📚 Work and Energy in GCSE Maths | GCSE 数学：功和能量考点精讲
In GCSE Maths, work and energy problems often appear in the context of mechanics and applied mathematics. You are expected to use formulas confidently, substitute values correctly, and interpret real‑world situations. Understanding how to calculate work done, kinetic energy, and gravitational potential energy will help you solve multi‑step problems and answer exam questions accurately.
在 GCSE 数学中，功和能量问题经常出现在力学和应用数学的背景中。你需要能够熟练运用公式、正确代入数值并解读实际情景。理解如何计算做功、动能和重力势能，将帮助你解决多步骤问题并在考试中准确答题。
1. What is Work? | 什么是功？
Work is done when a force moves an object in the direction of the force. In maths, we treat this as a scalar quantity so we only consider the magnitude of the force and the distance moved in the same direction.
当一个力使物体沿着力的方向移动时，就做了功。在数学中，我们把功视为标量，因此只考虑力的大小和沿同一方向移动的距离。
If the force is not parallel to the displacement, only the component of the force in the direction of the displacement does work. However, for GCSE, most questions assume the force and distance are in the same straight line.
如果力与位移不平行，只有沿位移方向的分力做功。不过，在 GCSE 阶段，大多数题目假设力和距离在同一直线上。
2. Work Done Formula and Units | 功的公式和单位
The work done by a constant force is given by: Work Done = Force × Distance, or W = F d.
恒力所做的功由以下公式给出：做功 = 力 × 距离，即 W = F d。
Force is measured in newtons (N), distance in metres (m), so work is measured in newton‑metres (N m), also called joules (J). 1 joule = 1 newton‑metre.
力的单位是牛顿 (N)，距离的单位是米 (m)，因此功的单位是牛顿·米 (N m)，也称为焦耳 (J)。1 焦耳 = 1 牛顿·米。
You must always check that distance is in metres before using the formula. If a question gives distance in cm or km, convert first.
在使用公式之前，你务必检查距离是否以米为单位。如果题目给出的距离是厘米或千米，要先进行单位换算。
3. Calculating Work Done | 计算做功
To calculate work done, multiply the size of the force by the distance the object moves. For example, if a force of 20 N pushes a box 5 m across a floor, the work done is W = 20 × 5 = 100 J.
要计算做功，用力的数值乘以物体移动的距离。例如，如果一个 20 N 的力把箱子在地板上推动了 5 m，所做的功为 W = 20 × 5 = 100 J。
Sometimes you will need to convert mass to weight using Weight = mass × gravitational field strength (g = 9.8 m/s² or 10 m/s²) before finding the work done.
有时你需要先用 重量 = 质量 × 重力场强度 (g = 9.8 m/s² 或 10 m/s²) 把质量转换成重量，再求做功。
Example: W = 150 N × 3 m = 450 J
4. What is Energy? | 能量是什么？
Energy is the capacity to do work. In GCSE Maths, you will mainly use kinetic energy (energy of motion) and gravitational potential energy (energy due to height). Both are measured in joules (J).
能量是做功的能力。在 GCSE 数学中，你主要会用到动能（运动的能量）和重力势能（因高度而具有的能量）。两者都以焦耳 (J) 为单位。
Energy can be transferred or converted, but the total energy in a closed system is conserved. This principle allows you to link work, kinetic energy, and potential energy in problems.
能量可以转移或转化，但封闭系统中的总能量是守恒的。这一原理让你在解题时可以把功、动能和势能联系起来。
5. Kinetic Energy | 动能
Kinetic energy (KE) is the energy an object possesses due to its motion. The formula is: KE = ½ m v², where m is mass in kilograms (kg) and v is speed in metres per second (m/s).
动能 (KE) 是物体由于运动而具有的能量。公式为：动能 = ½ m v²，其中 m 是质量，单位为千克 (kg)；v 是速度，单位为米/秒 (m/s)。
Notice that velocity is squared, so changes in speed have a large effect on kinetic energy. Always square the speed before multiplying by mass and ½.
注意速度是平方项，因此速度的变化对动能影响很大。务必先计算速度的平方，再乘以质量和 ½。
- Mass must be in kg, speed in m/s.
- 质量必须用 kg，速度必须用 m/s。
- KE is directly proportional to mass, and proportional to speed squared.
- 动能与质量成正比，与速度的平方成正比。
KE = ½ × 800 kg × (15 m/s)² = 90 000 J
6. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object due to its height above the ground. The formula is: GPE = m g h, where m is mass (kg), g is gravitational field strength (N/kg or m/s²), and h is height (m).
重力势能 (GPE) 是物体因离地高度而储存的能量。公式为：GPE = m g h，其中 m 是质量 (kg)，g 是重力场强度 (N/kg 或 m/s²)，h 是高度 (m)。
On Earth, g is approximately 9.8 m/s², but many exam questions use 10 m/s² to simplify calculations. Always check which value is specified in the question.
在地球表面，g 大约为 9.8 m/s²，但许多考题会使用 10 m/s² 以简化计算。一定要看清题目指定的数值。
Example: GPE = 2 kg × 10 m/s² × 5 m = 100 J
7. Conservation of Energy | 能量守恒
The principle of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. In mathematical problems, you can set the initial total energy equal to the final total energy.
能量守恒定律指出，能量既不会凭空产生，也不会凭空消失，只会从一种形式转化为另一种形式。在数学问题中，你可以令初始总能量等于最终总能量。
For a falling object, the loss in GPE equals the gain in KE (assuming no air resistance). This gives mgh = ½ mv², which simplifies to v² = 2gh when mass cancels out.
对于一个下落的物体，减少的重力势能等于增加的动能（假设没有空气阻力）。于是有 mgh = ½ mv²，质量约掉后可得 v² = 2gh。
You can be asked to find speed, height, or mass using this energy balance. Always write down the equation linking the energies first.
题目可能会要求你利用能量平衡求速度、高度或质量。一定要先写出能量之间的等式。
8. Power | 功率
Power is the rate at which work is done or energy is transferred. The formula is: Power = Work Done ÷ Time, or P = W / t.
功率是做功或能量转移的速率。公式为：功率 = 做的功 ÷ 时间，即 P = W / t。
Power is measured in watts (W), where 1 watt is 1 joule per second (J/s). Time must be in seconds for the unit to be watts.
功率的单位是瓦特 (W)，1 瓦特等于 1 焦耳每秒 (J/s)。时间必须以秒为单位，得到的才是瓦特。
Sometimes you need to calculate work done first (e.g., lifting a weight) and then divide by the time taken. Power questions often combine work, energy, and time.
有时你需要先计算所做的功（例如举起重物），然后除以所用时间。功率题往往综合了功、能量和时间。
| Quantity | Formula | SI Unit |
| Work Done | W = F d | J |
| Kinetic Energy | KE = ½ m v² | J |
| GPE | GPE = m g h | J |
| Power | P = W / t | W |
9. Efficiency | 效率
Efficiency measures how much of the total input energy is converted into useful output. It is a ratio often expressed as a decimal or percentage.
效率衡量的是总输入能量中有多少转化为有用的输出。它是一个比值，通常用小数或百分数表示。
The formula is: Efficiency = Useful Output Energy ÷ Total Input Energy. No machine is 100 % efficient because some energy is always lost to heat or friction.
公式为：效率 = 有用的输出能量 ÷ 总输入能量。没有机器能做到 100% 的效率，因为总会有能量以热能或摩擦的形式损耗掉。
In GCSE maths, you may be asked to calculate efficiency or to find the wasted energy from given values. Make sure to give your answer in the form requested (decimal or %).
在 GCSE 数学中，你可能会被要求计算效率，或者根据已知数值求浪费的能量。记得按照题目要求的格式（小数或百分数）给出答案。
10. Work-Energy Problems in Context | 情景应用题
Exam questions often describe everyday situations: a car braking, a weightlifter lifting a barbell, a cyclist going uphill, or water falling from a dam. You need to identify which energy transfers are happening.
考试题目经常描述日常情景：汽车刹车、举重运动员举起杠铃、骑自行车上坡、或者大坝泄水。你需要判断发生了哪种能量转移。
Draw a simple diagram and label the forms of energy at the start and end. Write down the relevant formulas and see if any quantity remains constant or cancels out.
画一个简单的示意图，标出开始和结束时能量的形式。写下相关公式，看看是否有哪个量保持不变或可以被约掉。
For example, a braking car loses kinetic energy, which is converted into thermal energy by the brakes. The work done by the braking force equals the change in kinetic energy.
例如，刹车时汽车失去动能，这些动能被刹车转化为热能。制动力所做的功等于动能的变化量。
Braking force × distance = ½ m v² (initial kinetic energy)
11. Rearranging Equations | 公式重组
Being able to rearrange work and energy formulas is crucial. You should practise making every variable the subject: F = W / d, m = KE / (½ v²), h = GPE / (m g), and so on.
能够重组功和能量公式至关重要。你应该练习把每个变量都当作公式的主语：F = W / d，m = KE / (½ v²)，h = GPE / (m g) 等等。
Always show the steps of your rearrangement in the exam. This helps you avoid mistakes and often earns method marks even if the final answer is wrong.
在考试中始终展示你的重组步骤。这有助于避免错误，而且即便最终答案错误，也常常能获得方法分。
- W = F d → F = W / d, d = W / F
- KE = ½ m v² → m = 2KE / v², v = √(2KE / m)
- GPE = m g h → m = GPE / (g h), h = GPE / (m g)
- P = W / t → W = P t, t = W / P
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always write down the formula first, even if it is given on the formula sheet. Substituting numbers into a formula without writing it down increases the risk of errors.
始终先写出公式，即使试卷的公式表里已经有。不写公式直接代入数据会增加出错的风险。
Check your units: convert g to kg, cm to m, minutes to seconds if needed. A common pitfall is forgetting to convert mass from grams to kilograms.
检查单位：必要时把 g 换算成 kg，cm 换算成 m，分钟换算成秒。一个常见的陷阱是忘记把质量从克换算成千克。
For kinetic energy questions, squaring a speed in km/h without converting to m/s first will give a wrong answer. Always convert to m/s before using the formula.
在动能问题中，如果先不把 km/h 换算成 m/s 就对速度平方，会得到错误答案。使用公式前务必先换算成 m/s。
Read the question carefully: it may ask for the final answer in kJ or MJ. Remember that 1 kJ = 1000 J, and 1 MJ = 1 000 000 J.
仔细读题：题目可能要求最终答案以 kJ 或 MJ 为单位。记住 1 kJ = 1000 J，1 MJ = 1 000 000 J。
Finally, if a question involves more than one step (e.g., find work done, then power), plan your solution and show each step clearly. Marks are awarded for method, not just the answer.
最后，如果题目涉及多步骤（例如先求做功、再求功率），规划好解题思路并清晰地展示每一步。评分不仅看答案，也看方法过程。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply