📚 A-Level CCEA Maths: Mechanics Revision Guide | A-Level CCEA 数学：力学考点精讲
Welcome to your essential revision guide for A-Level CCEA Mathematics Mechanics. This article breaks down the core topics you need to master, from kinematics to momentum and collisions, with clear explanations and key formulas.
欢迎阅读 A-Level CCEA 数学力学核心考点精讲。本文梳理了从运动学到动量与碰撞的关键主题，配备清晰解释和重要公式，助你高效备考。
1. Kinematics in One Dimension | 一维运动学
Kinematics describes motion using quantities such as displacement (s), velocity (v) and acceleration (a). Displacement and velocity are vector quantities, having both magnitude and direction, while distance and speed are scalars.
运动学使用位移 (s)、速度 (v) 和加速度 (a) 等物理量描述运动。位移和速度是矢量，既有大小又有方向；而路程和速率是标量。
Average speed = total distance / total time. Instantaneous velocity is the rate of change of displacement.
平均速率 = 总路程 ÷ 总时间。瞬时速度是位移的变化率。
Acceleration a = (v – u) / t, where u is initial velocity and v is final velocity. Uniform acceleration leads to the SUVAT equations.
加速度 a = (v – u) / t，其中 u 为初速度，v 为末速度。匀加速度引出 SUVAT 方程。
2. Equations of Motion (SUVAT) | 运动方程 (SUVAT)
For motion with constant acceleration in a straight line, the following equations hold:
1. v = u + at
2. s = ut + ½ at²
3. v² = u² + 2as
4. s = ½ (u + v) t
These link s (displacement), u (initial velocity), v (final velocity), a (acceleration) and t (time).
对于匀变速直线运动，以下方程成立：
1. v = u + at
2. s = ut + ½ at²
3. v² = u² + 2as
4. s = ½ (u + v) t
它们关联位移 s、初速度 u、末速度 v、加速度 a 和时间 t。
Remember: the equations apply only when acceleration is constant. Choose the equation that contains the unknown you need and all known variables.
记住：方程仅在加速度恒定时适用。选择包含所求未知量且所有量已知的方程。
3. Vertical Motion under Gravity | 重力作用下的竖直运动
Near Earth’s surface, all objects fall with constant acceleration g = 9.8 m/s² downwards. Use SUVAT with a = -g if upward direction is taken as positive.
在地球表面附近，所有物体以恒定加速度 g = 9.8 m/s² 下落。若取向上为正方向，则 a = -g，应用 SUVAT 方程。
When an object is thrown upward, at maximum height v = 0. Time to reach max height: t = u/g. Maximum displacement: s = u²/(2g).
物体竖直上抛时，在最高点速度 v = 0。到达最高点的时间 t = u/g。最大位移 s = u²/(2g)。
For free fall or projection, remember that acceleration is always g downwards, regardless of the direction of motion.
无论物体的运动方向如何，加速度始终为向下的 g。
4. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law: An object remains at rest or in uniform motion unless acted upon by a resultant external force.
牛顿第一定律：任何物体都将保持静止或匀速直线运动状态，直到外力迫使其改变。
Newton’s Second Law: F = ma, where F is the resultant force in newtons (N), m is mass (kg), and a is acceleration (m/s²).
牛顿第二定律：F = ma，F 为合外力 (N)，m 为质量 (kg)，a 为加速度 (m/s²)。
Newton’s Third Law: For every action, there is an equal and opposite reaction. These forces act on different bodies and are of the same type.
牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力。作用力和反作用力作用在不同物体上，且属于同种类型。
5. Connected Particles | 连接质点
In pulley or tow-bar problems, draw clear diagrams. Treat each particle separately and apply F = ma. The tension in a light inextensible string is the same on both sides of a smooth pulley.
在滑轮或牵引杆问题中，需绘制清晰的受力图。分别对每个质点应用 F = ma。轻质不可伸长的绳子在光滑滑轮两侧的张力大小相等。
For a system with two masses m₁ and m₂ connected over a pulley, if m₂ > m₁ the acceleration a = (m₂ – m₁)g/(m₁ + m₂) and the tension T = 2m₁m₂g/(m₁ + m₂).
对于通过滑轮连接的两个质量 m₁ 和 m₂，若 m₂ > m₁，系统加速度 a = (m₂ – m₁)g/(m₁ + m₂)，绳中张力 T = 2m₁m₂g/(m₁ + m₂)。
6. Moments and Equilibrium | 力矩与平衡
The moment of a force about a point = force × perpendicular distance from the point. It is measured in Nm, and can be clockwise or anticlockwise.
力对某点的力矩 = 力 × 力到该点的垂直距离。单位为 Nm，方向可为顺时针或逆时针。
For a body in equilibrium, the resultant force is zero and the sum of clockwise moments equals the sum of anticlockwise moments (principle of moments).
物体平衡时，合力为零，且顺时针力矩之和等于逆时针力矩之和（力矩平衡原理）。
When solving beam problems, identify all forces and their distances from a pivot. Take moments about a point that eliminates an unknown force.
解决梁的问题时，找出所有力及其到支点的距离。对能消去未知力的点求力矩。
7. Work, Energy and Power | 功、能与功率
Work done = force × distance moved in direction of force: W = F s cos θ. Energy is the capacity to do work, measured in joules (J).
做功 = 力 × 沿力方向移动的距离：W = F s cos θ。能量是做功的能力，单位为焦耳 (J)。
Kinetic energy (KE) = ½ m v². Gravitational potential energy (GPE) = m g h. The work-energy principle: net work done = change in kinetic energy.
动能 (KE) = ½ m v²，重力势能 (GPE) = m g h。功-能定理：合力做的功等于动能的变化量。
Power = work done / time = F v for constant velocity motion. Power is measured in watts (W).
功率 = 功 ÷ 时间 = 匀速运动时的 F v。功率单位为瓦特 (W)。
8. Momentum and Impulse | 动量与冲量
Momentum p = m v, a vector quantity measured in kg m/s. Impulse = change in momentum = F Δt = m v – m u.
动量 p = m v，为矢量，单位 kg m/s。冲量 = 动量的变化量 = F Δt = m v – m u。
In collision problems, impulse is the force multiplied by the time of contact, and also equals the area under a force-time graph.
在碰撞问题中，冲量等于力乘以作用时间，也等于力-时间图像下的面积。
9. Collisions and Conservation of Momentum | 碰撞与动量守恒
In the absence of external forces, total momentum before collision = total momentum after collision: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
在外力为零的情况下，碰撞前总动量 = 碰撞后总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
The coefficient of restitution e = (speed of separation) / (speed of approach). For perfectly elastic collisions, e = 1 and KE is conserved; for inelastic, 0 ≤ e < 1.
恢复系数 e = 分离速度 / 接近速度。完全弹性碰撞时 e = 1，动能守恒；非弹性碰撞时 0 ≤ e < 1。
10. Friction | 摩擦力
Friction opposes motion or attempted motion. The maximum frictional force F_max = μ R, where μ is the coefficient of friction and R is the normal reaction.
摩擦力阻碍运动或运动趋势。最大静摩擦力 F_max = μ R，μ 为摩擦系数，R 为法向反作用力。
If a surface is smooth, μ = 0. For limiting equilibrium, the friction force equals μ R and the body is just about to move.
若表面光滑，则 μ = 0。在极限平衡状态下，摩擦力等于 μ R，物体即将开始运动。
11. Projectiles | 抛体运动
A projectile moves under uniform gravity with initial velocity at an angle θ. Resolve motion horizontally and vertically. Horizontally: constant velocity u cosθ, a = 0. Vertically: initial velocity u sinθ, acceleration -g.
抛体以初速度 u、仰角 θ 抛出，在重力作用下运动。将运动分解为水平和竖直方向。水平方向：匀速，速度 u cosθ，加速度 0。竖直方向：初速度 u sinθ，加速度 -g。
Time of flight T = (2 u sinθ) / g. Maximum height H = (u² sin²θ) / (2g). Range R = (u² sin 2θ) / g.
飞行时间 T = (2 u sinθ) / g。最大高度 H = (u² sin²θ) / (2g)。射程 R = (u² sin 2θ) / g。
At any time t, vertical displacement y = (u sinθ) t – ½ g t², horizontal displacement x = (u cosθ) t.
任意时刻 t，竖直位移 y = (u sinθ) t – ½ g t²，水平位移 x = (u cosθ) t。
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