📚 Further Mechanics: Momentum, Collisions, Energy and Circular Motion | 进阶力学：动量、碰撞、能量与圆周运动
Further Mechanics extends the principles of classical mechanics met in the single Mathematics course. It covers momentum, collisions, circular motion, energy, centre of mass and simple harmonic motion. Mastery of these topics is essential for success in A Level Further Mathematics and for university study in physics or engineering.
进阶力学拓展了在数学单科课程中遇到的经典力学原理。它涵盖动量、碰撞、圆周运动、能量、质心和简谐运动。掌握这些主题对于在 A Level 进阶数学中取得成功以及未来大学物理或工程学习至关重要。
1. Momentum and Impulse | 动量与冲量
Linear momentum is defined as the product of mass and velocity. It is a vector quantity, so direction must be specified. The SI unit is kg m s⁻¹.
线性动量定义为质量与速度的乘积。它是一个矢量，因此必须指定方向。其国际单位是 kg m s⁻¹。
p = mv
Impulse is the change in momentum produced by a force acting over a time interval. Impulse can be calculated as force multiplied by time, or as the difference between final and initial momentum.
冲量是由力在一段时间内作用而产生的动量变化。冲量可以计算为力乘以时间，或者计算为末动量与初动量之差。
I = FΔt = mv - mu
In many exam problems, a force-time graph may be given, and the impulse is the area under the graph. This links impulse to variable forces as well.
在许多考试题目中，可能会给出力-时间图像，此时冲量就是图像下方的面积。这也将冲量与变力联系起来。
2. Conservation of Linear Momentum | 线性动量守恒
lyse collisions and explosions. For two bodies, the equation is m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
当系统不受外力作用时，总线性动量保持恒定。这一原理用于分析碰撞和爆炸。对于两个物体，方程为 m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
It is vital to choose a positive direction and assign signs to velocities accordingly. A common mistake is to mix directions when substituting values.
选择一个正方向并根据该方向给速度赋予正负号至关重要。一个常见错误是在代入数值时混淆方向。
3. Coefficient of Restitution and Collisions | 恢复系数与碰撞
Newton's experimental law of restitution defines the coefficient e as the ratio of the speed of separation to the speed of approach along the line of impact. Its value lies between 0 and 1 inclusive.
牛顿的实验恢复定律将恢复系数 e 定义为沿碰撞线分离速度与接近速度之比。其数值介于 0 和 1 之间（含端点）。
e = (v₂ - v₁) / (u₁ - u₂)
- e = 1: perfectly elastic collision; kinetic energy is conserved.
- 0 < e < 1: inelastic collision; some kinetic energy is lost.
- e = 0: perfectly inelastic collision; bodies coalesce and move together.
- e = 1：完全弹性碰撞；动能守恒。
- 0 < e < 1：非弹性碰撞；损失部分动能。
- e = 0：完全非弹性碰撞；物体粘在一起共同运动。
4. One-Dimensional Collision Problems | 一维碰撞问题
To solve a one-dimensional collision, write down the conservation of momentum equation and the restitution equation. Solve the two equations simultaneously to find the unknown velocities.
要解决一维碰撞问题，列出动量守恒方程和恢复系数方程。联立求解这两个方程以求出未知速度。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ and e = (v₂ - v₁) / (u₁ - u₂)
other, the final speed of separation cannot exceed the initial speed of approach if e ≤ 1.
始终检查你求出的末速度在物理上是否合理。例如，在一个物体追上另一个物体的碰撞中，如果 e ≤ 1，分离速度不会超过接近速度。
5. Oblique Collisions and Vector Methods | 斜碰撞与矢量方法
In oblique collisions, the impact is not along the line joining the centres. Resolve velocities into components along the common normal and along the common tangent. The tangential components remain unchanged, while the normal components obey the restitution law.
在斜碰撞中，冲击方向并不沿着两中心连线。将速度分解为沿公共法线和公共切线的分量。切向分量保持不变，而法向分量遵循恢复定律。
e = (v₂ₙ - v₁ₙ) / (u₁ₙ - u₂ₙ)
The momentum conservation is applied component-wise in the normal direction. A vector diagram or unit vectors can be used to organise the solution.
动量守恒在法线方向上按分量应用。可以使用矢量图或单位矢量来组织求解过程。
6. Work, Energy and Power | 功、能量与功率
os θ, where θ is the angle between the force and the displacement. Kinetic energy is ½mv² and gravitational potential energy is mgh.
恒力所做的功为 W = Fs cos θ，其中 θ 是力与位移之间的夹角。动能为 ½mv²，重力势能为 mgh。
W = Fs cos θ; KE = ½mv²; PE = mgh
ems involving non-conservative forces, use the work-energy principle: total work done equals the change in kinetic energy.
功率是做功的速率。对于作用在速度为 v 的物体上的恒力，功率为 P = Fv。在涉及非保守力的问题中，使用功能原理：总功等于动能的变化量。
P = Fv; Wtotal = ΔKE
7. Circular Motion: Angular Speed and Acceleration | 圆周运动：角速度与加速度
speed is related by v = rω. The acceleration towards the centre has magnitude v²/r or rω².
对于在半径为 r 的圆周上运动的质点，角速度 ω 是角度每秒的变化率（弧度制）。线速度与角速度的关系为 v = rω。指向圆心的加速度大小为 v²/r 或 rω²。
v = rω; a = v²/r = rω²
not a new force but the resultant of existing forces such as tension, friction or gravity.
指向圆心的合力称为向心力，其大小为 F = mv²/r = mrω²。这不是一种新的力，而是现有力（如张力、摩擦力或重力）的合力。
F = mv²/r = mrω²
8. Horizontal Circular Motion | 水平圆周运动
vertical. Resolving vertically gives T cos θ = mg, and horizontally gives T sin θ = mv²/r.
在水平圆周运动中，一个常见的例子是圆锥摆。一个小球系在绳上，在水平面内做圆周运动，绳与竖直方向成角度 θ。竖直方向分解给出 T cos θ = mg，水平方向分解给出 T sin θ = mv²/r。
T cos θ = mg; T sin θ = mv²/r
Another typical problem is a car rounding a banked curve or a particle on a rough horizontal turntable. The horizontal component of the normal reaction or friction provides the centripetal force.
另一个典型问题是汽车在倾斜弯道上转弯，或质点在粗糙水平转盘上运动。法向反力的水平分量或摩擦力提供向心力。
9. Vertical Circular Motion | 竖直圆周运动
At the lowest point, tension minus weight provides the centripetal force: T - mg = mv²/r.
竖直圆周运动更为复杂，因为在重力作用下速度大小和方向都发生变化。在最高点，向心力由张力加重力提供：T + mg = mv²/r。在最低点，张力减重力提供向心力：T - mg = mv²/r。
最高点 T + mg = mv²/r; 最低点 T - mg = mv²/r
For a particle attached to a string, the minimum speed at the top to keep the string taut is √gr, achieved when
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图