 10⁻⁴ m，面积 πr² ≈ 1.96 × 10⁻⁷ m²。许多考生直接用 mm² 或 cm² 代入而未换算导致失分。
Additionally, when asked to find the Young modulus from a stress–strain graph, candidates sometimes inverted the axes or used the wrong interval. The gradient Δσ/Δε must be taken from the straight-line section. If strain is plotted on the x‑axis and stress on the y‑axis, the gradient directly yields E. Never use data points beyond the linear region.
此外，要求从应力–应变图求杨氏模量时，部分考生倒置坐标轴或选取错误区间。梯度 Δσ/Δε 必须取自直线段。若应变在 x 轴、应力在 y 轴，梯度即直接给出 E。切勿使用线性区域以外的数据点。
5. Applying Conservation of Momentum in Collisions and Explosions | 碰撞与爆炸中的动量守恒应用
Momentum is a vector quantity, yet many students treated it as a scalar in the June 2022 PH02 paper. For one-dimensional collision problems, the conservation law is written as: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where u and v are initial and final velocities, and a consistent sign convention must be adopted. When objects stick together, the final combined mass moves with a common velocity v: m₁u₁ + m₂u₂ = (m₁+m₂)v.
动量是矢量，但在2022年6月PH02考试中许多学生将其当作标量处理。一维碰撞问题的守恒定律写作 m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂，其中 u 和 v 为初、末速度，须统一规定正方向。当物体粘合在一起时，合质量以共同速度 v 运动：m₁u₁ + m₂u₂ = (m₁+m₂)v。
In explosion problems, the initial total momentum is zero (since the system is stationary). After the explosion, fragments move apart such that their momenta are equal in magnitude but opposite in direction: 0 = m₁v₁ + m₂v₂. Students often forgot to assign a negative sign to one of the final velocities, leading to incorrect magnitudes. Always draw a sketch with labelled velocity directions and define your positive direction before writing equations.
6. Energy Principles and Work Done Against Friction | 能量原理与克服摩擦力做功
Work–energy principles were poorly applied when friction was present. The statement ‘work done by the resultant force is equal to the change in kinetic energy’ holds, but students often equated the work done by the driving force alone to the gain in K.E., ignoring the negative work done by friction. The correct energy balance is: W_driving + W_friction = ΔK.E., where W_friction = –F_f d (work done against friction is negative). This can be rearranged to W_driving = ΔK.E. + F_f d, showing that part of the input work is dissipated as thermal energy.
For objects moving on slopes, the change in gravitational potential energy (mgΔh) must be included. The full conservation equation with non‑conservative forces becomes: W_in – F_f d = ΔK.E. + ΔG.P.E. Many students incorrectly omitted the work done against friction when using ‘loss in G.P.E. = gain in K.E. + work against friction’. Be systematic: identify all forces, calculate work done by each, and equate their sum to the change in mechanical energy.
7. Drawing and Analysing Free‑body Diagrams | 受力分析图的绘制与解析
Free‑body diagrams were often incomplete or incorrectly annotated. A proper diagram shows only the body under consideration, with all forces acting on it represented as arrows originating from the body. Weight must act downward from the centre of mass, normal reaction perpendicular to the contact surface, and friction parallel to the surface opposing relative motion or impending motion. Tension forces in ropes/cables act away from the body along the line of the rope.
When resolving forces on an inclined plane, students frequently confused the components of weight. The weight mg should be resolved into mg sin θ parallel to the slope (downward) and mg cos θ perpendicular to the slope. The normal reaction balances mg cos θ unless there is additional vertical force. Avoid swapping sin and cos; check with the extreme case θ = 0° (slope horizontal) where the parallel component should vanish.
斜面问题中分解重力时，学生常混淆分量。重力 mg 应分解为沿斜面向下的 mg sin θ 和垂直于斜面的 mg cos θ。除非有额外竖直力，支持力通常平衡 mg cos θ。避免交换正弦和余弦；可用极端情况 θ = 0°（水平面）检验，此时平行分量应为零。
8. The Principle of Moments and Rotational Equilibrium | 力矩原理与转动平衡
Many struggled with moment calculations when the pivot was not at the end of a beam. The principle of moments states that for an object in rotational equilibrium, the sum of clockwise moments about any point equals the sum of counter‑clockwise moments. The moment of a force is the product of the force and the perpendicular distance from the pivot to the line of action: M = F × d_perp. If the force is not perpendicular to the lever arm, use the perpendicular component of the force or multiply by the lever arm length and the sine of the angle between them.
当支点不在杆端时，许多学生力矩计算出现问题。力矩原理指出，转动平衡的物体对任意支点满足顺时针力矩之和等于逆时针力矩之和。力矩等于力乘以支点到力作用线的垂直距离：M = F × d_perp。若力不垂直于力臂，则应使用力的垂直分量，或用力臂长度乘以力与力臂夹角的正弦。
Common examination errors involved omitting the weight of the beam itself. A uniform beam’s weight acts at its centre of gravity. When taking moments about a support, include the beam’s weight with its appropriate perpendicular distance. Choose a pivot that eliminates an unknown force (usually the point where an unknown reaction force acts) to simplify the equation.
9. Diffraction, Interference and Coherence | 衍射、干涉与相干性
The wave concepts of diffraction and interference were frequently muddled. Diffraction refers to the spreading of waves when they pass through a gap or around an obstacle. Maximum diffraction occurs when the gap width is comparable to the wavelength. For light, a narrow single slit produces a broad central maximum. Interference patterns from two slits require the sources to be coherent (constant phase difference and same frequency). Many answers in PH02 wrongly assumed that a single slit could produce a two‑source interference pattern.
Describe Young’s double‑slit experiment using the formula w = λD / s, where w is the fringe spacing, λ the wavelength, D the distance from slits to screen, and s the slit separation. Ensure all lengths are in metres. The fringe spacing increases with longer wavelength or with smaller slit separation. Coherence is typically achieved by using a single source illuminating both slits, not by using ‘laser light’ alone without explaining its coherence property.
描述杨氏双缝实验使用公式 w = λD / s，其中 w 为条纹间距，λ 为波长，D 为双缝到屏幕距离，s 为双缝间距，所有长度单位用米。条纹间距随波长增长或缝距减小而增大。通常通过单光源照亮双缝实现相干，而非仅说“激光”，需解释激光的相干特性。
10. Refractive Index and Snell’s Law Applications | 折射率与斯涅尔定律应用
When applying Snell’s law n₁ sin θ₁ = n₂ sin θ₂, many students used angles measured from the interface rather than from the normal. The angle of incidence and angle of refraction are always measured between the ray and the normal to the surface. For light travelling from medium 1 to medium 2, the refractive index n₁ is for the incident medium and n₂ for the refracting medium. Total internal reflection (TIR) occurs when light travels from a denser to a rarer medium (n₁ > n₂) and the angle of incidence exceeds the critical angle θ_c given by sin θ_c = n₂ / n₁.
应用斯涅尔定律 n₁ sin θ₁ = n₂ sin θ₂ 时，许多学生从界面量取角度而非从法线。入射角和折射角始终是光线与界面法线之间的夹角。光从介质1进入介质2时，n₁ 为入射介质折射率，n₂ 为折射介质。全内反射 (TIR) 发生在光从光密介质射向光疏介质 (n₁ > n₂) 且入射角大于临界角 θ_c 时，临界角满足 sin θ_c = n₂ / n₁。
A particularly common mistake was misusing the sine function or confusing which refractive index belongs to which medium. In a multiple‑choice or calculation setting, if light enters glass from air, n₁ = 1.0 and n₂ ≈ 1.5. If the ray bends towards the normal, the angle in the glass is smaller. Always check that your answer is physically sensible: light entering an optically denser medium slows down and bends towards the normal.
📚 AS Physics: Momentum Key Points Revision | AS 物理：动量考点精讲
Welcome to this comprehensive revision guide on momentum for AS-level Physics. Momentum is a fundamental concept that links mass and velocity, and its conservation is one of the most powerful principles in mechanics. Understanding impulse, collisions, and explosions is essential for solving a wide variety of exam problems. This article will walk you through the key points, common pitfalls, and typical question types.
欢迎阅读这份 AS 物理动量考点精讲。动量是将质量与速度联系起来的基本概念，动量守恒定律是力学中最强大的原理之一。理解冲量、碰撞和爆炸对于解决各类考试问题至关重要。本文将带你梳理关键考点、常见误区和典型题型。
1. What is Momentum? | 什么是动量？
Momentum (p) is defined as the product of an object’s mass and its velocity. It is a measure of the quantity of motion an object possesses. The formula is p = m v, where m is mass in kilograms (kg) and v is velocity in metres per second (m/s). The SI unit of momentum is kg m s⁻¹. Momentum is directly proportional to both mass and velocity: a heavier object moving at the same speed has more momentum, and a faster object of the same mass has more momentum.
动量（p）定义为物体质量与速度的乘积，它是衡量物体运动量的物理量。公式为 p = m v，其中 m 是质量（千克，kg），v 是速度（米/秒，m/s）。动量的国际单位是 kg·m·s⁻¹。动量与质量和速度成正比：相同速度下质量越大的物体动量越大，相同质量下速度越快的物体动量也越大。
p = m v
2. Momentum as a Vector Quantity | 动量是矢量
Momentum is a vector quantity because velocity is a vector. This means momentum has both magnitude and direction. When solving problems, you must account for direction, often using positive and negative signs. For example, if an object moves to the right with momentum +5 kg m s⁻¹ and then bounces back to the left, its final momentum might be -4 kg m s⁻¹. The change in momentum is crucial. The direction of momentum is the same as the direction of velocity.
3. Newton’s Second Law in Terms of Momentum | 用动量表述牛顿第二定律
Newton’s second law is often written as F = m a, but it was originally stated in terms of momentum: the net force acting on an object is equal to the rate of change of its momentum. Mathematically, F = Δp / Δt. This form is more general because it applies even when mass changes (e.g., rocket propulsion). For constant mass, F = Δ(m v)/Δt = m Δv/Δt = m a. This equation shows that a given force will produce a larger change in momentum if applied over a longer time.
牛顿第二定律常写作 F = m a，但其原始表述涉及动量：作用在物体上的合力等于物体动量的变化率。数学表达式为 F = Δp / Δt。这种形式更为普遍，因为即使质量变化（如火箭推进）也适用。对于质量不变的情况，F = Δ(m v)/Δt = m Δv/Δt = m a。该公式表明，在相同力作用下，作用时间越长，动量的变化越大。
F = Δp / Δt
4. Impulse and the Impulse-Momentum Theorem | 冲量与冲量-动量定理
Impulse (J) is defined as the product of the average force acting on an object and the time interval over which it acts. J = F Δt. Impulse is a vector; its direction is the same as the force. The impulse-momentum theorem states that the impulse applied to an object equals the change in its momentum: J = Δp = m v – m u, where u is initial velocity and v is final velocity. This theorem is extremely useful for solving problems involving collisions, especially when the force varies, as the total impulse equals the area under a force–time graph.
冲量（J）定义为作用在物体上的平均力与作用时间的乘积，即 J = F Δt。冲量是矢量，方向与力相同。冲量-动量定理指出，物体所受冲量等于其动量的变化量：J = Δp = m v – m u，其中 u 为初速度，v 为末速度。该定理在解决碰撞问题时非常有用，尤其是当力变化时，总冲量等于力-时间图下的面积。
J = F Δt = Δp = m v – m u
5. Force-Time Graphs and Impulse | 力-时间图与冲量
A force–time graph plots the force acting on an object against time. The area under the curve (integral of force with respect to time) gives the impulse. For a constant force, the area is a rectangle: F × Δt. For a varying force, the area can be estimated using geometrical shapes or counting squares. In many exam questions, a graph is provided, and you must calculate the impulse to then find the change in momentum or the final velocity. Remember that the impulse can be positive or negative depending on the direction of the force relative to the chosen positive direction.
The principle of conservation of linear momentum states that in a closed system (no external forces), the total momentum before an interaction (collision or explosion) equals the total momentum after the interaction. For two objects, m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂. This law is derived from Newton’s third law: the forces two objects exert on each other are equal and opposite, and they act for the same time, so the impulses are equal and opposite, leading to zero net change in total momentum. It applies to all types of collisions and explosions, provided no external resultant force acts.
In an elastic collision, both momentum and kinetic energy are conserved. This means the total kinetic energy before the collision equals the total kinetic energy after. For two objects, ½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂². Elastic collisions are idealised; in practice, only collisions between atomic particles or very hard objects (like billiard balls) are nearly elastic. In AS-level problems, you might be given the relative speed of approach equals the relative speed of separation for elastic collisions: v₂ – v₁ = u₁ – u₂ (in one dimension). This relationship, combined with momentum conservation, can solve for final velocities.
在弹性碰撞中，动量和动能都守恒。即碰撞前的总动能等于碰撞后的总动能。对于两个物体，½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂²。弹性碰撞是一种理想化模型；实际上，只有原子粒子之间的碰撞或非常坚硬的物体（如台球）近似弹性。在 AS 考试中，弹性碰撞可能会给出相对接近速度等于相对分离速度的关系：v₂ – v₁ = u₁ – u₂（一维情况）。该关系结合动量守恒可以求解末速度。
½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂²
v₂ – v₁ = u₁ – u₂
8. Inelastic Collisions | 非弹性碰撞
In an inelastic collision, momentum is conserved but kinetic energy is not. Some kinetic energy is transformed into other forms, such as heat, sound, or deformation. A perfectly inelastic collision (or completely inelastic) is one where the objects stick together after collision and move with a common velocity. In that case, the maximum kinetic energy is lost. The equation m₁ u₁ + m₂ u₂ = (m₁ + m₂) v is used, where v is the common final velocity. The loss in kinetic energy can be calculated. Most everyday collisions are inelastic.
在非弹性碰撞中，动量守恒，但动能不守恒。一部分动能转化为其他形式，如热能、声能或形变能。完全非弹性碰撞是指碰撞后物体粘在一起并以共同速度运动的情况。此时动能损失最大。方程 m₁ u₁ + m₂ u₂ = (m₁ + m₂) v 用于求解，其中 v 为共同末速度。可以计算动能的损失。大多数日常碰撞都是非弹性的。
m₁ u₁ + m₂ u₂ = (m₁ + m₂) v
9. Explosions | 爆炸
An explosion is the reverse of an inelastic collision. Initially, a single object (or system) is at rest or moving, and then it breaks into two or more fragments. The total momentum is conserved. If the system is initially at rest, the total momentum after the explosion must be zero, so the fragments move apart with equal and opposite momenta. For example, a stationary nucleus emitting an alpha particle: the alpha and the daughter nucleus have momenta equal in magnitude but opposite in direction. Equation: 0 = m₁ v₁ + m₂ v₂, thus v₁/v₂ = – m₂/m₁. Energy comes from the conversion of potential energy (e.g., chemical or nuclear) into kinetic energy.
This comprehensive revision guide covers the most essential topics, equations, and practical skills for the AQA GCSE Physics exam. Use it to organise your final review, spot common mistakes, and reinforce key concepts. Each section pairs concise English explanations with Chinese translations so you can grasp the ideas quickly and accurately.
Energy exists in different stores: kinetic, gravitational potential, elastic potential, thermal, chemical, magnetic, electrostatic, and nuclear. In any system change, energy moves between stores along four pathways – mechanical work, electrical work, heating, and radiation – but the total energy in a closed system never changes (conservation of energy).
When a force moves an object, work is done: W = F s. Rate of energy transfer is power: P = E / t. Kinetic energy depends on mass and speed squared, while gravitational potential energy rises with height.
力使物体移动时做功：W = F s。能量转移速率即功率：P = E / t。动能取决于质量和速度的平方，而重力势能随高度增加。
Kinetic energy: Ek = ½ m v² | Gravitational potential energy: Ep = m g h
Always convert to SI units: mass in kg, speed in m/s, height in m, force in N, energy in J. Sankey diagrams show useful and wasted energy as arrows of different widths.
Electric current is the rate of flow of charge. In a single closed loop, the current is the same everywhere; potential difference (voltage) across the supply is shared between components in series. In parallel branches, the supply voltage appears across each branch while the total current splits.
Charge: Q = I t | Ohm’s law: V = I R | Power: P = I V = I² R
Resistance of a wire increases with length and decreases with cross-sectional area. Required practical: investigate how the length of a resistance wire affects its resistance using an ammeter, voltmeter, and variable resistor.
Straight line through origin – current proportional to voltage
Filament lamp
Curve flattening at high voltage – resistance increases as it heats up
Diode
Very high resistance in reverse – only conducts in forward bias above ~0.6 V
元件：定值电阻（恒温）的 I–V 曲线为过原点的直线，电流与电压成正比；白炽灯丝曲线在高电压处变平，因为温度升高电阻增大；二极管反向电阻极大，仅在正向偏压约 0.6 V 以上导通。
In the UK, the mains supply is 230 V, 50 Hz alternating current. Live wire is brown, neutral blue, earth wire green-and-yellow. Fuses and circuit breakers prevent overheating.
Density measures how tightly packed the particles are. The particle model explains changes of state: melting, boiling, condensing, freezing, sublimation. During a change of state, the temperature stays constant while internal energy increases or decreases.
Density: ρ = m ÷ V | Specific heat capacity: ΔE = m c Δθ
Specific latent heat is the energy needed to change state without changing temperature. Lf for fusion, Lv for vaporisation: E = m L.
比潜热是改变物态而不改变温度所需的能量。熔化潜热 Lf，汽化潜热 Lv：E = m L。
In a gas, pressure is caused by particles colliding with walls. If temperature is constant, doubling the volume halves the pressure – p V = constant. Increasing temperature at constant volume raises pressure because particles move faster and hit harder.
在气体中，压强由粒子撞击容器壁引起。恒温下，体积加倍则压强减半——p V = 常数。定容升温时，压强增大，因为粒子运动更快、撞击更猛烈。
4. Atomic Structure and Radioactivity | 原子结构与放射性
The atom contains a tiny, positively charged nucleus made of protons and neutrons, surrounded by electrons in energy levels. Atomic number Z = number of protons, mass number A = protons + neutrons. Isotopes share the same Z but differ in neutron number.
原子含有一个极小的带正电的原子核，由质子和中子组成，周围电子在能级上运动。原子序数 Z = 质子数，质量数 A = 质子 + 中子。同位素质子数相同而中子数不同。
Unstable nuclei emit radiation: alpha (α: ₂⁴He nucleus, heavy, strong ionisation, stopped by paper), beta (β⁻: fast electron, medium ionisation, stopped by thin aluminium), gamma (γ: electromagnetic wave, weak ionisation, reduced by thick lead). Radioactive decay is random and each isotope has a fixed half-life.
Nuclear equations balance total mass and atomic numbers. Applications include medical tracers, radiotherapy, and smoke alarms. Nuclear fission splits a large nucleus; nuclear fusion joins light nuclei – the Sun’s energy comes from fusion.
Scalars have magnitude only (speed, distance, energy). Vectors have magnitude and direction (velocity, displacement, force, momentum). A resultant force causes acceleration according to F = m a and weight is a force due to gravity: W = m g (g = 9.8 N/kg on Earth).
标量只有大小（速率、路程、能量）。矢量既有大小也有方向（速度、位移、力、动量）。合力产生加速度，遵循 F = m a；重力为重量：W = m g（地表 g = 9.8 N/kg）。
Velocity: v = s / t | Acceleration: a = Δv / t | Momentum: p = m v
In a closed system, momentum is conserved: total momentum before = total momentum after. Stopping distance = thinking distance + braking distance; factors include speed, reaction time, road conditions, and tyre state.
For springs obeying Hooke’s law, extension is proportional to force up to the limit of proportionality: F = k e. Elastic potential energy: Ee = ½ k e². Required practical: measure force-extension for a spring and calculate the spring constant.
对遵守胡克定律的弹簧，在比例极限范围内伸长量与力成正比：F = k e。弹性势能：Ee = ½ k e²。必做实验：测量弹簧的力-伸长关系并计算劲度系数。
6. Waves and the Electromagnetic Spectrum | 波动与电磁波谱
Waves transfer energy without transferring matter. Transverse waves (e.g. light, water waves) vibrate perpendicular to energy transfer; longitudinal waves (e.g. sound) vibrate parallel. The wave equation links speed, frequency, and wavelength: v = f λ.
波动传递能量而不传递物质。横波（如光波、水波）振动方向与能量传递方向垂直；纵波（如声波）振动方向平行。波动方程联系波速、频率和波长：v = f λ。
Electromagnetic waves form a continuous spectrum: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma – in order of increasing frequency and decreasing wavelength. All travel at 3.0 × 10⁸ m/s in a vacuum.
Reflection follows the law: angle of incidence = angle of reflection. Refraction occurs when a wave crosses a boundary and changes speed. Convex lenses converge light to a real or virtual image depending on object distance.
Required practical: measure the speed of waves on water or in a solid, and investigate reflection and refraction with a ray box.
必做实验：测量水波或固体中的波速，并用光线盒研究反射和折射。
7. Magnetism and Electromagnetism | 磁学与电磁学
A current-carrying wire produces a magnetic field (right-hand grip rule). An electromagnet uses a soft iron core to strengthen the field. The motor effect: a conductor in a magnetic field experiences a force when current flows, given by Fleming’s left-hand rule.
Force on a conductor: F = B I L (when perpendicular)
Electromagnetic induction: moving a wire through a magnetic field (or changing the field) induces a potential difference. Generators rotate a coil in a magnetic field to produce alternating current; dynamos produce direct current via a split-ring commutator.
Transformers change the size of an alternating voltage. For an ideal transformer:
Vp / Vs = np / ns | Pp = Ps
Step-up transformers increase voltage and decrease current to minimise energy losses in the National Grid.
升压变压器提高电压、降低电流，以减小国家电网中的能量损耗。
8. Space Physics | 空间物理
Our solar system includes the Sun, planets, dwarf planets, moons, asteroids, and comets. Orbits result from the balance between gravitational pull and the object’s velocity. For a stable circular orbit, the gravitational force provides the centripetal force, and speed changes with radius.
The life cycle of a star depends on its mass. A Sun-like star evolves: nebula → protostar → main sequence → red giant → white dwarf. More massive stars end as supernova then neutron star or black hole. Fusion in stars produces elements up to iron; heavier elements form in supernovae.
Redshift of light from distant galaxies shows they are moving away; more distant galaxies have greater redshift, revealing the Universe is expanding. This supports the Big Bang theory. Key evidence also includes cosmic microwave background radiation.
Understanding thermal physics is essential for IGCSE Edexcel Physics. This article breaks down the key concepts of thermodynamics – from molecular motion and internal energy to heat transfer and latent heat – all tailored to the Edexcel specification. Each section pairs clear English explanations with Chinese translations, supported by examples, equations, and practical tips to help you master the exam.
1. Kinetic Particle Model and States of Matter | 分子运动论与物质状态
The kinetic particle model explains the behaviour of solids, liquids and gases. In solids, particles are closely packed in a regular arrangement, vibrating about fixed positions. They have strong intermolecular forces and fixed shape and volume.
In liquids, particles are still close together but can slide past each other. They take the shape of the container, have a fixed volume, and forces between particles are weaker than in solids.
In gases, particles are far apart and move randomly at high speeds. They have negligible intermolecular forces, no fixed shape or volume, and can be compressed easily.
气体中粒子间距很大，作高速无规则运动。分子间作用力可忽略，既无固定形状也无固定体积，容易被压缩。
2. Temperature vs Heat Energy | 温度与热能的区别
Temperature is a measure of the average kinetic energy of the particles in a substance. It is measured in degrees Celsius (°C) or Kelvin (K). Absolute zero (0 K = -273°C) is the temperature at which particles have minimum kinetic energy.
温度衡量物质中粒子的平均动能，单位是摄氏度 (°C) 或开尔文 (K)。绝对零度 (0 K = -273°C) 是粒子动能达到最小时的温度。
Heat (thermal energy) is the energy transferred from a hotter object to a colder one because of a temperature difference. It is measured in joules (J). An object does not ‘contain’ heat; it has internal energy. Heat is energy in transit.
Key distinction: temperature reflects the strength of particle motion, whereas heat refers to the total energy transferred. Two objects at the same temperature can have different internal energies if their masses differ.
Internal energy is the total kinetic energy (due to particle motion) and potential energy (due to intermolecular forces and bond energies) of all the particles in a substance. When a substance is heated, its internal energy increases.
During a change of state, the temperature remains constant even though heat is being supplied. The added energy increases the potential energy of particles, breaking bonds without increasing kinetic energy. Thus internal energy continues to rise.
Formula reminder: change in internal energy ΔU = heat supplied Q minus work done by the system W. For many IGCSE situations, work done is negligible, so ΔU ≈ Q.
4. Heat Capacity and Specific Heat Capacity | 热容与比热容
Heat capacity C is the energy required to raise the temperature of an entire object by 1°C. It is measured in J/°C. The relation is Q = C × Δθ, where Δθ is the temperature change.
热容 C 是将整个物体的温度升高 1°C 所需的能量，单位是 J/°C。公式 Q = C × Δθ，其中 Δθ 是温度变化量。
Specific heat capacity c is the energy required to raise the temperature of 1 kg of a substance by 1°C, unit J/(kg °C). Formula: Q = m c Δθ. Water has a high specific heat capacity (4200 J/(kg °C)), which makes it an excellent coolant and stabilises Earth’s climate.
比热容 c 是将 1 kg 物质升高 1°C 所需的能量，单位 J/(kg °C)。公式 Q = m c Δθ。水的比热容很大 (4200 J/(kg °C))，因此是极好的冷却剂，也能稳定地球气候。
Example: To heat 2 kg of aluminium (c = 900 J/(kg °C)) from 20°C to 100°C, energy needed Q = 2 × 900 × 80 = 144,000 J.
Specific latent heat is the energy required to change the state of 1 kg of a substance without temperature change. The formula is Q = m l, where l is the specific latent heat (J/kg).
比潜热是使 1 kg 物质在不改变温度的情况下完成物态变化所需的能量。公式 Q = m l，其中 l 是比潜热 (J/kg)。
Specific latent heat of fusion lf applies to melting or freezing. For water, lf = 334,000 J/kg. Specific latent heat of vaporisation lv applies to boiling or condensing; for water, lv = 2,260,000 J/kg. Notice vaporisation requires much more energy because particles must completely separate.
In a heating curve, the flat sections represent state changes where temperature remains constant. The slope sections show sensible heating (temperature rise). The energy input during plateaus is latent heat.
Conduction is the transfer of thermal energy through a solid (or between objects in direct contact) via particle vibrations and free electron movement, without bulk movement of the material. Metals are good conductors because free electrons rapidly pass kinetic energy along the metal.
Poor conductors (insulators) like wood, plastic, and air trap energy because they lack free electrons. Conduction rate increases with temperature difference and cross-sectional area, and decreases with thickness.
Real-world applications: saucepan bases are made of metal for fast conduction; handles are plastic or wood to prevent burns; double-glazed windows trap air to reduce conduction.
实际应用：锅底用金属以加快传导；手柄用塑料或木头防烫；双层玻璃窗留住空气以减少传导。
7. Heat Transfer: Convection | 热传递：对流
Convection is the transfer of heat in fluids (liquids and gases) by the movement of the fluid itself. Warmer, less dense fluid rises, while cooler, denser fluid sinks, creating a convection current.
Convection cannot occur in solids because particles are not free to move. Common examples include sea breezes, radiators heating a room, and boiling water where hot water rises from the bottom.
In a domestic hot water system, the boiler heats water at the bottom; hot water rises to the storage tank while cold water sinks to be reheated. This natural circulation is driven by density changes.
8. Heat Transfer: Radiation and Black Body | 热传递：辐射与黑体
Radiation is the transfer of energy by electromagnetic waves, mainly infrared. It does not require a medium and can travel through a vacuum. All objects above absolute zero emit thermal radiation.
A black body is a perfect absorber and emitter of radiation. Dark, matt surfaces are better absorbers and emitters than light, shiny surfaces. Shiny silvered surfaces are poor emitters and good reflectors.
Application: vacuum flasks have silvered inner and outer walls to minimise radiation; car radiators are black to emit heat effectively; white clothing keeps people cooler by reflecting sunlight.
应用：保温瓶的内外壁镀银以减少辐射；汽车散热器涂黑以有效散发热量；白色衣服反射阳光，使人更凉爽。
9. Gas Pressure and Temperature (Qualitative) | 气体压强与温度 (定性)
In a sealed container, gas particles collide with the walls, exerting a force. Pressure is force per unit area, p = F/A. The pressure of a fixed mass of gas increases with temperature at constant volume because particles move faster and hit the walls more frequently and more forcefully.
在密闭容器中，气体粒子撞击器壁产生力，压强 p = F/A。等体积下，一定质量的气体温度升高，压强增大，因为粒子运动更快，撞击器壁更频繁、更有力。
If the volume can change (e.g. a balloon), the gas expands when heated. For a fixed mass of gas at constant pressure, volume is directly proportional to absolute temperature: V/T = constant (Charles’ law, though Edexcel IGCSE does not require formula manipulation – understand the trend).
Explaining in terms of particles: heating increases average kinetic energy; particles move faster, push outward more, so volume increases until pressure balances external pressure.
Exam questions often combine specific heat capacity and latent heat. For example, heating ice at -10°C to steam at 120°C involves five stages: (1) warming ice, Q₁ = m cice Δθ; (2) melting ice, Q₂ = m lf; (3) warming water, Q₃ = m cwater Δθ; (4) boiling water, Q₄ = m lv; (5) warming steam, Q₅ = m csteam Δθ. Sum all Q for total energy.
考题常结合比热容和潜热。例如将 -10°C 的冰加热到 120°C 的水蒸气，需五步：(1) 冰升温，Q₁ = m c冰 Δθ；(2) 冰熔化，Q₂ = m lf；(3) 水升温，Q₃ = m c水 Δθ；(4) 水沸腾，Q₄ = m lv；(5) 水蒸气升温，Q₅ = m c汽 Δθ。总能量为各 Q 之和。
Assume m = 0.5 kg, cice = 2100 J/(kg °C), cwater = 4200 J/(kg °C), csteam = 2000 J/(kg °C), lf = 334,000 J/kg, lv = 2,260,000 J/kg. Calculate each Q and total. This systematic approach helps avoid mistakes.
11. Practical Skills: Measuring Specific Heat Capacity and Latent Heat | 实验技能：测量比热容与潜热
To determine the specific heat capacity of a solid, a common method is the electrical method: use an immersion heater (known power P) in a metal block, measure temperature rise Δθ over time t. Energy supplied Q = P × t, and c = Q / (m Δθ). Insulating the block reduces heat loss.
测量固体比热容常用电加热法：将已知功率 P 的加热器插入金属块，记录时间 t 内的温升 Δθ。供给能量 Q = P × t，c = Q / (m Δθ)。包裹隔热材料可减少热损失。
For the specific latent heat of fusion of ice, use a funnel, crushed ice at melting point, and an electric heater of known power. Melt ice, collect water, measure mass m melted in time t. Then lf = P t / m. Ensure ice is melting and water is at 0°C to avoid sensible heating errors.
测量冰的熔化潜热，可用漏斗、0°C 碎冰和已知功率的电热器。加热使冰熔化，收集水，测量时间 t 内熔化质量 m。则 lf = P t / m。确保冰正处于熔点，水温为 0°C，避免显热误差。
To determine specific latent heat of vaporisation, boil water with an immersion heater, collect condensed steam and measure mass m in time t. lv = P t / m. Lag the flask and use a shield to minimise condensation losses.
测量水的汽化潜热，用电热器沸腾水，收集冷凝蒸汽并测时间 t 内凝结质量 m。lv = P t / m。包扎烧瓶并使用挡板以减少冷凝损失。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Confusing temperature and heat. Remember, temperature measures average kinetic energy; heat is energy transferred. Don’t say ‘heat rises’ – it is hot air that rises by convection.
Mistake 2: Using Celsius instead of Kelvin in gas law relationships. For proportionality (e.g. V ∝ T), convert to Kelvin: T(K) = θ(°C) + 273.
错误二：在气体定律比例关系中使用摄氏度而非开尔文。正比关系 (如 V ∝ T) 中必须转为开尔文：T(K) = θ(°C) + 273。
Mistake 3: Forgetting that during a state change, temperature stays constant even though heat is added. Energy goes into breaking bonds, not increasing kinetic energy.
错误三：忘记物态变化时，尽管加熱温度不变。能量用于打破分子间键，而非提高动能。
Exam tip: Always show full working – write Q = m c Δθ, substitute values, then calculate. State the principle before calculation (e.g. ‘energy gained by cold water = energy lost by metal block’). Include units throughout.
考试技巧：解题时写出完整步骤 – 先写 Q = m c Δθ，代入数值，再计算。计算前先陈述原理 (如”冷水吸收能量 = 金属块放出能量”)。全程带单位。
Practical-based questions: be ready to describe an experiment, identify safety precautions (hot objects, steam burns), suggest improvements (insulation, stirrer, take multiple readings). Use clear diagrams and mention measuring instruments (thermometer, stopwatch, balance).
Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. For AQA A-Level Physics, you need to be confident with vector and scalar quantities, the equations of uniform acceleration (SUVAT), graphical analysis of motion, and the special case of projectile motion. This article breaks down every essential concept, common pitfalls, and exam technique so you can tackle any kinematics question with clarity.
A scalar quantity has magnitude only, for example distance, speed, mass and time. A vector quantity has both magnitude and direction, such as displacement, velocity, acceleration and force. In kinematics, the distinction is crucial because adding vector quantities requires consideration of direction.
When adding vectors, you can use tip-to-tail diagrams or resolve them into perpendicular components. For one-dimensional motion, a positive or negative sign often indicates direction. In exam questions, always define your positive direction clearly before writing any equation.
2. Displacement, Velocity and Acceleration | 位移、速度和加速度
Displacement (s) is the straight-line distance in a given direction from the starting point – it is a vector. Velocity (v) is the rate of change of displacement. Acceleration (a) is the rate of change of velocity. Both velocity and acceleration are vector quantities, so a change in direction means a change in velocity even if the speed is constant.
Instantaneous velocity is the velocity at a specific instant, while average velocity = total displacement / total time. Do not confuse speed (scalar) with velocity (vector). A car moving around a roundabout at constant speed is accelerating because its direction changes.
Three main graphs describe motion: displacement–time, velocity–time and acceleration–time. For AQA, you must be able to sketch, interpret and calculate quantities from slopes and areas.
On a displacement–time graph, the gradient gives velocity. A straight line means constant velocity; a curved line means acceleration is present. On a velocity–time graph, the gradient gives acceleration, and the area under the graph gives displacement. For an acceleration–time graph, the area gives change in velocity.
Beware: a horizontal line on a velocity–time graph means constant velocity (a = 0). A sloping straight line means constant acceleration. If the graph is curved, acceleration is changing.
The SUVAT equations apply only when acceleration is constant. The five variables are:
SUVAT方程仅在加速度恒定时适用。五个变量分别是：
s – displacement (m)
u – initial velocity (m s⁻¹)
v – final velocity (m s⁻¹)
a – acceleration (m s⁻²)
t – time (s)
The four standard equations are:
四个标准方程为：
v = u + a t
s = u t + ½ a t²
s = ½ (u + v) t
v² = u² + 2 a s
To solve problems, list the known quantities and the one you need. Choose the equation that does not involve the unknown variable you are not asked for. Often you can avoid solving quadratics by selecting the right equation.
解题时，列出已知量和所求量。选择不包含多余未知量的方程。通常，选对公式可以避免解二次方程。
Always check that your signs are consistent with your chosen positive direction. If an object decelerates, acceleration is negative. If it moves backwards, displacement and velocity become negative. In vertical motion, taking upward as positive gives a = –g.
务必确保符号与你选定的正方向一致。如果物体减速，加速度为负。如果它向后运动，位移和速度均为负。在竖直运动中，若取向上为正，则 a = –g。
5. Free Fall and Acceleration due to Gravity | 自由落体和重力加速度
All objects near the Earth’s surface fall with the same uniform acceleration, g, provided air resistance is negligible. The standard value is g = 9.81 m s⁻². In calculations, you often use g = 9.8 m s⁻² or 9.81 m s⁻² as specified in the question.
在地球表面附近，所有物体在没有空气阻力的情况下都以相同的匀加速度g下落。标准值为 g = 9.81 m s⁻²。计算时，按题目要求常取 g = 9.8 m s⁻² 或 9.81 m s⁻²。
When an object is thrown vertically upwards, it decelerates at 9.81 m s⁻² until momentarily stationary at the top, then accelerates downwards at 9.81 m s⁻². The time to rise to the highest point equals the time to fall back to the same level if air resistance is ignored. The symmetry of the motion can save you a lot of work in problems.
当物体竖直上抛时，它以 9.81 m s⁻² 减速，直至最高点瞬间静止，然后以 9.81 m s⁻² 向下加速。如果不计空气阻力，上升到最高点的时间等于落回同一水平面的时间。运动的对称性可以为解题节省不少时间。
For a dropped object, initial velocity u = 0. The equations simplify to v = g t, s = ½ g t² and v² = 2 g s.
对于下落的物体，初速度 u = 0。方程简化为 v = g t，s = ½ g t² 和 v² = 2 g s。
6. Projectile Motion | 抛体运动
A projectile is any object moving under the influence of gravity alone after being launched. For AQA, you treat the horizontal and vertical motions independently. The horizontal component of velocity remains constant because there is no horizontal acceleration (ignoring air resistance). The vertical motion is uniformly accelerated with a = g downwards.
抛体是任何被抛出后只受重力作用的物体。在AQA考试中，你需将水平和垂直运动分开处理。速度的水平分量保持不变，因为没有水平加速度（忽略空气阻力）。竖直运动是匀加速运动，加速度 a = g 向下。
Resolve the initial velocity u into horizontal and vertical components:
将初速度 u 分解为水平和竖直分量：
uₓ = u cos θ
u_y = u sin θ
where θ is the angle of projection measured from the horizontal.
其中 θ 是抛射角（与水平方向的夹角）。
The time of flight depends only on the vertical motion. Use s_y = u_y t – ½ g t² with s_y = 0 for a projectile landing on the same horizontal level. For the range, multiply the constant horizontal velocity by the total time of flight: range = uₓ × t_total. The maximum height is found from vertical motion with final vertical velocity v_y = 0.
飞行时间只取决于竖直运动。对于落回同一水平面的抛体，设 s_y = 0 使用 s_y = u_y t – ½ g t² 求解。射程为水平速度乘以总时间：射程 = uₓ × t_total。最大高度可由竖直末速度 v_y = 0 求得。
7. Resolving Vectors | 矢量分解
Many kinematics problems involve vectors not aligned with the axes. Resolving a vector into two perpendicular components (usually horizontal and vertical) simplifies calculations. Use trigonometry:
where V is the magnitude and θ is the angle to the horizontal.
其中 V 是矢量的大小，θ 是与水平方向的夹角。
When combining components back to find the resultant vector, use Pythagoras’ theorem for magnitude and trigonometry for direction. In projectile questions, always draw a diagram showing the velocity components at launch, at the highest point, and just before landing.
A common practical question involves measuring g using free fall. One method drops a steel ball from a known height and measures the time of fall using a trapdoor and electronic timer. Use s = ½ g t², which rearranges to g = 2 s / t². Repeating for several heights and plotting s against t² gives a straight line with gradient g/2.
一个常见的实验题涉及用自由落体测量g。一种方法是将钢球从已知高度释放，用电磁装置和电子计时器测量下落时间。使用 s = ½ g t²，变形为 g = 2 s / t²。多次改变高度，绘制 s 对 t² 的图线，得到一条直线，斜率为 g/2。
Another technique uses a light gate and a double-interrupt card to measure instantaneous velocity at two points and uses v² = u² + 2 a s. A more modern approach uses video analysis or a pendulum, but for kinematics the free fall method is standard. The main sources of uncertainty are reaction time, parallax when measuring height, and air resistance.
另一种方法使用光电门和双遮光板测量两点的瞬时速度，利用方程 v² = u² + 2 a s。更现代的方法包括视频分析或单摆，但运动学中自由落体法是标准方法。主要的误差来源是反应时间、测量高度时的视差以及空气阻力。
To reduce reaction time errors, use a mechanical release and a pressure switch or light gate. Repeating measurements and averaging reduces random errors. Always quote g to an appropriate number of significant figures and compare with the accepted value, calculating the percentage difference.
For a projectile launched at an angle, write separate SUVAT equations for horizontal (x) and vertical (y) directions. Horizontal: s_x = uₓ t, v_x = uₓ. Vertical: v_y = u_y – g t, s_y = u_y t – ½ g t², v_y² = u_y² – 2 g s_y.
对于斜抛体，分别为水平（x）和竖直（y）方向列出SUVAT方程。水平方向：s_x = uₓ t，v_x = uₓ。竖直方向：v_y = u_y – g t，s_y = u_y t – ½ g t²，v_y² = u_y² – 2 g s_y。
The velocity at any instant is the vector sum of the two components: magnitude = √(v_x² + v_y²), direction θ = tan⁻¹(v_y / v_x). The path of a projectile is a parabola, which you can derive by eliminating t from the x and y equations.
When solving problems where the launch height and landing height differ, set s_y to the vertical displacement between start and finish. Solve the quadratic for t, discarding any negative root. This skill is frequently tested.
AQA examiners often report that students confuse speed and velocity, forget that acceleration can be negative, or misapply the SUVAT equations by using the average velocity where instantaneous velocity is needed. Read the question carefully to identify which direction is positive.
Always write down the known quantities with correct signs before selecting your equation. For multi-step problems, a clear diagram with labelled vectors is invaluable. If a question asks for an algebraic expression, do not substitute numbers prematurely.
Unit errors lose marks: displacement in metres (m), time in seconds (s), velocity in m s⁻¹, acceleration in m s⁻². When using g, ensure all quantities use the same sign convention. Check that your final answer is reasonable – for instance, a student’s calculated time of flight of 200 s for a ball thrown across a field is implausible.
Finally, practise drawing and interpreting velocity–time graphs. Simple errors like confusing area with gradient are unfortunately common. Remember: gradient = acceleration, area = displacement.
Materials physics in GCSE CCEA Physics brings together ideas about density, forces in springs, deformation and the properties that make different materials suitable for specific applications. This revision guide walks through every core concept, from Hooke’s law to stress–strain curves, to help you build a solid foundation for exam success.
1. Density – Definition and Measurement | 密度——定义与测量
Density is the mass per unit volume of a substance. The formula is ρ = m / V, where ρ (rho) is density (kg/m³), m is mass (kg) and V is volume (m³).
密度是物质单位体积的质量，公式为 ρ = m / V，其中 ρ 为密度（kg/m³），m 为质量（kg），V 为体积（m³）。
For a regular solid, volume can be calculated from geometric dimensions. For an irregular solid, volume is found by the displacement method using a measuring cylinder and water. For a liquid, mass can be measured by subtracting the mass of an empty container from the mass of the container plus liquid.
Ensure you can rearrange this formula and convert between units such as g/cm³ and kg/m³ (1 g/cm³ = 1000 kg/m³).
务必能熟练变形该公式，并换算单位，例如 1 g/cm³ = 1000 kg/m³。
2. Hooke’s Law | 胡克定律
Hooke’s law states that the extension of a spring is directly proportional to the applied force, provided the elastic limit is not exceeded. It can be written as F = k × e, where F is force (N), e is extension (m) and k is the spring constant (N/m) – a measure of stiffness.
胡克定律指出，只要不超过弹性极限，弹簧的伸长量与所受外力成正比。数学表达式为 F = k × e，其中 F 为力（N），e 为伸长量（m），k 为弹簧常数（N/m），反映了弹簧的刚度。
F = k × e
A stiffer spring has a larger k value, meaning more force is needed to produce the same extension.
弹簧越硬，k 值越大，产生相同伸长量所需的力也越大。
3. Force–Extension Graphs | 力–伸长量关系图
A force–extension graph for an elastic spring shows a straight line through the origin up to the limit of proportionality. The slope of this line equals the spring constant k.
弹性弹簧的力–伸长量图在比例极限内是一条过原点的直线，该直线的斜率即为弹簧常数 k。
Beyond the limit of proportionality, the graph begins to curve; this indicates non-linear behaviour. If the force is removed before reaching the elastic limit, the spring returns to its original length. Once the elastic limit is exceeded, permanent (plastic) deformation occurs and the spring will not return to its original shape.
Many exam questions ask you to interpret loading and unloading curves – if the unloading line does not retrace the loading line and there is residual extension, plastic deformation has taken place.
许多考题要求解释加载与卸载曲线——若卸载路径与加载路径不一致且出现剩余伸长，说明发生了塑性形变。
4. Elastic Limit and Plastic Deformation | 弹性极限与塑性变形
Elastic deformation is reversible: the material returns to its original dimensions when the load is removed. Plastic deformation is permanent: the material does not spring back and has undergone a structural change at the atomic level.
The elastic limit is the maximum stress or force that a material can experience and still return to its original shape. Beyond this point, some atoms slip past each other and a new, permanent shape is formed.
弹性极限是材料在卸载后仍能恢复原状的最大应力或力。超过此点后，部分原子发生滑移，形成新的永久形状。
5. Elastic Potential Energy | 弹性势能
When a spring is stretched or compressed, work is done and energy is stored as elastic potential energy. For a spring obeying Hooke’s law, the energy stored is equal to the area under the force–extension graph, which is a triangle.
This formula only applies when the spring has not been stretched beyond its elastic limit. Always use consistent units (joules, newtons, metres).
该公式仅适用于弹簧未超出弹性极限的情况。计算时务必统一单位（焦耳、牛顿、米）。
6. Stress and Strain – Basic Concepts | 应力与应变基础
Stress σ is defined as the force applied per unit cross‑sectional area: σ = F / A, measured in pascals (Pa). Strain ε is the extension per unit original length: ε = e / L₀ and has no units.
应力 σ 定义为单位截面积上施加的力：σ = F / A，单位是帕斯卡（Pa）；应变 ε 是单位原长的伸长量：ε = e / L₀，无单位。
σ = F ÷ A ε = ΔL ÷ L₀
Using stress and strain instead of force and extension makes material comparisons fair, because they account for the dimensions of the sample. A thick metal bar and a thin wire of the same material will have the same stress–strain curve.
7. Stress–Strain Curves and Material Stiffness | 应力–应变曲线与材料刚度
A stress–strain graph shows how a material responds to loading. The initial straight-line section represents elastic behaviour; the gradient of this part is the Young modulus E = σ / ε, which measures stiffness.
应力–应变曲线展示了材料在受载时的响应。起始的直线段对应弹性行为，该段的斜率即为杨氏模量 E = σ / ε，表征材料的刚度。
Materials with a high Young modulus (e.g. steel) are very stiff and exhibit small strains for large stresses. Materials with a low Young modulus (e.g. rubber) stretch easily.
杨氏模量高的材料（如钢）刚度大，在较大应力下应变仍然很小；杨氏模量低的材料（如橡胶）很容易伸长。
After the linear region, many ductile metals show a yield point, followed by a large plastic region and necking before fracture. Brittle materials break suddenly with little plastic deformation.
8. Key Material Properties: Toughness, Brittleness and Malleability | 关键材料性质：韧性、脆性与延展性
Tough materials can absorb a lot of energy before fracturing, up to a large strain. They have a large area under the stress–strain curve. Brittle materials break at small strains and absorb little energy.
Malleable materials can be hammered or rolled into thin sheets without cracking (e.g. gold, copper). Ductile materials can be drawn into wires (e.g. copper). Both are linked to large plastic deformation.
Resistance to elastic deformation; high Young modulus
Steel
Toughness
Ability to absorb energy up to fracture (large area under σ–ε curve)
Low‑carbon steel
Brittleness
Fractures with little or no plastic deformation
Glass, ceramic
Malleability
Can be shaped by compressive forces (e.g. hammering)
Gold
Ductility
Can be drawn into a wire under tension
Copper
9. Practical – Investigating Hooke’s Law | 实验——验证胡克定律
A classic required practical involves hanging masses on a spring, measuring the resulting extension, and plotting a force–extension graph. Key steps: record the initial length of the spring without load, add known weights, measure the new length each time, and calculate extension = new length – original length.
A pointer and a metre rule with a set‑square help reduce parallax errors. Repeat measurements and calculate a mean for reliability. The spring constant k can be determined from the gradient of the best‑fit straight line.
使用指针、米尺和三角板有助于减小视差；重复测量并取平均值以提高可靠性。弹簧常数 k 可通过最佳拟合直线的斜率求得。
10. Common Exam Mistakes and Tips | 常见错误与应试技巧
Confusing limit of proportionality with elastic limit: the limit of proportionality is the point where force and extension stop being proportional (graph curves), while the elastic limit is the point beyond which permanent deformation occurs. They are often very close but not identical.
Forgetting to convert extension to metres when calculating energy or spring constant. Always work in SI units unless specified otherwise.
计算弹性势能或弹簧常数时忘记将伸长量单位转换为米。除非题目明确要求，均应采用国际单位制。
When describing practical results, always relate observations to the behaviour of atoms or layers of atoms sliding past each other in plastic deformation.
描述实验结果时，要将现象与原子的行为联系起来——塑性形变中原子层发生了滑移。
In stress–strain questions, remember that the Young modulus is the gradient of the initial straight‑line region and is only valid for the elastic range.
涉及应力–应变的问题中，记住杨氏模量是初始直线段的斜率，且仅适用于弹性范围。
Published by TutorHao | Physics Revision Series | aleveler.com
📚 Key Formula Derivations in IAL Physics Unit 2: Insights from the Jan 2021 Examiner’s Report | IAL物理Unit 2关键公式推导：2021年1月考官报告解析
The January 2021 examiner’s report for International A-Level Physics Unit 2 highlighted that many candidates struggled to secure full marks on questions requiring step-by-step formula derivation or a deep understanding of underlying assumptions. Common pitfalls included poor handling of small-angle approximations, unit inconsistencies, and confusion between standing wave harmonic patterns for different boundary conditions. This article revisits the essential derivations, clarifies the physics behind them, and draws directly on the examiners’ observations to help you avoid the most frequent mistakes.
The core of the derivation lies in relating the path difference between light from the two slits to the geometry on the screen. For a point P at a vertical distance y from the central maximum, the path difference Δ = S₂P – S₁P ≈ a sinθ, where a is the slit separation. Using the small-angle approximation sinθ ≈ tanθ = y / D (with D the slit-to-screen distance), we obtain Δ ≈ a y / D.
推导的核心在于将两缝光到达屏幕某点的光程差与几何关系联系起来。对于距离中央极大竖直距离为 y 的点 P，光程差 Δ = S₂P – S₁P ≈ a sinθ，其中 a 为缝间距。利用小角度近似 sinθ ≈ tanθ = y / D（D 为缝屏间距），得到 Δ ≈ a y / D。
Constructive interference occurs when Δ = nλ, so a y / D = nλ. Hence the position of the n-th bright fringe is yₙ = nλD / a. The fringe separation (distance between adjacent bright fringes) is then Δy = yₙ₊₁ – yₙ = λD / a. The examiner’s report noted that many candidates lost marks by forgetting to state the small-angle approximation explicitly or by mixing up y (distance from centre) and Δy (fringe width).
相长干涉条件为 Δ = nλ，故 a y / D = nλ。因此第 n 级亮纹的位置为 yₙ = nλD / a。相邻亮纹间距则为 Δy = yₙ₊₁ – yₙ = λD / a。考官报告指出，很多考生因为未明确写出小角度近似，或混淆了 y（距中心的距离）与 Δy（条纹宽度）而丢分。
Δy = λD / a
2. Diffraction Grating Equation | 衍射光栅方程推导
For a diffraction grating with line spacing d, the path difference between waves from adjacent slits is d sinθ. Constructive interference (principal maxima) arises when this path difference is an integer multiple of the wavelength: d sinθ = nλ, where n = 0, ±1, ±2, … This relation governs the angular positions of bright fringes.
对于线间距为 d 的衍射光栅，相邻缝发出的波之间的光程差为 d sinθ。当该光程差等于波长的整数倍时，出现相长干涉（主极大）：d sinθ = nλ，其中 n = 0, ±1, ±2, … 该关系决定了亮纹的角位置。
The examiner’s report underlined that students often treated the grating as a double-slit, overlooking that for a grating maxima are extremely sharp and that the small-angle approximation sinθ ≈ θ (in radians) could be used only when θ is small. Also, many failed to convert the line spacing from lines per mm to metres, leading to gross errors. Always express d in metres (d = 1/N, where N is lines/m).
3. Standing Waves on a Stretched String | 张紧弦上的驻波频率推导
A stretched string fixed at both ends supports standing waves when its length L equals an integer number of half-wavelengths: L = n(λ/2), with n = 1, 2, 3, … Hence λ = 2L/n. Using the wave speed equation v = fλ, we obtain the resonant frequencies f = v/λ = n v / (2L).
两端固定的张紧弦，当其长度 L 等于半波长的整数倍时，可形成驻波：L = n(λ/2)，n = 1, 2, 3, … 故 λ = 2L/n。利用波速公式 v = fλ，得到共振频率 f = v/λ = n v / (2L)。
The fundamental frequency (first harmonic) is f₁ = v/(2L). The n-th harmonic is fₙ = n f₁. The examiner’s report revealed that many candidates misidentified the number of antinodes shown in a diagram, counting incorrectly, or they attempted to use the pipe formula for a string by mistake. Remember: string fixed at both ends → all harmonics (n = 1,2,3…) are possible.
基频（一次谐波）为 f₁ = v/(2L)。第 n 次谐波为 fₙ = n f₁。考官报告显示，许多考生错误识别了图中所显示的波腹数量，计数有误，或者错误地将管中的公式用于弦。请记住：两端固定的弦 → 所有谐波（n = 1,2,3…）都可能存在。
fₙ = n v / (2L) where n = 1, 2, 3, …
4. Standing Waves in Pipes | 管中的驻波频率推导
For a pipe open at both ends, both ends are antinodes, leading to the same harmonic sequence as a string: L = n(λ/2), so fₙ = n v/(2L), n = 1,2,3… For a pipe closed at one end, the closed end is a node and the open end an antinode. The length L corresponds to an odd number of quarter-wavelengths: L = (2n-1)λ/4, so λ = 4L/(2n-1) and fₙ = (2n-1) v/(4L). Only odd harmonics are present.
对于两端开口的管，两端均为波腹，其谐波序列与弦相同：L = n(λ/2)，故 fₙ = n v/(2L)，n = 1,2,3… 对于一端封闭的管，闭端为波节，开口端为波腹。长度 L 对应于奇数倍的四分之一波长：L = (2n-1)λ/4，因此 λ = 4L/(2n-1)，fₙ = (2n-1) v/(4L)。只有奇次谐波存在。
A common mistake flagged in the examiner’s report was confusing the boundary conditions for open and closed pipes, especially when a diagram showed a standing wave pattern. Students often applied the string formula to a closed pipe, ignoring that n must be odd. Another pitfall was using a value of v that was not adjusted for the correct medium (e.g., using the speed of sound in air but forgetting the temperature effect when given). Always use the v specified or recall v ≈ 340 m s⁻¹ for sound in air at room temperature.
考官报告指出的一个常见错误是混淆开管与闭管的边界条件，尤其是当题目给出驻波示意图时。学生常将弦公式套用到闭管上，忽略了 n 必须为奇数。另一个易错点是使用了未根据介质调整的波速 v（例如，使用空气中的声速但忘记题目给出的温度效应）。始终使用题目指定的 v，或记住室温下空气中声速约为 340 m s⁻¹。
Open pipe: fₙ = n v/(2L) ; Closed pipe: fₙ = (2n-1) v/(4L)
5. Snell’s Law from Huygens’ Principle | 由惠更斯原理推导斯涅尔定律
Huygens’ principle states that every point on a wavefront acts as a source of secondary wavelets. When a plane wavefront meets a boundary at an angle, the wavelets in the second medium travel at a different speed. Consider a wavefront AB incident on an interface. In time t, the wavelet from A travels a distance v₂t in medium 2, while the wavelet from B travels v₁t in medium 1. From the geometry, sinθ₁ = (v₁t) / AB and sinθ₂ = (v₂t) / AB. Dividing gives sinθ₁ / sinθ₂ = v₁ / v₂.
惠更斯原理指出，波前上的每一点都可视为子波的波源。当平面波前以一定角度遇到界面时，第二种介质中的子波以不同速度传播。考虑入射到界面的波前 AB。在时间 t 内，来自 A 的子波在介质 2 中传播距离 v₂t，而来自 B 的子波在介质 1 中传播 v₁t。由几何关系，sinθ₁ = (v₁t) / AB，sinθ₂ = (v₂t) / AB。两式相除得 sinθ₁ / sinθ₂ = v₁ / v₂。
Since the refractive index n = c / v, where c is the speed of light in vacuum, we can write v₁/v₂ = n₂/n₁. Substituting this gives n₁ sinθ₁ = n₂ sinθ₂, which is Snell’s law. The examiner’s report observed that when asked to outline this derivation, candidates often omitted the crucial step linking the ratio of speeds to the ratio of refractive indices, or they confused which index belonged to which medium.
由于折射率 n = c / v（c 为真空中光速），可写为 v₁/v₂ = n₂/n₁。代入后得到 n₁ sinθ₁ = n₂ sinθ₂，即斯涅尔定律。考官报告注意到，当要求概述此推导时，考生常遗漏将速度比与折射率比联系起来的关键步骤，或混淆了哪个折射率对应哪个介质。
n₁ sinθ₁ = n₂ sinθ₂
6. Resistivity and Resistance | 电阻率与电阻公式推导
The resistance R of a uniform wire is directly proportional to its length L and inversely proportional to its cross-sectional area A: R ∝ L / A. Introducing the constant of proportionality ρ (resistivity) gives R = ρL / A. This relation is fundamental to understanding how material property, geometry and temperature affect resistance.
均匀导线的电阻 R 与其长度 L 成正比，与其横截面积 A 成反比：R ∝ L / A。引入比例常数 ρ（电阻率），得到 R = ρL / A。这一关系对于理解材料属性、几何形状和温度如何影响电阻至关重要。
In the Jan 2021 exam, many students incorrectly calculated the cross-sectional area from a diameter, forgetting to halve the diameter to get the radius or using the diameter directly in A = πr². The examiner’s report stressed that area must be in m². Additionally, when rearranging the equation to find resistivity, candidates sometimes confused units and gave ρ in Ω cm rather than Ω m. Always keep units consistent: L in metres, A in m², R in ohms gives ρ in Ω m.
在2021年1月的考试中，许多学生由直径计算截面积时出错，忘记将直径除以二得到半径，或在 A = πr² 中直接使用直径。考官报告强调面积必须以 m² 为单位。此外，在变形公式求电阻率时，考生有时混淆单位，给出的 ρ 单位为 Ω cm 而非 Ω m。请始终保持单位一致：L 以米计，A 以 m² 计，R 以欧姆计，得到 ρ 的单位为 Ω m。
R = ρL / A
7. Series Resistors and the Potential Divider | 串联电阻与分压器公式推导
For resistors in series, the same current I flows through each. Ohm’s law gives V₁ = I R₁, V₂ = I R₂, … and the total p.d. V = I(R₁ + R₂ + …). Hence the total resistance Rtotal = R₁ + R₂ + … . More importantly, the ratio of voltages is V₁ / V₂ = R₁ / R₂. This is the foundation of the potential divider rule.
对于串联电阻，流过每个电阻的电流 I 相同。由欧姆定律得 V₁ = I R₁，V₂ = I R₂，… 总电压 V = I(R₁ + R₂ + …)。因此总电阻 Rtotal = R₁ + R₂ + … 。更重要的是，电压之比为 V₁ / V₂ = R₁ / R₂。这是分压器规则的基础。
The potential divider equation for two resistors R₁ and R₂ in series across a supply voltage Vin is Vout = Vin × R₂/(R₁ + R₂), where Vout is measured across R₂. A frequent error, noted by examiners, was placing R₁ in the numerator, leading to the wrong output voltage. Another mistake was assuming the current through the divider remained constant when a load was added; actually the effective resistance changes. Always check whether the circuit is unloaded or loaded.
For a real source of emf ε, the terminal p.d. V is less than ε when a current I is drawn, due to the internal resistance r. Energy conservation gives ε = I(R + r), where R is the external load. Rearranging: V = ε – I r. This linear equation shows that plotting V against I yields a straight line with gradient -r and y-intercept ε.
对于具有电动势 ε 的实际电源，当有电流 I 流出时，由于内阻 r 的存在，路端电压 V 会小于 ε。由能量守恒得 ε = I(R + r)，其中 R 为外接负载。变形得：V = ε – I r。这一线性关系表明，以 V 对 I 作图得到一条直线，斜率为 -r，y 轴截距为 ε。
Examiners reported that many candidates struggled to extract r correctly from a V-I graph, often ignoring the negative sign and simply quoting the gradient as r. They also confused the intercept, particularly when the graph was plotted as I-V (current on y-axis) instead of V-I. In such cases, the gradient becomes -1/r and the intercept ε/r. Be very careful with graph axes. Another classic error is using the nominal emf of a cell as exactly the y-intercept when the line does not clearly reach I = 0; interpolation may be needed.
考官报告指出，许多考生难以从 V-I 图中正确提取 r，常常忽略负号而直接将斜率当作 r。他们也会混淆截距，特别是当图像以 I-V（电流在 y 轴）的形式呈现时。此时斜率变为 -1/r，截距为 ε/r。务必留意坐标轴变量。另一个经典错误是当直线并未清晰延伸到 I = 0 时，仍直接把标称电动势当作 y 截距；可能需要进行外推。
V = ε – I r
9. Photoelectric Effect Equation and Graph Analysis | 光电效应方程与图像分析
The energy of a photon is E = hf, where h is Planck’s constant. When a photon strikes a metal surface, the maximum kinetic energy of the emitted photoelectron is Ek,max = hf – φ, where φ is the work function of the metal. This can also be written using the stopping potential Vs: e Vs = hf – φ, so Vs = (h/e) f – (φ/e).
光子能量为 E = hf，其中 h 为普朗克常数。当光子撞击金属表面时，逸出光电子的最大动能为 Ek,max = hf – φ，φ 为金属的逸出功。也可用遏止电势 Vs 表达：e Vs = hf – φ，即 Vs = (h/e) f – (φ/e)。
The graph of Vs against f is a straight line with gradient h/e and x-intercept equal to the threshold frequency f₀ = φ/h. The Jan 2021 report noted that many candidates could not link the gradient to Planck’s constant correctly, especially when the graph’s units were not standard (e.g., frequency in 10¹⁴ Hz). Always convert to Hz before calculating the gradient and use e = 1.60 × 10⁻¹⁹ C. Another pitfall was confusing Ek,max with the energy of the photon itself; make it clear that not all photon energy goes into kinetic energy — some is used to overcome the work function.
Vs 对 f 作图是一条直线，斜率为 h/e，x 截距为阈频率 f₀ = φ/h。2021年1月的报告指出，许多考生无法正确将斜率与普朗克常数关联，尤其当图像的单位非标准时（如频率以 10¹⁴ Hz 为单位）。在计算斜率前务必转换为 Hz，并使用 e = 1.60 × 10⁻¹⁹ C。另一个易错点是混淆 Ek,max 与光子本身的能量；须说明并非全部光子能量都转化为动能——一部分用于克服逸出功。
e Vs = hf – φ
10. Common Pitfalls and Unit Conversion Mastery | 常见失误与单位换算精通
Across all derivations, the examiner’s report repeatedly highlighted the importance of consistent SI units. For instance, using millimetres for slit separation or centimetres for screen distance in the Young’s slits formula will yield a fringe spacing off by factors of 10³ or 10². Always convert lengths to metres and frequencies to hertz before substituting into equations. Keep powers of ten clear and use standard form to avoid arithmetic slips.
Other generic mistakes included incomplete derivations—jumping straight to the final formula without showing the condition (e.g., constructive interference: Δ = nλ). Students also lost marks for not clearly stating assumptions such as the small-angle approximation or the medium being uniform. When a question asks “derive” or “show that”, each logical step must be presented. Simply writing the known result is insufficient.
The examiners also pointed out that candidates often confused the symbols for slit separation (a) and grating spacing (d), or wavelength (λ) and distance (D). Using a clear, labelled diagram in your answer can help demonstrate your understanding and avoid symbol mix-ups. Remember: a well-drawn sketch with annotations can earn marks even if the algebra goes slightly awry.
Welcome to this intensive revision guide focused on calculation questions for the CCEA IGCSE Physics specification. Mastering these numerical problems is essential for achieving a high grade, as they test not only your knowledge of formulae but also your ability to convert units, rearrange equations, and present final answers with appropriate significant figures. This article covers the most frequently examined calculation topics, providing step‑by‑step approaches and worked examples to build your confidence. Follow along, practise with the embedded questions, and remember that consistent application of the correct method will lead to success in the examination.
Speed, distance and time are linked by the equation v = s / t, where v is speed, s is distance and t is time. When an object moves at constant acceleration, we use a = (v – u) / t, with u being initial velocity, v final velocity and a acceleration. Always ensure that units are consistent: distance in metres (m), time in seconds (s), speed in m/s, and acceleration in m/s². In some exam questions you will need to convert km to m or minutes to seconds before substituting values.
速度、距离和时间由公式 v = s / t 联系，其中 v 是速度，s 是距离，t 是时间。当物体匀加速运动时，我们使用 a = (v – u) / t，u 是初速度，v 是末速度，a 是加速度。务必保持单位一致：距离用米 (m)，时间用秒 (s)，速度用 m/s，加速度用 m/s²。在一些考题中，你需要先将千米换算为米，或将分钟换算为秒，然后再代入数值。
Worked example: A sprinter accelerates from rest to 10 m/s in 4 seconds. Calculate the acceleration and the distance covered during this time. For acceleration: a = (10 – 0) / 4 = 2.5 m/s². To find distance, use the equation s = (u + v) × t / 2: s = (0 + 10) × 4 / 2 = 20 m. Notice how we choose the appropriate equation for the data given.
Newton’s second law states that the resultant force acting on an object is equal to the product of its mass and acceleration: F = m × a. Force is measured in newtons (N), mass in kilograms (kg), and acceleration in m/s². The weight of an object is a specific force caused by gravity: W = m × g, where g is the gravitational field strength (on Earth, approximately 9.8 N/kg, often taken as 10 N/kg in CCEA questions). Free‑body diagrams help to resolve forces and find the resultant before applying the formula.
牛顿第二定律指出，作用在物体上的合力等于其质量与加速度的乘积：F = m × a。力的单位是牛顿 (N)，质量是千克 (kg)，加速度是 m/s²。物体的重量是由重力引起的一种特殊的力：W = m × g，其中 g 是引力场强度（在地球上约为 9.8 N/kg，CCEA 考试中常取 10 N/kg）。受力示意图有助于分解力并先求出合力，再应用公式。
Often you will combine the two equations. For example, a 2 kg mass hangs on a rope. The tension T in the rope must balance the weight: T – W = m × a. If the mass is accelerating upward at 1.5 m/s², then T – (2 × 10) = 2 × 1.5, so T = 20 + 3 = 23 N. Always define the positive direction clearly in your working.
常常需要将两个方程结合。例如，一个 2 kg 的质量悬挂在绳子上。绳子张力 T 必须与重量平衡：T – W = m × a。如果该质量以 1.5 m/s² 向上加速，那么 T – (2 × 10) = 2 × 1.5，所以 T = 20 + 3 = 23 N。解题时始终要清晰地规定正方向。
3. Work, Energy and Power | 功、能与功率
Work done is defined as the force multiplied by the distance moved in the direction of the force: W = F × d, with work in joules (J). Energy can exist in different forms, and two important mechanical forms are kinetic energy, Eₖ = ½ × m × v², and gravitational potential energy, Eₚ = m × g × h. Power is the rate of doing work or transferring energy: P = W / t, measured in watts (W). When a question asks for the power of a machine, you often need to calculate the work done first and then divide by the time taken.
功定义为力乘以在力的方向上移动的距离：W = F × d，功的单位是焦耳 (J)。能量可以以不同形式存在，两种重要的机械能是动能 Eₖ = ½ × m × v² 和重力势能 Eₚ = m × g × h。功率是做功或转换能量的速率：P = W / t，单位是瓦特 (W)。当题目要求计算机器的功率时，你通常需要先算出所做的功，再除以所用的时间。
In a typical problem, a 0.5 kg ball is dropped from a height of 4 m. Calculate its speed just before hitting the ground, assuming no air resistance. Using conservation of energy: loss in Eₚ = gain in Eₖ. m × g × h = ½ × m × v², so v² = 2 × g × h = 2 × 10 × 4 = 80, therefore v = √80 ≈ 8.94 m/s. Notice that the mass cancels, so any object dropped from the same height reaches the same speed.
在典型问题中，一个 0.5 kg 的小球从 4 m 高处落下。假设无空气阻力，计算其刚好撞击地面前的速度。利用能量守恒：减少的重力势能 = 增加的动能。m × g × h = ½ × m × v²，所以 v² = 2 × g × h = 2 × 10 × 4 = 80，因此 v = √80 ≈ 8.94 m/s。注意质量被消去，因此任何物体从相同高度落下都会达到相同的速度。
4. Density and Pressure | 密度与压强
Density is mass per unit volume: ρ = m / V. The SI unit is kg/m³, but you may also encounter g/cm³. Remember that 1 g/cm³ = 1000 kg/m³. For regular solids, volume can be found from geometry; for irregular objects, use the displacement method. Pressure is force per unit area: P = F / A, with pascals (Pa) equal to N/m². In fluids, pressure increases with depth and can be calculated as P = ρ × g × h, where h is the depth below the surface.
密度是单位体积的质量：ρ = m / V。国际单位是 kg/m³，但你也会遇到 g/cm³。记住 1 g/cm³ = 1000 kg/m³。对于规则固体，体积可以通过几何方法求得；对于不规则物体，使用排水法。压强是单位面积上的力：P = F / A，单位帕斯卡 (Pa) 等于 N/m²。在流体中，压强随深度增加，可以用 P = ρ × g × h 计算，其中 h 是表面以下的深度。
A common examination task involves a rectangular block of dimensions 0.2 m × 0.1 m × 0.05 m and mass 2 kg resting on a table. Calculate the maximum and minimum pressure it can exert. Area of the smallest face = 0.1 × 0.05 = 0.005 m², so maximum pressure = F / A = (2 × 10) / 0.005 = 4000 Pa. For minimum pressure, use the largest face area = 0.2 × 0.1 = 0.02 m², giving P = 20 / 0.02 = 1000 Pa.
一种常见的考试题涉及一个尺寸为 0.2 m × 0.1 m × 0.05 m、质量为 2 kg 的长方体放在桌子上。计算它能产生的最大和最小压强。最小面的面积 = 0.1 × 0.05 = 0.005 m²，因此最大压强 = F / A = (2 × 10) / 0.005 = 4000 Pa。对于最小压强，使用最大面面积 = 0.2 × 0.1 = 0.02 m²，得出 P = 20 / 0.02 = 1000 Pa。
5. Current, Voltage and Resistance in Circuits | 电路中的电流、电压与电阻
Ohm’s law is the foundation of circuit calculations: V = I × R, where V is potential difference in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω). For components connected in series, the current is the same everywhere, the total voltage is shared, and the total resistance is the sum of individual resistances: Rtotal = R₁ + R₂ + …. In parallel circuits, the voltage across each branch is the same, the total current is the sum of branch currents, and the total resistance is found from 1 / Rtotal = 1 / R₁ + 1 / R₂ + ….
Consider a 6 Ω and a 3 Ω resistor connected in parallel, and this combination is connected in series with a 4 Ω resistor across a 12 V battery. First, find the parallel resistance: 1 / Rparallel = 1/6 + 1/3 = 1/2, so Rparallel = 2 Ω. Total circuit resistance = 2 + 4 = 6 Ω. Then total current from battery: Itotal = Vtotal / Rtotal = 12 / 6 = 2 A. The voltage across the parallel block = Itotal × Rparallel = 2 × 2 = 4 V. Therefore the current through the 6 Ω resistor = 4 V / 6 Ω ≈ 0.67 A. Breaking the problem into steps prevents confusion.
Electrical power can be expressed in three useful forms: P = I × V, P = I² × R, and P = V² / R. The appropriate version depends on the given quantities. Energy transferred in an electrical device is E = P × t, where t is time in seconds, giving energy in joules. In domestic contexts, kilowatt‑hours (kW h) may be used: energy (kW h) = power (kW) × time (h). One kW h equals 3.6 × 10⁶ J. Exam questions often require you to combine these with cost calculations.
电功率可以用三种有用的形式表达：P = I × V、P = I² × R 和 P = V² / R。选择哪一种取决于已知量。电器中转移的电能为 E = P × t，其中 t 是时间（秒），能量单位为焦耳。在家庭用电中，可以用千瓦时 (kW h)：电能 (kW h) = 功率 (kW) × 时间 (h)。1 kW h 等于 3.6 × 10⁶ J。考试题常要求你结合这些公式计算电费。
A worked example: a 230 V heater has a resistance of 50 Ω. Determine the power rating and the cost of running it for 3 hours if each kW h costs 15 pence. Using P = V² / R: P = (230)² / 50 = 52900 / 50 = 1058 W = 1.058 kW. Energy used in 3 h = 1.058 kW × 3 h = 3.174 kW h. Cost = 3.174 × 15 p = 47.61 p. Always check whether to give answers in pence or pounds.
7. Wave Speed, Frequency and Wavelength | 波速、频率与波长
All waves obey the wave equation: v = f × λ, where v is wave speed (m/s), f is frequency (Hz), and λ is wavelength (m). This relationship is fundamental for both transverse and longitudinal waves, including sound, light and water waves. In ripple tank experiments or electromagnetic spectrum questions, you may be given any two of the three quantities and asked to find the third. Do not forget that period T = 1 / f, and the wave equation can also be written as v = λ / T.
所有波都遵循波动方程：v = f × λ，其中 v 是波速 (m/s)，f 是频率 (Hz)，λ 是波长 (m)。这一关系对横波和纵波都基本适用，包括声波、光波和水波。在波纹槽实验或电磁波谱题目中，通常会给出三个量中的任意两个，要求你求出第三个。不要忘记周期 T = 1 / f，波动方程也可写作 v = λ / T。
A radio station transmits waves with frequency 100 MHz and wavelength 3 m. The speed v = (100 × 10⁶ Hz) × 3 m = 3 × 10⁸ m/s, which confirms that electromagnetic waves travel at the speed of light in a vacuum. If a water wave has a speed of 25 cm/s and a wavelength of 5 cm, its frequency is f = v / λ = 25 / 5 = 5 Hz. Always express speed and wavelength in the same length unit before dividing.
某广播电台发射频率为 100 MHz、波长为 3 m 的波。波速 v = (100 × 10⁶ Hz) × 3 m = 3 × 10⁸ m/s，这证实了电磁波在真空中以光速传播。如果一个水波波速为 25 cm/s、波长为 5 cm，其频率 f = v / λ = 25 / 5 = 5 Hz。在做除法之前，始终将波速和波长表示为相同的长度单位。
8. Specific Heat Capacity | 比热容
When an object is heated, the temperature rise depends on its mass, the material’s specific heat capacity and the energy supplied. The equation is Q = m × c × Δθ, where Q is heat energy (J), m is mass (kg), c is specific heat capacity (J/(kg °C)), and Δθ is temperature change (°C or K). Water has a high specific heat capacity of about 4200 J/(kg °C), which is a common value in numerical problems. Rearranging to find any unknown is a key skill.
Example: An electric heater supplies 10 000 J of energy to a 0.5 kg aluminium block (c = 900 J/(kg °C)). Find the temperature rise. Rearranging: Δθ = Q / (m × c) = 10000 / (0.5 × 900) = 10000 / 450 ≈ 22.2 °C. Some questions combine this with electrical power: if the heater operates at 50 W, the time taken can be found from Q = P × t. Then t = 10000 / 50 = 200 s. Always show the substitution step clearly.
例题：一个电加热器给一个 0.5 kg 的铝块（c = 900 J/(kg °C)）提供 10 000 J 的能量。求温度升高多少。变形公式：Δθ = Q / (m × c) = 10000 / (0.5 × 900) = 10000 / 450 ≈ 22.2 °C。有些题目会结合电功率来考：如果加热器功率为 50 W，那么所需时间可由 Q = P × t 求出。则 t = 10000 / 50 = 200 s。始终清晰地展示代入步骤。
9. Radioactive Decay and Half‑life | 放射性衰变与半衰期
The half‑life of a radioactive isotope is the time taken for half the nuclei in a sample to decay, or for the count rate to fall to half its original value. You can calculate the remaining mass or activity after a given number of half‑lives using: remaining amount = original amount × (½)ⁿ, where n is the number of half‑lives elapsed (n = total time / half‑life). Alternatively, sketch a decay curve and read values from the graph.
A sample of iodine‑131 has a half‑life of 8 days and an initial mass of 40 mg. Calculate the mass remaining after 24 days. Number of half‑lives, n = 24 / 8 = 3. Remaining mass = 40 mg × (½)³ = 40 × 1/8 = 5 mg. If a question asks for the time to decay to a certain fraction, rearrange: 40 × (½)ⁿ = 2.5 → (½)ⁿ = 1/16 → n = 4, so time = 4 × 8 = 32 days. This method is quick and reliable.
Efficiency measures how well a device converts input energy into useful output energy. It can be expressed as a decimal or a percentage: Efficiency = (useful output energy or power / total input energy or power) × 100%. No real machine is 100% efficient due to energy losses, often as heat due to friction or electrical resistance. When calculating efficiency, you must identify the ‘useful’ component correctly from the description.
For instance, an electric motor lifts a 20 N weight through a height of 3 m, doing useful work = 20 × 3 = 60 J. The motor draws 100 J of electrical energy. Efficiency = (60 / 100) × 100% = 60%. The remaining 40 J is wasted as heat and sound. In a power station context, overall efficiency is the product of individual efficiencies. Practising these multi‑step calculations is vital for CCEA exams, where such questions often combine different topics.
Nuclear physics is a cornerstone of both the IB Diploma and OCR A-Level Physics syllabi, offering a deep dive into the structure of the atomic nucleus, the forces that hold it together, and the energetic processes that can tear it apart. From the properties of alpha, beta and gamma radiation to the principles of nuclear fission and fusion, this topic blends theoretical insight with real-world applications such as power generation, medical imaging and carbon dating. Mastering these concepts demands a clear grasp of mass–energy equivalence, binding energy per nucleon, radioactive decay laws and the conservation rules governing nuclear reactions. This revision guide walks you through every key idea, pairing clear explanations in English with their precise Chinese counterparts, and is designed to help you tackle multiple‑choice, structured and data‑analysis questions with confidence.
The nucleus consists of protons and neutrons, collectively called nucleons. The number of protons Z is the atomic number, which defines the chemical element. The total number of nucleons A = Z + N (where N is the neutron number) is the mass number. A nuclide is denoted as ᴬ ZX, for example ²³⁸ ₉₂U represents uranium‑238 with 92 protons and 146 neutrons. Isotopes are nuclei with the same Z but different N, meaning they have identical chemical properties but differing nuclear stabilities. The radius of a nucleus is approximately R = r₀A^(1/3), with r₀ ≈ 1.2 × 10⁻¹⁵ m, indicating that nuclear volume is proportional to the mass number.
原子核由质子和中子（统称核子）组成。质子数 Z 即原子序数，决定了化学元素的种类。核子的总数 A = Z + N（N 为中子数）为质量数。核素用符号 ᴬ ZX 表示，例如 ²³⁸ ₉₂U 代表铀‑238，含有 92 个质子和 146 个中子。同位素是指 Z 相同但 N 不同的核素，它们化学性质相同，但核稳定性各异。原子核的半径近似为 R = r₀A^(1/3)，其中 r₀ ≈ 1.2 × 10⁻¹⁵ m，这表明核体积与质量数成正比。
2. Nuclear Forces and Binding Energy | 核力与结合能
The strong nuclear force binds nucleons together, acting over very short ranges (~10⁻¹⁵ m). It is attractive at nucleon separations of about 1–3 fm and repulsive at even shorter distances, preventing collapse. The mass of a nucleus is always less than the sum of the masses of its individual nucleons; this mass defect Δm corresponds to the binding energy via E = Δmc². The binding energy per nucleon is obtained by dividing the total binding energy by A. A graph of binding energy per nucleon against mass number peaks around iron‑56, meaning that energy can be released both by fusing light nuclei (fusion) and by splitting heavy nuclei (fission).
强核力将核子束缚在一起，其作用范围极短（约 10⁻¹⁵ m）。在核子间距约为 1–3 fm 时表现为吸引力，在更短距离上表现为排斥力，从而防止核坍缩。原子核的质量总是小于其各个核子质量之和；这一质量亏损 Δm 通过 E = Δmc² 与结合能对应。将总结合能除以 A 即可得到每核子结合能。每核子结合能随质量数变化的曲线在铁‑56 附近达到峰值，这意味着无论是轻核的融合（聚变）还是重核的分裂（裂变）都能释放能量。
3. Types of Radioactive Decay | 放射性衰变类型
Unstable nuclei spontaneously transform into more stable configurations through radioactive decay. Alpha (α) decay involves the emission of a helium‑4 nucleus (⁴₂He), reducing A by 4 and Z by 2. Beta‑minus (β⁻) decay occurs when a neutron converts into a proton, emitting an electron and an antineutrino: n → p + e⁻ + ν̄ₑ. This increases Z by 1 while A remains unchanged. Beta‑plus (β⁺) decay occurs when a proton converts into a neutron, emitting a positron and a neutrino: p → n + e⁺ + νₑ. Gamma (γ) emission is the release of high‑energy photons from an excited nucleus, leaving A and Z unchanged. Each decay mode is governed by conservation of charge, nucleon number, energy and momentum.
不稳定核素通过放射性衰变自发转变为更稳定的结构。α 衰变释放一个氦‑4 核（⁴₂He），使 A 减少 4、Z 减少 2。β⁻ 衰变中，一个中子转变为一个质子，释放一个电子和一个反中微子：n → p + e⁻ + ν̄ₑ，Z 增加 1，A 不变。β⁺ 衰变是一个质子转变为中子，释放一个正电子和一个中微子：p → n + e⁺ + νₑ。γ 辐射是激发态原子核释放高能光子，A 和 Z 均保持不变。所有衰变模式都必须遵守电荷、核子数、能量和动量守恒。
4. Decay Law and Half‑life | 衰变规律与半衰期
Radioactive decay is a random and spontaneous process. The number of undecayed nuclei N at time t obeys the exponential law N = N₀e^(−λt), where λ is the decay constant. The activity A = λN, measured in becquerels (Bq), where 1 Bq equals one decay per second. The half‑life T₁/₂ is the time taken for half the nuclei in a sample to decay, related to λ by T₁/₂ = ln 2 / λ. On an N–t graph, the half‑life is constant regardless of starting point. Practical determinations of half‑life often involve background subtraction and logarithmic plots to find λ from the gradient of ln A vs t.
放射性衰变是一种随机且自发的过程。未衰变核数 N 随时间 t 遵从指数规律 N = N₀e^(−λt)，其中 λ 是衰变常量。活度 A = λN，单位是贝克勒尔（Bq），1 Bq 表示每秒一次衰变。半衰期 T₁/₂ 是样品中一半核发生衰变所需的时间，与 λ 的关系为 T₁/₂ = ln 2 / λ。在 N–t 图上，无论从何处开始计时，半衰期都是恒定的。实验测定半衰期常需要扣除本底，并利用 ln A–t 图的斜率来求 λ。
5. Nuclear Reactions and Q‑value | 核反应与 Q 值
A nuclear reaction is written as a + X → Y + b, where a is the projectile and X the target nucleus. Reactions must conserve total charge, total nucleon number, energy and momentum. The reaction energy Q is the difference between the total rest masses of the reactants and products multiplied by c²: Q = (mᵢₙᵢₜᵢₐₗ − m_final)c². If Q is positive, the reaction is exothermic and releases kinetic energy; if negative, the reaction is endothermic and requires a threshold kinetic energy of the incident particle. Common examples include (α, p) and (n, γ) reactions.
核反应通常写作 a + X → Y + b，其中 a 是入射粒子，X 是靶核。反应必须遵守总电荷、总核子数、能量和动量守恒。反应能 Q 是反应物与生成物的总静止质量之差再乘以 c²：Q = (m_initial − m_final)c²。若 Q 为正，反应放热并释放动能；若 Q 为负，反应吸热，要求入射粒子具有阈动能。常见的核反应有 (α, p) 反应和 (n, γ) 反应等。
6. Mass–Energy Equivalence | 质能等价
Einstein’s relation E = mc² is the bedrock of nuclear energy calculations. It states that mass and energy are interchangeable: a system’s rest energy is its rest mass multiplied by the square of the speed of light in vacuum. In nuclear processes, the tiny mass defect is multiplied by a huge factor (9 × 10¹⁶ J/kg) to produce significant energy changes. The standard atomic mass unit u is defined as 1/12 the mass of a carbon‑12 atom, so 1 u = 1.661 × 10⁻²⁷ kg, and its energy equivalent is 931.5 MeV. This conversion factor is essential when computing binding energies and Q‑values from tabulated nuclear masses.
爱因斯坦关系式 E = mc² 是核能计算的基础。它表明质量与能量可以相互转化：一个系统的静能等于其静止质量乘以真空光速的平方。在核过程中，微小的质量亏损乘以巨大的因子（9 × 10¹⁶ J/kg）就能产生显著的能量变化。标准原子质量单位 u 定义为碳‑12 原子质量的 1/12，因此 1 u = 1.661 × 10⁻²⁷ kg，其能量当量为 931.5 MeV。当从核质量表计算结合能和 Q 值时，这一换算因子至关重要。
7. Nuclear Fission | 核裂变
Fission is the splitting of a heavy nucleus, such as uranium‑235, into two lighter fragment nuclei, accompanied by the release of several neutrons and a large amount of energy. A typical reaction is n + ²³⁵₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3n + energy. The released neutrons can induce further fissions, creating a chain reaction. The critical mass is the minimum mass of fissile material needed for a self‑sustaining chain reaction. Control rods (often containing boron or cadmium) absorb excess neutrons, while a moderator (such as water or graphite) slows down neutrons to thermal energies where they are more likely to cause fission in ²³⁵U. In an A‑level or IB exam, you may be asked to balance fission equations or explain the role of each component in a nuclear reactor.
Fusion is the joining of two light nuclei to form a heavier nucleus, again releasing energy because the binding energy per nucleon increases. The Sun’s energy comes from the proton–proton chain, which effectively converts four protons into a helium‑4 nucleus plus two positrons, two neutrinos and energy. For fusion to occur, nuclei must overcome the Coulomb repulsion between them, requiring extremely high temperatures (~10⁷–10⁸ K) to give them sufficient kinetic energy. This is called thermonuclear fusion. On Earth, magnetic confinement (tokamak) and inertial confinement are being explored as routes to controlled fusion power. In examination contexts, you should be able to calculate the energy released in simple fusion reactions, such as ²H + ³H → ⁴He + n, using the mass defect method.
Ionising radiation can be detected by devices that register the ionisation it produces. A Geiger–Müller (GM) tube connected to a counter records each ionising particle that enters the tube as a ‘count’. The cloud chamber and bubble chamber make particle tracks visible through condensation or boiling along the ionisation trail. Solid‑state detectors and scintillation counters are also used for precise energy measurements. When counting, the background count must be measured separately and subtracted from the total count. Detectors are often characterised by their efficiency and resolving time (dead time). In practical activities or exam data analysis, you may need to correct for background radiation or estimate the dead time of a GM tube.
10. Biological Effects and Radiation Dose | 生物效应与辐射剂量
Ionising radiation damages living tissue by breaking chemical bonds and creating reactive ions. The absorbed dose D is the energy absorbed per unit mass of tissue, measured in grays (Gy), where 1 Gy = 1 J/kg. To account for the different biological effects of different radiation types, the absorbed dose is multiplied by a radiation weighting factor w_R to give the equivalent dose H = w_R D, measured in sieverts (Sv). For example, α particles have a much higher w_R (≈20) than β or γ radiation (w_R = 1). Effective dose further accounts for the varying sensitivity of different organs. Understanding these quantities is crucial for evaluating the risks of medical procedures and for setting safety limits in nuclear industries.
11. Background Radiation and Half‑life Determinations | 背景辐射与半衰期测定
Background radiation originates from cosmic rays, terrestrial sources (e.g. radon gas), and even the building materials around us. Any measurement of radioactive sources must therefore include a background count, which is subtracted to obtain the corrected count rate. When a set of activity–time data is analysed for half‑life, the background should first be subtracted to ensure the exponential fit accurately reflects the decay constant of the source alone. Long or short half‑lives can be determined from the gradient of ln(corrected count rate) vs time. This technique is directly assessed in IB and OCR practical investigations, where students may use a GM tube and a source such as protactinium‑234 or a simulated radioactive decay.
Nuclear techniques pervade modern life. Radioactive tracers, using isotopes such as technetium‑99m (T₁/₂ = 6 hours), allow medical imaging of organs and blood flow. Carbon‑14 dating determines the age of archaeological specimens up to around 50,000 years by measuring the residual ¹⁴C activity. Industrial radiography uses gamma sources to inspect welds and metal castings for defects. In agriculture, radiation induces beneficial mutations in seeds. Nuclear power plants harness the controlled fission of uranium or plutonium to generate electricity without direct CO₂ emissions. Students should be able to discuss the benefits and risks, linking physical principles to societal and environmental contexts.
📚 A-Level OCR Science: Comparing Key Concepts Across Physics, Chemistry and Biology | A-Level OCR 科学：物理、化学与生物核心知识点对比
In OCR A-Level Sciences, students often encounter overlapping themes that span physics, chemistry and biology. Understanding how the same fundamental principle appears in different disciplinary contexts deepens comprehension and supports synoptic thinking. This article explores selected core ideas and contrasts their treatments across the physical and life sciences, following the OCR specification approach.
In physics, the first law of thermodynamics states that the change in internal energy equals heat added to the system minus work done by the system: ΔU = Q − W. In chemistry, this appears as ΔU = q + w, where sign conventions for work often differ. Biologists express energy transformations through ATP hydrolysis, which couples exergonic and endergonic reactions without explicitly calculating internal energy changes.
Physics: Focus on work done by expanding gases, efficiency of heat engines, and absolute temperature. Chemistry: Emphasises enthalpy changes ΔH, Hess’s law, and bond enthalpies. Biology: Describes energy in terms of activation energy, metabolic pathways, and ATP as the ‘energy currency’.
物理：关注气体膨胀做功、热机效率及绝对温度。化学：强调焓变 ΔH、盖斯定律及键焓。生物：用活化能、代谢途径及 ATP 作为“能量货币”来描述能量。
2. Structure of Matter: Atoms, Molecules and Cells | 物质结构：原子、分子与细胞
Physics explores atomic structure via the nuclear model, describing protons, neutrons, electrons and energy levels, with a focus on fundamental particles and quantum behaviour. Chemistry extends this into molecular bonding—ionic, covalent, metallic—and three-dimensional shapes using VSEPR theory. Biology builds upwards from macromolecules to organelles and cells, viewing structure-function relationships at the microscopic and ultrastructural levels.
Physics: Energy levels, absorption and emission spectra, electron diffraction.
Chemical: Hybridisation, molecular orbitals, polarity and isomerism.
Biology: Phospholipid bilayers, protein folding, organelle compartmentalisation.
物理：能级、吸收和发射光谱、电子衍射。
化学：杂化、分子轨道、极性与异构。
生物：磷脂双分子层、蛋白质折叠、细胞器区室化。
3. Forces and Interactions | 力与相互作用
Physicists categorise forces into gravitational, electromagnetic, strong and weak nuclear forces, and use Newton’s laws to predict motion. Chemists interpret intermolecular forces—hydrogen bonding, permanent dipole-dipole, London dispersion forces—to explain physical properties like boiling points. Biologists view interactions in terms of enzyme-substrate binding, receptor-ligand specificity, and ecological relationships such as predator-prey dynamics.
Physics provides the wave equation, superposition, interference and the electromagnetic spectrum. Chemistry applies electromagnetic radiation in spectroscopy: IR causes bond vibrations, UV-visible excites electrons, NMR uses radio waves to probe nuclear spin. Biology harnesses light for photosynthesis, vision, and uses microscopy that relies on wave properties such as resolution limits.
Physics examines rates primarily through radioactive decay, expressed by the exponential law N = N₀e⁻λᵗ and half-life. Chemistry studies reaction kinetics: rate equations, activation energy from Arrhenius plots, and catalysts. Biology analyses enzyme kinetics (Michaelis-Menten), population growth, and nerve impulse transmission rates, each often modelled with saturation curves or exponentials.
物理主要通过放射性衰变研究速率，以指数规律 N = N₀e⁻λᵗ 和半衰期表示。化学研究反应动力学：速率方程、通过阿伦尼乌斯图获得的活化能以及催化剂。生物分析酶动力学（米氏方程）、种群增长和神经冲动传导速率，通常用饱和曲线或指数曲线建模。
Physics: Activity, background radiation correction. Chemistry: Order of reaction, rate-determining step, heterogeneous vs homogeneous catalysis. Biology: Vₘₐₓ and Kₘ, effect of temperature and pH on enzymes, logistic growth model.
A physical equilibrium occurs when the net force and net torque on a body are zero; thermal equilibrium is when two objects reach the same temperature. Chemical equilibrium is dynamic, with forward and reverse rates equal, and Le Chatelier’s principle predicting shifts. Biological homeostasis maintains internal constancy—blood glucose, temperature, pH—through negative feedback loops, analogous to restorative forces.
Physics: Principle of moments, centre of mass, zeroth law of thermodynamics. Chemistry: Equilibrium constant Kc, effect of temperature, pressure, concentration. Biology: Insulin and glucagon, thermoregulation, osmoregulation, chemoreceptors.
In chemistry, redox reactions involve electron transfer between species, quantified by oxidation states and standard electrode potentials. In biology, the electron transport chain in mitochondria and chloroplasts transfers electrons through a series of protein complexes, generating a proton gradient for ATP synthesis. Physics describes electric current as a flow of charge, often electrons, through conductors, linking to potential difference and resistance.
8. Quantum Effects and Discrete Energy Levels | 量子效应与离散能级
Physics introduces quantisation through the photoelectric effect, line spectra and energy levels in atoms, where electrons exist in discrete shells. Chemistry uses quantum numbers to describe atomic orbitals (s, p, d, f) and explains bonding via orbital overlap and hybridisation. Biology encounters quantum effects in photosynthesis, where exciton transfer and the discrete absorption of photons are crucial for light-harvesting complexes.
Physics: hf = φ + KEₘₐₓ, energy level diagrams. Chemistry: Electron configuration, Hund’s rule, orbital shapes. Biology: Photosystems I and II, quantum yield, fluorescence resonance energy transfer.
物理：hf = φ + KEₘₐₓ，能级图。化学：电子排布、洪特规则、轨道形状。生物：光系统 I 和 II、量子产率、荧光共振能量转移。
9. Measurements and Uncertainties | 测量与不确定性
All OCR sciences require rigorous treatment of measurements. Physics details absolute, fractional and percentage uncertainties, combining them through propagation rules. Chemistry applies uncertainties to titrations, weighing and temperature readings, often emphasising systematic vs random errors. Biology uses uncertainty in ecological sampling, microscope calibration, and statistical tests like standard deviation and confidence limits.
Physics: Using a micrometer, repeated readings, error bars on graphs.
Chemistry: Burette and pipette uncertainties, endpoint detection.
Biology: Simpson’s index of diversity, standard error, chi-squared tests.
物理：使用千分尺、重复读数、图上的误差棒。
化学：滴定管和移液管的不确定性、终点检测。
生物：辛普森多样性指数、标准误差、卡方检验。
10. Models and Scientific Thinking | 模型与科学思维
Models simplify complex phenomena. Physics uses idealised models—point masses, frictionless surfaces, ideal gases—to derive fundamental laws. Chemistry employs models like the kinetic theory of gases, the ideal solution, and reaction mechanisms to predict behaviour. Biology constructs models of the fluid mosaic membrane, nervous system circuits, and population dynamics, acknowledging limitations and refining them with evidence.
The contrasting approaches highlight a key message: physics often seeks universal laws, chemistry explains substance transformations, and biology embraces complexity and adaptation. Recognising these differences while seeing the underlying unity is the hallmark of a mature scientist.
Mastering calculation problems is the cornerstone of success in both IB Physics and AQA A-level Physics. In these syllabuses, quantitative problem-solving accounts for a significant portion of the exam, requiring students to apply principles from mechanics to modern physics with precision and confidence. This article provides a systematic drill on key calculation types, covering essential equations, common pitfalls, and strategic approaches. Whether you’re preparing for Paper 1 multiple-choice, Paper 2 extended response, or the internal assessment, strengthening your numerical skills will elevate your overall performance.
Kinematics deals with the description of motion without considering its causes. The five SUVAT equations link initial velocity (u), final velocity (v), acceleration (a), displacement (s), and time (t). When solving problems, first list the three known quantities and the one unknown, then pick the equation that does not contain the irrelevant variable. Remember that acceleration due to gravity g = 9.81 m s⁻² acts downwards, so assign a sign convention (usually positive upward or downward).
运动学研究对运动的描述而不考虑其成因。五个 SUVAT 方程连接了初速度(u)、末速度(v)、加速度(a)、位移(s)和时间(t)。解题时，先列出三个已知量和待求量，然后选择不含无关变量的方程。记住重力加速度 g = 9.81 m s⁻² 方向向下，因此要规定符号惯例（通常取向上或向下为正）。
The five equations are listed below. Always check that all quantities are in SI units (m, s, m s⁻¹, m s⁻²).
五个方程列于下方。务必检查所有物理量均使用国际单位（m, s, m s⁻¹, m s⁻²）。
v = u + a t
s = ½ (u + v) t
s = u t + ½ a t²
s = v t – ½ a t²
v² = u² + 2 a s
v = u + a t
s = ½ (u + v) t
s = u t + ½ a t²
s = v t – ½ a t²
v² = u² + 2 a s
For projectile motion, treat the horizontal and vertical components independently. The horizontal velocity remains constant (aₓ = 0), while vertical motion follows SUVAT with a = g downward.
对于抛体运动，应独立处理水平与竖直分量。水平速度保持不变（aₓ = 0），竖直运动则使用 SUVAT 并取 a = g 向下。
2. Newton’s Laws and Force Resolution | 牛顿定律与力的分解
Newton’s second law, F = m a, is the bedrock of dynamics. Always draw a free-body diagram, label all forces, and resolve vectors along convenient axes. The net force in a given direction equals mass times acceleration in that direction. On inclined planes, the weight mg is resolved into components: mg sin θ parallel to the slope and mg cos θ perpendicular.
牛顿第二定律 F = m a 是动力学的基础。务必画出隔离体图，标出所有力，并沿方便的坐标轴分解矢量。某个方向的合外力等于质量乘以该方向的加速度。在斜面上，重力 mg 分解为：沿斜面的分量 mg sin θ 和垂直斜面的分量 mg cos θ。
In equilibrium problems, net force is zero, so balanced force equations are written. For coupled bodies, consider the whole system to find acceleration, then isolate one part to find internal forces. Typical pitfalls include forgetting normal force, mislabeling tension, or using the wrong mass in F = ma.
在平衡问题中，合外力为零，因此列出力的平衡方程。对于连接体，可先用整体法求加速度，再隔离一个物体求内力。常见的错误包括遗漏法向力、错标张力、或在 F = ma 中使用了错误的质量。
3. Energy Conservation and Work | 能量守恒与功
The work-energy principle states that the net work done on an object equals its change in kinetic energy: W_net = ΔEₖ. Kinetic energy is Eₖ = ½ m v² and gravitational potential energy is Eₚ = m g h. If only conservative forces act, mechanical energy is conserved. In the presence of friction, the work done against friction reduces the total mechanical energy, often converted to thermal energy.
功能原理指出，合外力对物体做的功等于其动能的变化量：W_net = ΔEₖ。动能 Eₖ = ½ m v²，重力势能 Eₚ = m g h。如果只有保守力做功，机械能守恒。存在摩擦时，克服摩擦做的功会使机械能减少，通常转化为内能。
Power is the rate of doing work: P = W / t. For a constant force moving at velocity v, instantaneous power is P = F v. In efficiency calculations, useful output power divided by input power gives the efficiency, often expressed as a percentage.
功率是做功的速率：P = W / t。对于一个以速度 v 运动的恒定力，瞬时功率为 P = F v。在效率计算中，有用输出功率除以输入功率得到效率，通常以百分比表示。
4. Momentum and Collisions | 动量与碰撞
Linear momentum p = m v is conserved in isolated systems with no external resultant force. For a two-body collision: m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂. In elastic collisions, kinetic energy is also conserved. For an elastic head-on collision, the relative speed of approach equals the relative speed of separation. In inelastic collisions, momentum is conserved but kinetic energy is not.
线动量 p = m v 在无外合力的孤立系统中守恒。对于两个物体的碰撞：m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂。在弹性碰撞中，动能也守恒。对于弹性正碰，相对接近速度等于相对分离速度。在非弹性碰撞中，动量守恒但动能不守恒。
Impulse is the change in momentum, J = Δp = F_avg Δt. Force–time graphs yield impulse as area under the curve. This is particularly useful in impact problems where forces vary rapidly.
An object moving in uniform circular motion experiences a centripetal acceleration directed towards the centre: a = v² / r = ω² r. The centripetal force is F = m v² / r = m ω² r. The angular speed ω = Δθ / Δt is related to linear speed by v = ω r. The period T = 2π / ω = 2π r / v.
做匀速圆周运动的物体具有指向圆心的向心加速度：a = v² / r = ω² r。向心力为 F = m v² / r = m ω² r。角速度 ω = Δθ / Δt 与线速度的关系为 v = ω r。周期 T = 2π / ω = 2π r / v。
In vertical circular motion, speed varies, and the tension or normal reaction is greatest at the bottom of the loop. Use energy conservation to find speed at different points, then apply Newton’s second law in the radial direction.
Newton’s law of gravitation: F = G M m / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻². The gravitational field strength g = F / m = G M / r². On Earth’s surface, g ≈ 9.81 N kg⁻¹. For a satellite in circular orbit, centripetal force is provided by gravity: G M m / r² = m v² / r, leading to v = √(G M / r) and Kepler’s third law T² ∝ r³.
牛顿万有引力定律：F = G M m / r²，其中 G = 6.67 × 10⁻¹¹ N m² kg⁻²。引力场强度 g = F / m = G M / r²。在地球表面，g ≈ 9.81 N kg⁻¹。对于圆形轨道上的卫星，向心力由引力提供：G M m / r² = m v² / r，得出 v = √(G M / r) 和开普勒第三定律 T² ∝ r³。
Gravitational potential V = – G M / r, and gravitational potential energy U = m V. The escape velocity from a celestial body is v_esc = √(2 G M / r). When dealing with binary stars or energy of orbiting systems, always be attentive to negative signs.
引力势 V = – G M / r，引力势能 U = m V。天体的逃逸速度为 v_esc = √(2 G M / r)。在处理双星或轨道系统能量时，务必注意负号。
7. Electric Fields | 电场
Coulomb’s law: F = k Q q / r², where k = 1/(4π ε₀) ≈ 8.99 × 10⁹ N m² C⁻². Electric field strength E = F / q. For a point charge, E = k Q / r². In a uniform electric field between parallel plates, E = V / d. The force on a charge in an electric field is F = q E.
库仑定律：F = k Q q / r²，其中 k = 1/(4π ε₀) ≈ 8.99 × 10⁹ N m² C⁻²。电场强度 E = F / q。对于点电荷，E = k Q / r²。在平行板间的匀强电场中，E = V / d。电荷在电场中所受的力为 F = q E。
Work done in moving a charge is W = q ΔV, where ΔV is the potential difference. The path integral in a uniform field is simplified as W = q E d cos θ. In electron guns, acceleration through a potential difference V gives the kinetic energy: e V = ½ m v².
移动电荷所做的功为 W = q ΔV，ΔV 为电势差。在匀强电场中，路径积分简化为 W = q E d cos θ。在电子枪中，经过电势差 V 加速后，动能满足：e V = ½ m v²。
8. Magnetic Fields and Forces | 磁场与力
A charged particle moving in a magnetic field experiences the Lorentz force: F = q v B sin θ, where θ is the angle between v and B. The direction is given by the Fleming’s left-hand rule for conventional current. When θ = 90°, the force is maximum and the particle moves in a circular path with radius r = m v / (q B).
带电粒子在磁场中运动时会受到洛伦兹力：F = q v B sin θ，其中 θ 为 v 与 B 的夹角。方向由左手定则判定（针对正电荷）。当 θ = 90° 时，力最大，粒子做圆周运动，半径 r = m v / (q B)。
For a current-carrying wire of length L in a magnetic field, the force is F = B I L sin θ. In electric motor and mass spectrometer calculations, combining F = B q v with centripetal force or balancing electric and magnetic forces (q v B = q E) is common.
对于磁场中长为 L 的通电导线，受力为 F = B I L sin θ。在电动机和质谱仪计算中，常将 F = B q v 与向心力结合，或平衡电场力与洛伦兹力 (q v B = q E) 以确定速度。
9. Electrical Circuits | 电路
Ohm’s law: V = I R. Resistance in series: R_total = R₁ + R₂ + … ; in parallel: 1/R_total = 1/R₁ + 1/R₂ + … . Kirchhoff’s current law states that the sum of currents entering a junction equals the sum leaving. Kirchhoff’s voltage law states that the sum of EMFs is equal to the sum of p.d.s around a closed loop.
Electrical power P = V I = I² R = V² / R. In internal resistance problems, terminal voltage V = E – I r, and total power supplied by the battery is E I. For potential dividers, the output voltage across a resistor is V_out = (R₂ / (R₁ + R₂)) V_in.
电功率 P = V I = I² R = V² / R。在内阻问题中，路端电压 V = E – I r，电池提供的总功率为 E I。对于分压器，电阻上的输出电压为 V_out = (R₂ / (R₁ + R₂)) V_in。
10. Radioactive Decay and Half-life | 放射性衰变与半衰期
The number of undecayed nuclei N decays exponentially: N = N₀ exp(–λ t). The decay constant λ is related to half-life by λ = ln 2 / T₁/₂. Activity A = λ N, which also follows A = A₀ exp(–λ t). The SI unit of activity is the becquerel (Bq). In calculus-based problems, the rate of decay dN/dt = –λ N.
未衰变的原子核数目 N 随时间指数衰减：N = N₀ exp(–λ t)。衰变常量 λ 与半衰期的关系为 λ = ln 2 / T₁/₂。活度 A = λ N，同样满足 A = A₀ exp(–λ t)。活度的国际单位是贝可勒尔 (Bq)。在涉及微积分的问题中，衰变速率 dN/dt = –λ N。
When using the exponential formula, ensure time and half-life are in the same units. For half-life calculations, the fraction remaining after n half-lives is (1/2)ⁿ. In carbon dating or radioisotope applications, use the ratio of current activity to initial activity to solve for age.
使用指数公式时，确保时间和半衰期单位一致。对于半衰期计算，经过 n 个半衰期后剩余的比例为 (1/2)ⁿ。在碳定年或放射性同位素应用中，利用当前活度与初始活度之比求解年代。
11. Quantum Physics and Photon Energy | 量子物理与光子能量
Photon energy E = h f = h c / λ, where Planck’s constant h = 6.63 × 10⁻³⁴ J s. In the photoelectric effect, Einstein’s equation states h f = Φ + Eₖ_max, where Φ is the work function of the metal. The stopping potential V_s is given by e V_s = Eₖ_max. The threshold frequency f₀ = Φ / h.
光子能量 E = h f = h c / λ，普朗克常量 h = 6.63 × 10⁻³⁴ J s。在光电效应中，爱因斯坦方程：h f = Φ + Eₖ_max，其中 Φ 是金属的逸出功。遏止电压 V_s 满足 e V_s = Eₖ_max。截止频率 f₀ = Φ / h。
When analyzing electron energy level transitions, the energy of the emitted or absorbed photon equals the difference between two levels: ΔE = E₂ – E₁ = h f. Wavelength can then be found via λ = h c / ΔE. In electron diffraction, the de Broglie wavelength is λ = h / p = h / (m v).
分析电子能级跃迁时，发射或吸收的光子能量等于两能级之差：ΔE = E₂ – E₁ = h f。再通过 λ = h c / ΔE 求波长。在电子衍射中，德布罗意波长 λ = h / p = h / (m v)。
12. Common Pitfalls and Problem-Solving Strategies | 常见错误与解题策略
Successful calculation practice goes beyond memorising formulas. Develop a structured approach: (1) Read the problem and identify given quantities and the target variable. (2) Convert all units to SI. (3) Choose the fundamental principle (Newton’s laws, energy conservation, etc.) and the appropriate equation. (4) Solve algebraically before plugging in numbers to minimise rounding errors. (5) Check the final answer’s order of magnitude and units.
Weight = m g, not ‘m’ alone in force equations | 重量 = m g，力的方程中不能直接以 m 代替。
Applying wrong energy conservation conditions | 错误应用能量守恒条件
Account for work done by non-conservative forces | 计及非保守力做功。
Incorrect substitution in exponential decay | 指数衰减代入错误
Ensure time and half-life have same unit; use N/N₀ or A/A₀ correctly | 确保时间与半衰期单位一致；正确使用 N/N₀ 或 A/A₀。
Practice deliberately with past paper questions, time yourself, and review errors. Strengthening calculation competence will boost both your confidence and your final grade.
有意识地用历年真题练习，计时并分析错误。强化计算能力将提升你的信心和最终成绩。
Published by TutorHao | Physics Revision Series | aleveler.com