IB & WJEC Maths: Mechanics Key Points Review | IB WJEC 数学：力学考点精讲

📚 IB & WJEC Maths: Mechanics Key Points Review | IB WJEC 数学：力学考点精讲

Welcome to this bilingual revision guide covering key mechanics topics for IB and WJEC mathematics courses. Mechanics bridges pure mathematics and physical intuition, requiring a solid grasp of modelling assumptions, vector quantities, and problem-solving strategies. This article summarises the essential concepts, formulas and typical problem types in kinematics, dynamics, energy and statics.

欢迎阅读本双语复习指南，涵盖 IB 和 WJEC 数学课程的核心力学主题。力学连接纯数学与物理直觉，需要扎实掌握建模假设、矢量量和解题策略。本文总结了运动学、动力学、能量和静力学中的基本概念、公式及典型问题类型。

1. Kinematics: Displacement, Velocity and Acceleration | 运动学：位移、速度和加速度

Kinematics describes motion using three key vectors: displacement (s) from a fixed origin, velocity (v) as the rate of change of displacement, and acceleration (a) as the rate of change of velocity. In one-dimensional models, direction is indicated by a positive or negative sign. The instantaneous velocity is given by the derivative v = ds/dt, and acceleration by a = dv/dt or the second derivative d²s/dt².

运动学用三个关键矢量描述运动：位移 (s) 从固定原点开始，速度 (v) 是位移的变化率，加速度 (a) 是速度的变化率。在一维模型中，方向用正负号表示。瞬时速度由导数 v = ds/dt 给出，加速度由 a = dv/dt 或二阶导数 d²s/dt² 给出。

For constant acceleration, the motion can be analysed using the SUVAT equations without calculus. Always define a positive direction and check that quantities are consistent in sign. In IB, you may also need to integrate acceleration vectors to find velocity and position functions.

对于匀加速度，运动可以用 SUVAT 方程分析，无需微积分。务必设定正方向并检查各量的符号是否一致。在 IB 课程中，还可能需对加速度向量积分以求速度和位置函数。

2. SUVAT Equations for Constant Acceleration | 匀加速运动的 SUVAT 方程

The five standard equations linking displacement s, initial velocity u, final velocity v, acceleration a and time t are fundamental. Memorise them and understand when each is most useful:

联系位移 s、初速度 u、末速度 v、加速度 a 和时间 t 的五个标准方程是基础。牢记它们并理解各自的最佳使用场景：

v = u + at

s = ut + ½ at²

s = ½ (u + v) t

v² = u² + 2as

s = vt − ½ at²

These equations apply only when acceleration is constant. In WJEC M1 and IB problems, you will often use vertical motion under gravity, where a = ±g. Always include units and consider the sign convention.

这些方程仅在加速度恒定时适用。在 WJEC M1 和 IB 问题中，常会用到重力下的竖直运动，其中 a = ±g。务必包含单位并考虑符号约定。

3. Vertical Motion under Gravity | 重力作用下的竖直运动

When a particle moves vertically under gravity alone, the acceleration is g ≈ 9.8 m s⁻² downwards. Taking upward as positive gives a = −g. For an object thrown upward, the velocity at the highest point is zero. Time to maximum height is found from v = u + at setting v = 0.

当质点仅在重力下竖直运动时，加速度为向下的 g ≈ 9.8 m s⁻²。取向上为正时 a = −g。对于上抛物体，最高点速度为零。到达最大高度的时间可通过令 v=0 由 v = u + at 求得。

The total time of flight for a symmetric vertical launch is twice the time to the top. Displacement, not distance, is used in SUVAT. If an object falls from rest, u = 0 and s becomes negative if upward is positive. Questions often require combining two stages of motion.

对称竖直抛体的总飞行时间是上升时间的两倍。SUVAT 中使用的是位移而非路程。若物体从静止下落，u = 0；若向上为正，则位移为负。题目经常要求将两段运动结合起来。

4. Projectile Motion | 抛体运动

Projectile motion can be analysed by resolving into horizontal and vertical components. Horizontally, acceleration is zero, so speed u cos θ is constant. Vertically, acceleration is −g, and the vertical motion obeys SUVAT equations. The initial velocity components are ux = u cos θ and uy = u sin θ.

抛体运动可通过分解为水平和竖直分量进行分析。水平方向加速度为零，因此速度 u cos θ 恒定。竖直方向加速度为 −g，竖直运动遵循 SUVAT 方程。初速度分量为 ux = u cos θ 和 uy = u sin θ。

The time of flight is determined entirely by vertical motion: solve y = uy t − ½ gt² for y = 0 (if landing at same height). Range R = ux × total time. Maximum height H = (uy²)/(2g). For WJEC, you may need to derive the Cartesian equation of the trajectory y = x tan θ − (g x²)/(2u² cos² θ). In IB HL, calculus methods using vectors are also common.

飞行时间完全由竖直运动决定：令 y = uy t − ½ gt²，并在 y=0 (若落回同一高度) 时求解。射程 R = ux × 总时间。最大高度 H = (uy²)/(2g)。在 WJEC 中，可能需推导轨迹的笛卡儿方程 y = x tan θ − (g x²)/(2u² cos² θ)。在 IB HL 中，使用向量的微积分方法也很常见。

5. 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 net external force. Second Law: The resultant force F on a body equals the rate of change of momentum; for constant mass, F = ma. Third Law: If body A exerts a force on body B, then B exerts an equal and opposite force on A.

牛顿第一定律：物体在没有净外力作用时保持静止或匀速直线运动。第二定律：作用在物体上的合力等于动量的变化率；对于质量恒定的情况，F = ma。第三定律：如果物体 A 对物体 B 施加一个力，那么 B 会对 A 施加一个大小相等、方向相反的力。

When analysing forces, draw a clear free-body diagram, resolve forces along perpendicular axes, and apply ΣF = ma in each direction. In connected particle problems, treat each particle separately and use the same magnitude of tension and acceleration. In IB and WJEC, pulleys and slopes are common contexts.

分析受力时，要画出清晰的受力图，沿垂直轴分解力，并在各方向应用 ΣF = ma。在连接体问题中，隔离每个质点处理，并使用相同的拉力大小和加速度。IB 和 WJEC 中常出现滑轮和斜面情境。

6. Friction and Inclined Planes | 摩擦与斜面

Friction is modelled as F ≤ μR for static friction and F = μR for kinetic friction, where R is the normal contact force and μ is the coefficient of friction. On

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IB & WJEC Maths: Mechanics Key Points Review | IB WJEC 数学：力学 考点精讲