📚 IGCSE CIE Maths: Mechanics Key Points | IGCSE CIE 数学：力学考点精讲
Mechanics in the IGCSE Additional Mathematics syllabus (CIE 0606) covers the fundamental study of motion, forces, and energy. Mastering these concepts requires a clear understanding of kinematic equations, Newton’s laws, momentum, and energy principles. This article summarises every essential topic you need for the exam, with worked examples and key formulae.
IGCSE 附加数学 (CIE 0606) 中的力学部分涵盖运动、力与能量的基本研究。掌握这些概念需要清晰理解运动学方程、牛顿定律、动量与能量原理。本文总结了你备考所需的所有核心主题，配有范例和关键公式。
1. Constant Acceleration Formulae | 匀加速运动公式
The four SUVAT equations describe motion in a straight line with constant acceleration. They link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). You must memorise and become fluent in applying them.
四个 SUVAT 方程描述匀加速直线运动，它们联结了位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)。你必须熟记并能熟练应用它们。
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
Always list the known quantities and the unknown before selecting an equation. The positive direction must be defined consistently.
在选择方程之前，务必列出已知量和未知量。必须一致地定义正方向。
2. Motion under Gravity | 重力下的运动
When an object moves vertically under gravity, the acceleration is g = 9.8 m/s² downwards (often taken as 10 m/s² for simplicity). The SUVAT equations still apply, with a = -g if upward is positive, or a = +g if downward is positive.
当物体在重力作用下竖直运动时，加速度为向下的 g = 9.8 m/s² (为简化常取 10 m/s²)。SUVAT 方程仍然适用，若向上为正则 a = -g，向下为正则 a = +g。
For an object projected upwards, the velocity at the highest point is zero. The time to reach maximum height is u/g, and the maximum height is u²/(2g).
对于上抛物体，最高点速度为零。到达最大高度的时间为 u/g，最大高度为 u²/(2g)。
3. Velocity-Time Graphs | 速度-时间图
A velocity-time graph shows how velocity changes with time. The gradient gives acceleration, and the area under the graph gives displacement. A straight line implies constant acceleration; a horizontal line indicates constant velocity.
速度-时间图展示速度如何随时间变化。斜率表示加速度，图线下面积表示位移。直线表示匀加速；水平线表示匀速。
Distinguish carefully between displacement (area, considering signs) and distance (total area, treating all areas as positive).
仔细区分位移（带符号的面积）和路程（所有面积取正的总和）。
4. Force, Mass, and Acceleration | 力、质量与加速度
Newton’s Second Law states that the resultant force acting on an object is equal to the product of its mass and acceleration: F = ma. Force is measured in newtons (N), mass in kilograms (kg), and acceleration in m/s².
牛顿第二定律指出，作用在物体上的合力等于其质量与加速度的乘积：F = ma。力以牛顿 (N) 为单位，质量以千克 (kg) 为单位，加速度以 m/s² 为单位。
Always resolve forces in the direction of motion or perpendicular to it. Use free-body diagrams to identify all forces acting on an object.
始终沿运动方向或垂直于运动方向分解力。运用受力图识别作用在物体上的所有力。
5. Weight and Tension | 重量与张力
Weight is the gravitational force on a mass: W = mg. Tension is the pulling force transmitted through a string, cable, or rod. In equilibrium, the tension is constant throughout a light inextensible string passing over a smooth pulley.
重量是作用在质量上的引力：W = mg。张力是通过绳子、缆绳或杆传递的拉力。在平衡状态下，经过光滑滑轮的轻质不可伸长绳中张力处处相等。
When solving pulley problems, draw separate diagrams for each mass and apply F = ma to each, linking them through the common tension and acceleration.
在求解滑轮问题时，为每个质量分别画图，对每个质量应用 F = ma，并通过共同的张力和加速度将它们联系起来。
6. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law: an object remains at rest or moves with constant velocity unless acted upon by a resultant external force. Newton’s Third Law: if object A exerts a force on object B, then B exerts an equal and opposite force on A.
牛顿第一定律：物体保持静止或匀速直线运动，除非受到净外力的作用。牛顿第三定律：若物体 A 对物体 B 施加力，则 B 对 A 施加大小相等、方向相反的力。
These principles are essential for analysing equilibrium and interaction forces, especially in lift problems and collisions.
这些原理对于分析平衡和相互作用力至关重要，尤其是在电梯问题和碰撞中。
7. Momentum and Impulse | 动量与冲量
Momentum is the product of mass and velocity: p = mv. It is a vector quantity measured in kg m/s. Impulse is the change in momentum, given by force multiplied by time: Impulse = Ft = mv – mu.
动量是质量与速度的乘积：p = mv。它是矢量，单位为 kg m/s。冲量是动量的变化，等于力乘以时间：冲量 = Ft = mv – mu。
The area under a force-time graph also represents impulse. In collisions, the impulse on each body is equal and opposite.
力-时间图下的面积也表示冲量。在碰撞中，每个物体受到的冲量大小相等、方向相反。
8. Conservation of Momentum | 动量守恒
In the absence of external forces, the total momentum of a system is conserved. For a collision or explosion, total momentum before equals total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
在没有外力的情况下，系统的总动量守恒。对于碰撞或爆炸，碰撞前总动量等于碰撞后总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
This principle is used to find unknown velocities after impacts. Always define a positive direction and treat velocities with appropriate signs.
这一原理用于求碰撞后未知的速度。务必规定正方向，并用适当的符号处理速度。
9. Work and Energy | 功与能量
Work is done when a force moves its point of application: Work = Force × distance moved in the direction of the force (Joules). Kinetic energy (KE) is ½mv² and potential energy (PE) is mgh.
当力使其作用点移动时做功：功 = 力 × 沿力方向移动的距离 (焦耳)。动能 (KE) 为 ½mv²，势能 (PE) 为 mgh。
The work-energy principle states that the net work done on an object equals its change in kinetic energy. In systems with no friction, mechanical energy is conserved.
功能原理指出，对物体做的净功等于其动能的变化。在无摩擦的系统中，机械能守恒。
10. Power | 功率
Power is the rate of doing work or transferring energy: P = Work done / time taken (watts, W). For a constant force moving an object at constant speed v, power is also given by P = Fv.
功率是做功或传递能量的速率：P = 做功 / 所用时间 (瓦特, W)。对于恒力驱动物体以恒定速度 v 运动，功率也可表示为 P = Fv。
Students often use this relationship when dealing with cars moving up slopes or against resistances, linking engine power, driving force, and velocity.
学生在处理汽车上坡或克服阻力运动时经常用到此关系，将发动机功率、驱动力和速度联系起来。
11. Connected Particles | 连接体
Connected particles problems typically involve two masses linked by a light inextensible string over a smooth pulley or being towed. Both masses share the same magnitude of acceleration and the same tension in the string.
连接体问题通常涉及两个物体，由轻质不可伸长绳通过光滑滑轮连接，或被拖曳。两物体的加速度大小相同，绳中张力相同。
Write F = ma for each particle along the direction of motion. Solve the simultaneous equations to find acceleration and tension. For towed objects, include driving forces and resistances.
对每个物体沿运动方向写 F = ma。解联立方程求出加速度和张力。对于拖曳物体，要包含驱动力和阻力。
12. Friction and Limiting Equilibrium | 摩擦力与极限平衡
Friction opposes relative motion between surfaces. The maximum static friction is given by Fmax = μR, where μ is the coefficient of friction and R is the normal reaction force. When a body is on the point of moving, it is in limiting equilibrium, and friction equals μR.
摩擦力阻碍表面间的相对运动。最大静摩擦由 Fmax = μR 给出，其中 μ 是摩擦系数，R 是法向反作用力。当物体即将运动时，它处于极限平衡，摩擦力等于 μR。
Kinetic friction (during motion) is slightly less than limiting friction, but at IGCSE level it is often assumed equal to μR as well. Resolve forces perpendicular and parallel to the surface to find R and use F = ma.
动摩擦（运动时）略小于极限摩擦，但在 IGCSE 阶段常假设也等于 μR。分解垂直和平行于表面的力以求出 R，并运用 F = ma。
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