📚 International A-level Unit FM2 Mechanics Revision | 国际A-level单元FM2力学复习
This revision guide covers the core topics in AQA International A-level Further Mathematics Unit FM2 (Mechanics), including dimensional analysis, vector kinematics, energy methods, impulse and momentum, collisions, circular motion, simple harmonic motion and centres of mass. Each section gives an English explanation followed by a paired Chinese translation to support bilingual learning and exam practice.
本复习指南涵盖 AQA 国际 A-level 进阶数学单元 FM2（力学）的核心主题，包括量纲分析、矢量运动学、能量方法、冲量与动量、碰撞、圆周运动、简谐运动以及质心。每一节先给出英文讲解，再配以中文翻译，帮助双语学习与备考。
1. Dimensional Analysis | 量纲分析
Dimensional analysis treats mass M, length L and time T as independent base dimensions. Every mechanical quantity can be expressed in terms of these three dimensions, which makes it possible to check whether an equation is homogeneous.
量纲分析将质量 M、长度 L 和时间 T 视为独立的基本量纲。任何力学量都可以用这三个基本量纲表示，因此可以检验方程是否为齐次式。
velocity: [v] = [L][T]⁻¹
Acceleration is the rate of change of velocity, so it has dimensions [L][T]⁻². Force is mass multiplied by acceleration, giving [M][L][T]⁻².
加速度是速度的变化率，因此其量纲为 [L][T]⁻²。力等于质量乘以加速度，因此其量纲为 [M][L][T]⁻²。
| Quantity | Dimensions |
|---|---|
| Velocity | [L][T]⁻¹ |
| Acceleration | [L][T]⁻² |
| Force | [M][L][T]⁻² |
| Energy or work | [M][L]²[T]⁻² |
| Power | [M][L]²[T]⁻³ |
For an equation to be valid, every term must have the same dimensions. This principle is called homogeneity, and it is often used in FM2 questions to find unknown powers or to identify mistakes in a formula.
一个有效的方程必须满足每一项都具有相同的量纲。这一原则称为量纲齐次性，在 FM2 考题中常用于求解未知指数或识别公式中的错误。
example, if a resistive force is given by F = k v², then k must have dimensions [M][L]⁻¹.
量纲分析还可以帮助推导常数的单位。例如，若阻力为 F = k v²，则 k 的量纲必须是 [M][L]⁻¹。
2. Vector Kinematics and Projectiles | 矢量运动学与抛体运动
In FM2, position, velocity and acceleration are often written as vectors using the unit vectors i and j. This makes it easier to handle two-dimensional motion without losing information about direction.
在 FM2 中，位置、速度和加速度通常使用单位矢量 i 和 j 写成矢量形式。这样可以更方便地处理二维运动，同时保留方向信息。
v = u + a t
vertical acceleration is constant at g downwards, so the vector acceleration is a = −g j.
对于抛体运动，通常忽略空气阻力。水平方向加速度为零，而竖直方向加速度恒为向下的 g，因此矢量加速度为 a = −g j。
x = u cos θ t and y = u sin θ t − ½ g t²
The horizontal component of velocity remains constant throughout the flight. The vertical component changes linearly with time, just like one-dimensional motion under gravity.
速度的水平分量在整个飞行过程中保持不变。竖直分量随时间线性变化，就像重力作用下的一维运动一样。
The total time of flight is found by setting the vertical displacement to zero. The maximum height occurs when the vertical velocity is zero, and the range is the horizontal distance travelled in the total flight time.
总飞行时间通过令竖直位移为零来求得。最大高度出现在竖直速度为零的时刻，而射程是总飞行时间内的水平位移。
R = (u² sin 2θ) / g
The maximum range for a given initial speed is achieved when the angle of projection is 45°. Keep in mind that this result assumes level ground and no air resistance.
给定初速度时，当发射角为 45° 时可获得最大射程。请记住，该结论假设地面水平且无空气阻力。
3. Work, Energy and Power | 功、能量与功率
Work is done when a force moves its point of application in the direction of the force. If the force and displacement are not parallel, only the component of the force in the direction of displacement does work.
当力使其作用点沿力的方向移动时，力就做了功。如果力与位移不平行，只有力在位移方向上的分量做功。
W = F s cos θ
Kinetic energy is the energy a body has because of its motion, and gravitational potential energy is the energy stored because of a body's height above a chosen reference level. The work-energy principle states that the total work done by all forces equals the change in kinetic energy.
动能是物体由于运动而具有的能量，重力势能是物体由于位于参考水平面以上而储存的能量。功能原理指出，所有力所做的总功等于动能的变化量。
KE = ½ m v² and ΔE = W
Power is the rate at which work is done. For a constant force acting on an object moving at constant speed, power can be written as the product of force and velocity.
功率是做功的快慢。对于作用在匀速运动物体上的恒力，功率可以写成力与速度的乘积。
P = F v
Energy methods are especially useful when acceleration is not constant. In FM2 problems, you should always consider whether conservation of mechanical energy can simplify a solution.
当加速度不恒定时，能量方法尤其有用。在 FM2 问题中，始终应考虑机械能守恒是否可以简化求解。
4. Impulse and Momentum in One and Two Dimensions | 一维与二维冲量和动量
Momentum is a vector quantity defined as mass multiplied by velocity. Impulse is the change in momentum, and it is equal to the average force multiplied by the time for which the force acts.
动量是定义为质量乘以速度的矢量。冲量是动量的变化量，等于平均力乘以力的作用时间。
I = m(v − u) = F Δt
When two or more bodies interact in an isolated system, the total momentum before the interaction is equal to the total momentum after the interaction. This conservation law applies separately in perpendicular directions.
当两个或多个物体在孤立系统中相互作用时，相互作用前的总动量等于相互作用后的总动量。该守恒定律分别适用于垂直方向。
m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂
In two-dimensional problems, it is usually best to resolve velocities into i and j components, apply conservation of momentum in each direction, and then combine the components to find the required vector.
在二维问题中，通常最好将速度分解为 i 和 j 分量，在每个方向上应用动量守恒，然后合成各分量以求出所需矢量。
Impulse-momentum diagrams can be very helpful. Draw the velocities before and after the impulse, label all vectors clearly, and use the vector form to avoid sign errors.
冲量-动量图非常有帮助。画出冲量作用前后的速度，清楚标注所有矢量，并使用矢量形式以避免符号错误。
5. Collisions and Coefficient of Restitution | 碰撞与恢复系数
The coefficient of restitution, e, measures how elastic a collision is. It is defined as the ratio of the speed of separation to the speed of approach along the line of impact.
恢复系数 e 衡量碰撞的弹性程度。它被定义为沿碰撞线方向上分离速度与接近速度之比。
e = (v₂ − v₁) / (u₁ − u₂)
elastic collision, e = 0, and the particles coalesce. Most real collisions have 0 < e < 1.
对于完全弹性碰撞，e = 1，动能守恒。对于完全非弹性碰撞，e = 0，两物体粘在一起。大多数真实碰撞满足 0 < e < 1。
In a direct collision, you can solve problems by writing two equations: conservation of momentum and Newton's experimental law. These simultaneous equations give the two unknown final velocities.
在正碰中，可以通过写出两个方程来求解：动量守恒和牛顿实验定律。这两个联立方程可以解出两个未知的末速度。
For oblique collisions between smooth spheres, only the velocity components along the line of centres change. The components perpendicular to the line of centres remain unchanged, and restitution is applied only along the line of centres.
对于光滑球体之间的斜碰，只有沿球心连线方向的速度分量发生变化。垂直于球心连线的分量保持不变，且恢复系数只沿球心连线方向应用。
6. Circular Motion: Horizontal and Vertical Circles | 圆周运动：水平与竖直圆周
Circular motion requires a resultant force directed towards the centre of the circle. This force is called centripetal force, and it changes the direction of the velocity without changing its magnitude in uniform circular motion.
圆周运动需要一个指向圆心的合力。这个力称为向心力，在匀速圆周运动中它只改变速度的方向，不改变速度的大小。
v = r ω
The centripetal acceleration can be expressed in terms of linear speed or angular speed. Both forms are important in FM2, and you should choose the one that matches the given information.
向心加速度可以用线速度或角速度表示。这两种形式在 FM2 中都很重要，应选择与已知条件匹配的形式。
a = v²/r = r ω² and F = m v²/r = m r ω²
In horizontal circular motion, such as a conical pendulum, you often resolve vertically to balance weight and horizontally to provide the centripetal force. In vertical circular motion, use conservation of energy to find the speed at different points.
在水平圆周运动中，例如圆锥摆，通常竖直方向分解以平衡重力，水平方向分解以提供向心力。在竖直圆周运动中，使用能量守恒来求不同位置的速度。
For an object to complete a vertical circle, it must have enough speed at the top of the circle. At the top, the minimum speed condition is found by setting the tension or normal reaction to zero.
物体要完成竖直圆周运动，在
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