Further Mechanics 1: Comprehensive Key Concepts | 进阶力学1：知识点精讲

📚 Further Mechanics 1: Comprehensive Key Concepts | 进阶力学1：知识点精讲

This article provides a thorough revision of the key topics in Further Mechanics 1, a module commonly studied in A Level Further Mathematics. We cover momentum and impulse, work, energy and power, elastic strings and springs, elastic collisions, centres of mass, and further kinematics with vectors and calculus. Each concept is explained with essential formulas and practical applications to help you master the subject.

本文全面复习A Level进阶数学中常见模块Further Mechanics 1的核心知识点，涵盖动量与冲量、功、能和功率、弹性绳与弹簧、弹性碰撞、质心以及基于向量和微积分的进阶运动学。每个概念都配有关键公式和实际应用，帮助你掌握这门课程。

1. Momentum and Impulse | 动量与冲量

Momentum is defined as the product of mass and velocity: p = m v. It is a vector quantity measured in kg m s⁻¹. The direction of the momentum vector is the same as the direction of the velocity.

动量定义为质量与速度的乘积：p = m v。它是一个矢量，单位为 kg·m·s⁻¹，方向与速度方向相同。

p = m v

The impulse of a force is the product of the force and the time interval during which it acts: I = F Δt. The impulse–momentum principle states that impulse equals the change in momentum: I = m v − m u.

力的冲量是力与作用时间的乘积：I = F Δt。冲量–动量定理指出，冲量等于动量的变化量：I = m v − m u。

I = F Δt = m v − m u

When no external forces act, the total momentum of a system is conserved. For two objects colliding along a straight line, m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂, where u₁ and u₂ are velocities before impact, and v₁, v₂ are velocities after impact.

当无外力作用时，系统的总动量守恒。对于沿直线运动的两个物体发生碰撞，有 m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂，其中 u₁、u₂ 为碰撞前速度，v₁、v₂ 为碰撞后速度。

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

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2. Newton’s Law of Restitution | 牛顿恢复定律

The coefficient of restitution e measures the elasticity of a collision. For a direct impact of two bodies moving in the same straight line, assuming positive direction to the right and u₁ > u₂, it is defined as the ratio of relative speed after collision to relative speed before collision.

恢复系数 e 衡量碰撞的弹性程度。对于沿同一直线运动的两个物体的正面碰撞，取向右为正方向并假设 u₁ > u₂，其定义为碰撞后相对速度与碰撞前相对速度之比。

e = (v₂ − v₁) / (u₁ − u₂)

The value of e lies between 0 and 1. A perfectly elastic collision has e = 1 and conserves kinetic energy. A perfectly inelastic collision has e = 0; the bodies coalesce and move with a common velocity.

e 的取值范围在 0 到 1 之间。完全弹性碰撞 e = 1，动能守恒。完全非弹性碰撞 e = 0，物体粘在一起并获得共同速度。

For a collision with a fixed smooth wall, if an object hits the wall with speed u perpendicularly and rebounds with speed v, then e = v / u and v = e u. The loss of kinetic energy is ½ m u² (1 − e²).

对于与固定光滑墙壁的碰撞，若物体以速度 u 垂直撞向墙壁并以速度 v 反弹，则 e = v / u，v = e u。动能损失为 ½ m u² (1 − e²)。

v = e u （方向相反记为 v = −e u）

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3. Direct Collisions and Conservation of Momentum | 正面碰撞与动量守恒

To solve collision problems involving two particles, write the conservation of momentum equation together with the restitution equation. These two equations can be solved simultaneously to find the unknown velocities after impact.

求解涉及两个质点的碰撞问题时，同时列出动量守恒方程和恢复系数方程。联立这两个方程可解出碰撞后的未知速度。

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂
e = (v₂ − v₁) / (u₁ − u₂)

In problems where a particle collides with a wall and then with another particle, treat each collision separately, applying the restitution and conservation laws sequentially. Always define a clear positive direction to avoid sign errors.

在一个质点先与墙壁碰撞再与另一质点碰撞的问题中，应分别处理每一次碰撞，依次应用恢复定律和守恒定律。务必明确正方向，以避免符号错误。

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4. Work, Energy and Power | 功、能量与功率

Work done by a constant force is W = F d cos θ, where θ is the angle between the force and the displacement. If the force is parallel to the displacement, W = F d.

恒定力做功为 W = F d cos θ，θ 为力与位移之间的夹角。若力与位移平行，则 W = F d。

W = F d cos θ

The work–energy principle states that the net work done on a particle equals its change in kinetic energy: W = ½ m v² − ½ m u².

动能定理表明，合力对质点做的功等于质点动能的变化量：W = ½ m v² − ½ m u²。

W = ½ m v² − ½ m u²

Gravitational potential energy (GPE) near Earth’s surface is GPE = m g h, where h is the vertical height above a reference level. Power is the rate of doing work: P = W / t = F v, where v is the velocity component in the direction of the force.

地表附近的重力势能为 GPE = m g h，h 为相对于参考平面的竖直高度。功率是做功的速率：P = W / t = F v，其中 v 是速度在力方向上的分量。

P = F v

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5. Elastic Strings and Springs | 弹性绳与弹簧

Hooke’s Law for an elastic string or spring: the tension T is proportional to the extension x, provided the elastic limit is not exceeded. The formula is T = (λ x) / l, where λ is the modulus of elasticity and l is the natural length.

弹性绳或弹簧的胡克定律：在弹性限度内，拉力 T 与伸长量 x 成正比。公式为 T = (λ x) / l，其中 λ 为弹性模量，l 为自然长度。

T = λ x / l

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