📚 IB & OCR Mathematics: Mechanics Essentials | IB 与 OCR 数学：力学考点精讲
Mechanics is a core applied module in both IB Mathematics (Analysis & Approaches / Applications & Interpretation) and OCR A Level Mathematics. It bridges pure mathematics with real‑world motion, forces and energy. In this guide we review the essential topics you must master – from SUVAT and Newton’s laws to moments, energy and variable acceleration – all explained with clear bilingual pairings.
力学是 IB 数学（分析与方法/应用与解释）以及 OCR A Level 数学的核心应用模块。它将纯数学与现实世界的运动、力和能量联系起来。本篇精讲梳理了必须掌握的关键考点——从 SUVAT 和牛顿定律到力矩、能量与变加速——全部采用清晰的中英双语对照讲解。
1. Kinematics and SUVAT Equations | 运动学与 SUVAT 方程
Kinematics describes motion without considering its causes. The five SUVAT equations apply when acceleration is constant along a straight line. They relate displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).
运动学描述运动而不追究其成因。五个 SUVAT 方程适用于匀加速直线运动，它们建立了位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 之间的关系。
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
s = vt − ½at²
Always list the known quantities and choose the equation that omits the unknown you are not asked for. If an object starts from rest, u = 0; if it comes to rest, v = 0.
先列出已知量，然后选择不含所求未知量的那个方程。若物体从静止开始，则 u = 0；若最终静止，则 v = 0。
In vertical motion, acceleration is g = 9.8 m s⁻² (downwards). Take a consistent sign convention.
在竖直运动中，加速度取 g = 9.8 m s⁻²（向下）。务必选取一致的符号规定。
| Quantity | Symbol | SI Unit |
|---|---|---|
| Displacement | s | m |
| Initial velocity | u | m s⁻¹ |
| Final velocity | v | m s⁻¹ |
| Acceleration | a | m s⁻² |
| Time | t | s |
2. Motion Graphs | 运动图像
Displacement–time (s–t) graphs: the gradient gives velocity. A straight line indicates constant velocity; a curve indicates acceleration.
位移–时间 (s–t) 图：斜率表示速度。直线代表匀速，曲线代表加速。
Velocity–time (v–t) graphs: the gradient gives acceleration, and the area under the graph gives displacement. A horizontal line means constant velocity.
速度–时间 (v–t) 图：斜率表示加速度，曲线下的面积表示位移。水平线代表匀速。
Acceleration–time (a–t) graphs: the area under the graph gives the change in velocity.
加速度–时间 (a–t) 图：曲线下的面积代表速度的变化量。
Always check whether the graph represents a one‑dimensional journey; use positive and negative signs for direction.
务必明确图像表示的是一维运动，用正负号表示方向。
3. Projectile Motion | 抛体运动
A projectile moves under constant vertical acceleration g and zero horizontal acceleration (air resistance negligible). The horizontal and vertical motions are independent.
抛体在恒定竖直加速度 g 和水平零加速度下运动（忽略空气阻力）。水平与竖直运动相互独立。
Horizontal: uₓ = u cos θ, vₓ = uₓ, sₓ = uₓ t. Vertical: uₙ = u sin θ, vₙ = uₙ − gt, sₙ = uₙ t − ½gt².
水平方向：uₓ = u cos θ，vₓ = uₓ，sₓ = uₓ t。竖直方向：uₙ = u sin θ，vₙ = uₙ − gt，sₙ = uₙ t − ½gt²。
The time of flight is found when the vertical displacement returns to its initial level (sₙ = 0). The maximum height occurs when vₙ = 0. The range is the horizontal distance travelled in that time.
飞行时间由竖直位移回到初始高度 (sₙ = 0) 求得。最大高度出现在 vₙ = 0 时。射程为这段时间内的水平位移。
Range = (u² sin 2θ)/g
Only use the range formula when launch and landing are at the same height.
射程公式仅当抛射与落地点等高时才可使用。
4. Newton’s Laws of Motion | 牛顿运动定律
First law: a body remains at rest or moves with constant velocity unless acted upon by a resultant force.
第一定律：除非受到合外力作用，物体将保持静止或匀速直线运动。
Second law: F = ma, where F is the resultant force in newtons, m is the mass in kg, and a is the acceleration in m s⁻².
第二定律：F = ma，F 为合外力（N），m 为质量（kg），a 为加速度（m s⁻²）。
Always resolve forces into components along the direction of motion and perpendicular to it. Use a clear force diagram.
始终将力沿运动方向和垂直运动方向分解。画出清晰的受力图。
Weight always acts vertically downwards (W = mg). Normal reaction is perpendicular to the contact surface. Tension in a light inextensible string is constant throughout its length.
重力始终竖直向下 (W = mg)。法向反力垂直于接触面。轻质不可伸长绳中的张力处处相等。
5. Connected Particles | 连接体问题
For two or more particles connected by a light inextensible string over a smooth pulley, treat each particle separately and apply F = ma. The acceleration magnitude is the same for all connected particles.
对于由轻质不可伸长绳跨过光滑滑轮连接的多个物体，需逐个隔离分析并应用 F = ma。所有连接体的加速度大小相等。
For a system on a smooth horizontal table with a hanging mass, use the overall equation: net accelerating force = total mass × acceleration.
对于水平光滑桌面连接悬挂重物的系统，可用整体方程：净加速力 = 总质量 × 加速度。
a = (m₂g) / (m₁ + m₂)
Always check which direction each mass moves and assign a positive direction consistently.
务必检查每个物体的运动方向，并一致规定正方向。
6. Momentum and Impulse | 动量与冲量
Momentum p = mv (kg m s⁻¹). It is a vector quantity. In a closed system, total momentum is conserved during collisions and explosions.
动量 p = mv (kg m s⁻¹)，是矢量。在封闭系统中，碰撞与爆炸前后总动量守恒。
Impulse = force × time = change in momentum: I = F t = Δp = m(v − u).
冲量 = 力 × 时间 = 动量的变化量：I = F t = Δp = m(v − u)。
For oblique collisions, resolve vectors and apply conservation separately in perpendicular directions.
对于斜碰撞，按垂直方向分解矢量并分别应用守恒定律。
The coefficient of restitution e = speed of separation / speed of approach, used when appropriate (OCR/IB questions may ask for impact analysis).
恢复系数 e = 分离速度 / 接近速度，在相关题目中会用到（OCR/IB 可能考查碰撞分析）。
7. Work, Energy and Power | 功、能与功率
Work done by a constant force: W = Fd cos θ, where θ is the angle between the force and the direction of motion. Unit: joule (J).
恒力做功：W = Fd cos θ，θ 为力与位移方向之间的夹角。单位：焦耳 (J)。
Kinetic energy: KE = ½mv². Gravitational potential energy: GPE = mgh (near Earth’s surface).
动能：KE = ½mv²。重力势能：GPE = mgh（地球表面附近）。
The work–energy principle: net work done = change in kinetic energy. If only conservative forces act, mechanical energy is conserved.
功能原理：合外力做功 = 动能的变化量。若只有保守力做功，机械能守恒。
Power is the rate of doing work: P = W/t. For a vehicle moving at constant speed, P = Fv.
功率是做功的快慢：P = W/t。对于匀速运动的车辆，P = Fv。
8. Moments and Equilibrium | 力矩与平衡
The moment of a force about a point = force × perpendicular distance from the point to the line of action. Unit: N m.
力对某点的力矩 = 力 × 该点到力作用线的垂直距离。单位：N m。
For a body in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot (principle of moments). The resultant force in any direction is zero.
物体平衡时，对任意支点的顺时针力矩之和等于逆时针力矩之和（力矩原理）；任意方向上合力为零。
Always draw all forces acting on the rigid body, including weight acting at the centre of mass, reactions at pivots and tension/contact forces.
始终画出刚体上所有作用力，包括作用在质心的重力、支点反力和张力/接触力。
9. Friction on Inclined Planes | 斜面与摩擦
When resolving forces on an inclined plane, align your axes parallel and perpendicular to the slope. The weight component along the plane is mg sin θ; perpendicular to the plane is mg cos θ.
分析斜面受力时，沿斜面与垂直斜面建立坐标轴。重力沿斜面的分量为 mg sin θ，垂直斜面为 mg cos θ。
Friction F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. The direction of friction always opposes motion or the tendency to move.
摩擦力 F ≤ μR，μ 为摩擦系数，R 为法向反力。摩擦力的方向始终与运动或运动趋势相反。
If a particle is in limiting equilibrium, F = μR. On a rough slope, use the condition of equilibrium or apply Newton’s second law.
若物体处于极限平衡状态，F = μR。在粗糙斜面上，运用平衡条件或牛顿第二定律。
10. Variable Acceleration and Calculus | 变加速与微积分
When acceleration is not constant, use calculus. If displacement s(t), velocity v(t) and acceleration a(t) are functions of time, then v = ds/dt and a = dv/dt = d²s/dt².
当加速度不恒定时，需使用微积分。若位移 s(t)、速度 v(t) 和加速度 a(t) 均为时间函数，则 v = ds/dt，a = dv/dt = d²s/dt²。
Conversely, s = ∫v dt and v = ∫a dt. Do not forget the constant of integration; use initial conditions to find its value.
反之，s = ∫v dt，v = ∫a dt。不要忘记积分常数，需利用初始条件确定其值。
If acceleration is given as a function of displacement, use a = v (dv/ds) to set up a differential equation that can be solved by separating variables.
若加速度表示为位移的函数，使用 a = v (dv/ds) 建立微分方程，并通过分离变量法求解。
IB and OCR papers frequently test these calculus methods with kinematics of a particle moving in a straight line.
IB 和 OCR 试卷经常通过直线运动粒子的运动学来考查这些微积分方法。
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