📚 IGCSE OCR Maths: Mechanics Exam Guide | IGCSE OCR 数学：力学考点精讲
Mechanics in IGCSE OCR Mathematics bridges physics and algebra, requiring you to interpret motion graphs, apply Newton’s laws, and solve problems about forces and momentum. This guide covers every essential topic with clear explanations, worked examples, and bilingual notes to strengthen both your mathematical skills and your understanding of physical principles. Whether you are targeting a grade 5 or aiming for a top grade 9, mastering these mechanics concepts will give you a solid advantage.
IGCSE OCR 数学中的力学部分连接了物理与代数，要求你解读运动图像、应用牛顿定律，并解决关于力和动量的问题。本指南涵盖每个重要考点，配合清晰的解释、范例和中英双语讲解，旨在夯实你的数学技能与物理直觉。不论你的目标是5分还是满分9分，吃透这些力学概念都会让你脱颖而出。
1. Speed, Distance and Time | 速度、距离与时间
The relationship between speed, distance and time is one of the first building blocks of mechanics. The fundamental formula is speed = distance / time. In symbols, s = d / t. Rearranging gives d = s × t and t = d / s. You must always use consistent units: metres per second (m/s), metres (m), seconds (s) or kilometres per hour (km/h), kilometres (km) and hours (h).
速度、距离和时间的关系是力学入门的第一块基石。基本公式为 速度 = 距离 / 时间，用符号表示为 v = d / t。变形可得 d = v × t 和 t = d / v。你必须始终使用一致的单位：米每秒（m/s）、米（m）、秒（s），或千米每小时（km/h）、千米（km）、小时（h）。
Average speed is used when the speed is not constant. For example, if a car travels 100 km in 2 hours, its average speed is 50 km/h, even if it moved faster or slower at times. In exam questions, be careful with mixed units—convert everything to metres and seconds before calculating motion quantities.
当速度不恒定时，使用平均速度。例如，一辆汽车 2 小时行驶 100 km，其平均速度为 50 km/h，即使过程中时快时慢。在考试题中，注意单位混用的陷阱——计算运动量前，先把所有单位统一为米和秒。
2. Interpreting Distance-Time Graphs | 解读距离-时间图
A distance-time graph plots distance moved from a starting point against time. The gradient of the line equals speed. A horizontal line (zero gradient) indicates the object is stationary. A straight sloped line indicates constant speed, while a curved line indicates changing speed (acceleration or deceleration).
距离-时间图描绘从起点开始移动的距离随时间变化。图像的斜率等于速度。水平线（零斜率）表示物体静止。倾斜的直线表示匀速运动，而曲线表示变速运动（加速或减速）。
To calculate speed from a straight section, pick two points and use gradient = change in distance / change in time. If the graph is a curve, the speed at an instant is the gradient of the tangent drawn at that point. Exam questions often ask you to describe a journey, so practise reading the story behind the graph: where the object moves, stops, returns, etc.
若图线为直线，可从直线上取两点计算速度：斜率 = 距离的变化量 / 时间的变化量。若图线为曲线，则某一瞬间的速度等于该点处切线的斜率。考题常要求你描述一段行程，因此要练习从图像读出“故事”：物体何时运动、何时静止、何时返回等。
3. Velocity and Acceleration | 速度与加速度
Velocity is speed in a given direction; in mechanics, direction matters. Acceleration is the rate of change of velocity. The formula for average acceleration is a = (v – u) / t, where u is initial velocity, v is final velocity and t is time taken. Acceleration can be positive (speeding up) or negative (slowing down, often called deceleration).
速度是带有方向的速率；在力学中，方向很重要。加速度是速度的变化率。平均加速度的公式为 a = (v − u) / t，其中 u 是初速度，v 是末速度，t 是所用时间。加速度可为正值（加速）或负值（减速，常称为负加速度）。
Remember that if an object moves in the opposite direction, velocity becomes negative. This is crucial when dealing with motion graphs or momentum. Always label a positive direction clearly in your working.
切记，若物体朝反方向运动，速度变为负值。在处理运动图像或动量问题时，这至关重要。在解题过程中一定要清楚地标明正方向。
4. Velocity-Time Graphs and Area | 速度-时间图与面积
A velocity-time graph shows how an object’s velocity changes over time. The gradient of the line gives acceleration. A horizontal line means constant velocity (zero acceleration). The area between the line and the time axis represents the distance travelled (displacement when direction is included).
速度-时间图展示物体速度随时间的变化。图线的斜率表示加速度。水平线代表匀速（加速度为零）。图线与时间轴之间所围成的面积代表行驶的距离（若考虑方向，则为位移）。
To calculate total distance, split the area into rectangles, triangles and trapeziums. For example, if a car accelerates uniformly for 10 seconds from rest to 20 m/s, the area under the triangle is ½ × 10 × 20 = 100 m. Exam questions often test your ability to find acceleration from the gradient and distance from the area on the same graph.
计算总距离时，可将面积拆分为矩形、三角形与梯形。例如，一辆汽车从静止匀加速 10 秒到 20 m/s，其下方三角形面积为 ½ × 10 × 20 = 100 m。考试中常在同一张图上考察你从斜率求加速度、从面积求距离的能力。
5. Forces and Newton’s Laws | 力与牛顿定律
Newton’s second law is central to mechanics: the resultant force on an object equals its mass times its acceleration, written as F = m × a. Force is measured in newtons (N), mass in kilograms (kg) and acceleration in m/s². This relationship shows that a larger force gives a larger acceleration, and a larger mass gives a smaller acceleration for the same force.
牛顿第二定律是力学的核心：作用在物体上的合力等于物体的质量乘以加速度，写作 F = m × a。力的单位是牛顿（N），质量单位是千克（kg），加速度单位是米每平方秒（m/s²）。这个关系表明，力越大加速度越大，而同样力下质量越大加速度越小。
Newton’s first law states that an object stays at rest or moves at constant velocity unless acted on by a resultant force. Newton’s third law says that when two objects interact, the forces they exert on each other are equal in magnitude and opposite in direction. These principles are tested through force diagrams and motion descriptions.
牛顿第一定律指出，除非受到合外力作用，物体将保持静止或匀速直线运动。牛顿第三定律表明，两个物体相互作用时，彼此施加的力大小相等、方向相反。这些原理通过受力图和运动描述进行考查。
6. Mass, Weight and Gravity | 质量、重量与重力
Weight is a force caused by gravity, given by the formula W = m × g, where m is mass and g is gravitational field strength (about 9.8 N/kg on Earth, but often rounded to 10 N/kg in exam problems). Weight is measured in newtons, while mass is measured in kilograms; they are not the same quantity.
重量是由重力产生的力，公式为 W = m × g，其中 m 是质量，g 是重力场强度（地球上约为 9.8 N/kg，但考试中常近似为 10 N/kg）。重量以牛顿为单位，而质量以千克为单位；两者并非同一个物理量。
In mechanics problems, weight always acts vertically downwards from the centre of mass. When an object is on a slope, the weight must be resolved into components parallel and perpendicular to the surface. A common mistake is to confuse mass and weight in F = ma…always remember F is the resultant force including weight and other forces.
在力学题目中，重量始终从重心竖直向下作用。当物体位于斜面上时，需将重量分解为沿斜面方向和垂直于斜面方向的分力。一个常见错误是在 F=ma 中混淆质量与重量……切记 F 是包括重量和其他力在内的合力。
7. Resultant Forces and Equilibrium | 合力与平衡
When multiple forces act on an object, the resultant force is the single force that has the same effect as all the original forces combined. If the resultant force is zero, the forces are balanced and the object is in equilibrium—it either stays still or continues at constant velocity.
当多个力作用在物体上时，合力是指与所有原始力共同作用效果相同的单一力。若合力为零，则力处于平衡，物体处于平衡状态——它要么保持静止，要么继续匀速运动。
To find the resultant of forces acting along a line, simply add forces in one chosen positive direction and subtract those in the opposite. For forces at an angle, use vector addition or tip-to-tail diagrams. OCR IGCSE often tests equilibrium by asking for an unknown force when an object is stationary or moving steadily.
求沿同一直线上力的合力时，只需将选定正方向的力相加，减去反方向的力。对于互成角度的力，可使用矢量加法或头尾相接作图法。OCR IGCSE 常通过物体静止或匀速运动来考查未知力的求解。
8. Friction and Drag Forces | 摩擦力与阻力
Friction is a force that opposes motion between two surfaces in contact. It acts parallel to the surfaces. Static friction prevents an object from starting to move; kinetic friction acts while it is moving. In many mechanics problems, friction is assumed constant or proportional to the normal contact force.
摩擦力是阻碍接触面之间相对运动的力，沿接触面方向作用。静摩擦力阻止物体开始运动；动摩擦力在物体运动时作用。在许多力学题目中，摩擦力被视为恒力或与法向接触力成正比。
Drag (air resistance or fluid resistance) increases with speed. In real-life motion, a falling object reaches terminal velocity when drag equals weight and resultant force becomes zero. These concepts connect directly to velocity-time graphs where acceleration gradually reduces as drag builds up.
阻力（空气阻力或流体阻力）随速度增加而增大。在真实运动中，下落的物体当阻力与重量相等、合力为零时达到终极速度。这些概念直接与速度-时间图关联——图中加速度随阻力增大而逐渐减小。
9. Momentum (Higher Tier) | 动量（高级）
Momentum is defined as the product of mass and velocity: p = m × v. It is a vector quantity, so direction matters. Momentum helps us analyse collisions and explosions, because in a closed system, total momentum before an event equals total momentum after — this is the principle of conservation of momentum.
动量定义为质量与速度的乘积：p = m × v。它是矢量，因此方向很重要。动量帮助我们分析碰撞与爆炸，因为在一个封闭系统中，事件发生前的总动量等于事件发生后的总动量——这就是动量守恒定律。
For a collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where u are initial velocities and v are final velocities. If objects stick together, their combined mass moves at the same final velocity. OCR Higher tier questions often ask you to find an unknown velocity or mass after an impact.
对于碰撞：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂，其中 u 代表初速度，v 代表末速度。若物体粘连在一起，它们的合体质量以相同的末速度运动。OCR 高级考卷中的题目常要求你在碰撞后求解未知速度或质量。
10. Impulse and Force–Time Graphs | 冲量与力-时间图
Impulse is the change in momentum caused by a force acting over a time interval. It is calculated as Impulse = F × t where F is a constant force, or the area under a force–time graph. Impulse also equals the change in momentum: F × t = mv – mu.
冲量是力在时间间隔内作用引起的动量变化。当力恒定时，计算公式为 冲量 = F × t；若力变化，则等于力-时间图下的面积。冲量也等于动量变化量：F × t = mv − mu。
In safety design, increasing the time of impact reduces the force for a given change in momentum (e.g. crumple zones, airbags). This is a direct application of the impulse equation. If given a force-time graph, practise finding impulse via counting squares or area formulas for triangles and rectangles.
在安全设计中，延长碰撞时间可在动量变化一定时减小受力（例如缓冲区和安全气囊）。这是冲量方程的直接应用。若给出力-时间图，应练习通过数格子或三角形、矩形面积公式求冲量。
11. Using Formula for Uniform Acceleration | 匀加速运动公式应用
For motion with constant acceleration, you can use the suvat equations. The OCR specification expects you to solve problems using these relationships, especially from velocity-time graphs, but also algebraically. The four main equations are:
对于匀加速运动，可使用匀变速运动公式。OCR 考纲要求使用这些关系解决问题，特别是从速度-时间图入手，也要会代数求解。四个主要方程为：
v = u + at
s = (u + v)t / 2
s = ut + ½ at²
v² = u² + 2as
Here, s is displacement, u is initial velocity, v is final velocity, a is constant acceleration, t is time. Always list the known quantities and the missing one before choosing the right equation. Negative acceleration is used if the object is slowing down.
其中 s 为位移，u 为初速度，v 为末速度，a 为恒定加速度，t 为时间。选择方程前，先列出已知量和待求量。若物体正减速，加速度取负值。
12. Problem-Solving with Mechanics | 力学问题求解技巧
Start every mechanics problem by drawing a diagram showing all forces, velocities, and directions. Choose and label a clear positive direction. Write down the data given with correct units and convert if necessary. Select the appropriate formula (e.g., F = ma, motion equations, momentum).
处理每个力学问题时，首先画出示力图，标明所有力、速度和方向。选定并明确标明正方向。写下给出的数据并带正确单位，必要时进行换算。选择合适的公式（如 F=ma、运动学公式、动量公式）。
After substituting numbers, solve algebraically step by step, keeping an eye on sign. Interpret the answer in context—does a negative velocity mean opposite direction? Is the force realistic? Practise with past OCR exam questions to become familiar with mixed-topic problems such as a ball thrown up, sliding block on slope, or collisions.
代入数值后，逐步进行代数求解，并注意符号。在上下文中解释答案——负速度是否意味着反方向？力的大小是否合理？通过练习 OCR 历年真题，熟悉综合题型，如上抛小球、斜面上滑行物体或碰撞问题。
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