📚 Mechanics Key Points | IB CIE 数学：力学 考点精讲
Mechanics plays a vital role in both IB and CIE A-Level Mathematics, testing your ability to model physical systems mathematically. This revision note covers key topics including SUVAT, forces, moments, energy, and variable acceleration, with essential formulas and problem-solving tips.
力学在 IB 与 CIE A-Level 数学中占有重要地位，考查将物理系统数学建模的能力。本复习笔记涵盖匀加速运动、力、力矩、能量与变加速等核心专题，并提供关键公式与解题思路。
1. Kinematics: SUVAT Equations | 运动学：匀加速运动方程
When acceleration is constant, the five SUVAT equations provide a direct link between displacement s, initial velocity u, final velocity v, acceleration a, and time t. Memorising these equations and knowing how to select the right one is crucial. The equations are: v = u + at; s = ut + ½at²; s = ½(u+v)t; v² = u² + 2as; s = vt – ½at². Always define a positive direction before applying them.
匀加速运动中，五个 SUVAT 方程将位移 s、初速度 u、末速度 v、加速度 a 和时间 t 联系起来。熟记公式并会选择合适方程是关键。五个方程为：v = u + at；s = ut + ½at²；s = ½(u+v)t；v² = u² + 2as；s = vt – ½at²。使用前务必规定正方向。
Note that these equations only apply when a = constant. For vertical motion under gravity, a = g ≈ 9.8 m s⁻², with direction taken as negative if upwards is positive.
注意这些方程仅适用于匀加速度。重力下的竖直运动 a = g ≈ 9.8 m s⁻²，若取向上为正，则加速度为负。
2. Projectile Motion | 抛体运动
Treat the horizontal and vertical components independently. Initial velocity u at angle θ to the horizontal gives ux = u cosθ, uy = u sinθ. Horizontally, velocity is constant, so x = u cosθ · t. Vertically, acceleration is -g (if upward positive), using y = u sinθ · t – ½gt². Time of flight T = 2u sinθ / g, maximum height H = u² sin²θ / (2g), and horizontal range R = u² sin2θ / g.
处理抛体运动需独立分析水平和竖直分量。初速度 u 与水平夹角 θ，则 ux = u cosθ，uy = u sinθ。水平方向匀速，x = u cosθ·t；竖直方向以向上为正时加速度为 -g，满足 y = u sinθ·t – ½gt²。飞行时间 T = 2u sinθ/g，最大高度 H = u² sin²θ/(2g)，水平射程 R = u² sin2θ/g。
The trajectory equation eliminates t: y = x tanθ – (g x²)/(2u² cos²θ), showing a parabola. The maximum range occurs at θ = 45° on level ground.
消去 t 得到轨迹方程 y = x tanθ – (g x²)/(2u² cos²θ)，为抛物线。水平面上最大射程对应发射角 45°。
3. Newton’s Laws of Motion | 牛顿运动定律
First law: an object remains at rest or uniform motion unless acted upon by a net external force. Second law: F = ma, where F is the resultant force in the direction of motion. Third law: action and reaction forces are equal in magnitude, opposite in direction, and act on different bodies. In mechanics problems, always draw a free-body diagram, resolve forces, and apply F = ma in each direction.
第一定律：物体在无合外力时保持静止或匀速运动。第二定律：F = ma，F 为运动方向上的合力。第三定律：作用力与反作用力大小相等、方向相反、作用在不同物体上。解题时务必画受力图、分解力、并在各方向上应用 F = ma。
4. Forces and Equilibrium | 力与平衡
An object is in equilibrium when the vector sum of all forces is zero and the sum of moments is zero. Resolve forces in mutually perpendicular directions, usually horizontally and vertically. Example: a mass on a smooth inclined plane of angle θ: the component of weight down the slope is mg sinθ, and normal reaction is mg cosθ. For equilibrium parallel to the slope, an external force must balance mg sinθ.
物体平衡时所有力的矢量和为零且合力矩为零。通常将力沿正交方向分解，如水平和竖直。示例：光滑斜面上质量为 m 的物体，斜面倾角 θ，重力沿斜面的分量为 mg sinθ，法向反作用力为 mg cosθ。若要平衡，需施加外力抵消 mg sinθ。
5. Moments and Torque | 力矩
Moment of a force = force × perpendicular distance from pivot. The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point. When solving beam or rod problems, take moments about a point where unknown forces act to simplify equations.
力矩 = 力 × 力臂（支点到力作用线的垂直距离）。力矩原理：物体转动平衡时，绕任一点顺时针力矩之和等于逆时针力矩之和。求解梁或杆问题时，可对未知力所在点取矩以简化方程。
6. Momentum and Impulse | 动量与冲量
Linear momentum p = mv, a vector. Impulse J = FΔt = Δp = m(v – u). The law of conservation of momentum: in a closed system with no external forces, total momentum before collision equals total momentum after collision. For a perfectly inelastic collision, objects stick together (e = 0). For an elastic collision, kinetic energy is conserved (e = 1). IB formula booklet includes these relationships.
动量 p = mv，为矢量。冲量 J = FΔt = Δp = m(v – u)。动量守恒定律：封闭系统无外力时，碰撞前总动量等于碰撞后总动量。完全非弹性碰撞中物体粘合（恢复系数 e=0）；完全弹性碰撞中动能守恒（e=1）。IB 公式表包含这些关系。
7. Work, Energy and Power | 功、能量与功率
Work done W = F s cosθ, measured in joules. Kinetic energy KE = ½mv², gravitational potential energy GPE = mgh. The work-energy principle: net work = change in kinetic energy. Conservation of mechanical energy applies when no external work is done. Power P = W/t = F v (at constant speed). Efficiency = useful output / total input.
功 W = F s cosθ，单位焦耳(J)。动能 KE = ½mv²，重力势能 GPE = mgh。功-能原理：合力做功等于动能变化。无外力做功时机械能守恒。功率 P = W/t = F v（匀速时）。效率 = 有用输出功 / 总输入功。
8. Friction | 摩擦力
Friction opposes relative motion. Static friction f ≤ μₜ R, where μₜ is the coefficient of static friction and R the normal reaction. Kinetic friction f = μₚ R. On an inclined plane, the object will slide if tanθ > μₜ. The angle of friction is given by tanφ = μ. Use F ≤ μR to test equilibrium.
摩擦力阻碍相对运动。静摩擦力 f ≤ μₜ R，μₜ 为静摩擦系数，R 为法向反力。动摩擦力 f = μₚ R。斜面上若 tanθ > μₜ，物体将下滑。摩擦角 φ 满足 tanφ = μ。用 F ≤ μR 检查是否能平衡。
9. Connected Particles and Pulleys | 连接体与滑轮
Systems of particles connected by light inextensible strings share the same magnitude of tension and acceleration. Treat each mass separately: write F = ma equations in the direction of motion. For a pulley, one mass will move up while the other moves down; set up simultaneous equations to find a and T. Always assume the string is light and the pulley is smooth (no friction).
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