📚 Modelling Collisions: Momentum, Impulse and Restitution | 碰撞建模：动量、冲量与恢复系数
Collisions appear across the A-level syllabus: car crashes, snooker balls, alpha particles striking nuclei, and gas molecules bouncing off container walls. We cannot easily track every tiny deformation, sound wave and thermal effect during impact, so physicists build a collision model. The model assumes a very short interaction, large internal forces, small external impulse, and a small set of measurable quantities such as mass, velocity, momentum and kinetic energy.
碰撞在 A-level 大纲中随处可见：汽车碰撞、斯诺克球、α 粒子撞击原子核，以及气体分子在容器壁上的反弹。我们很难追踪碰撞过程中每一个微小的形变、声波和热效应，因此物理学家建立了一种碰撞模型。该模型假设相互作用时间极短、内力很大、外力冲量很小，并且只使用少量可测量量，如质量、速度、动量和动能。
1. What Is a Collision Model? | 什么是碰撞模型
A collision is a short interaction between two or more bodies in which large contact forces act for a brief time. During the impact, external forces such as friction, air resistance and gravity may still be present, but their effects over the very short collision time are usually much smaller than the internal contact impulses. This allows us to treat the colliding bodies as an isolated system for the duration of the impact.
碰撞是两个或多个物体之间的一种短时间相互作用，在此期间有较大的接触力作用。碰撞过程中，摩擦力、空气阻力和重力等外力可能仍然存在，但在极短的碰撞时间内，它们产生的冲量通常远小于内部接触冲量。这使我们能够在碰撞持续时间内把相互碰撞的物体近似看作一个孤立系统。
The modelling process replaces complicated deformation, vibration and sound generation with idealised objects. Typical approximations include point masses, rigid bodies, smooth surfaces and instantaneous contact. These simplifications are not perfectly true, but they give excellent predictions for final velocities and allow conservation laws to be applied cleanly.
建模过程用理想化物体来代替复杂的形变、振动和声音产生。常见的近似包括质点、刚体、光滑表面和瞬时接触。这些简化并非完全真实，但它们能很好地预测末速度，并使守恒定律可以被清晰地应用。
2. Momentum as the Central Quantity | 作为核心量的动量
Linear momentum is defined as the product of an object’s mass and its velocity. Momentum is a vector quantity, so its direction must be stated. The SI unit is kilogram metre per second, which is also equivalent to newton second.
动量被定义为物体质量与速度的乘积。动量是矢量，因此必须说明其方向。动量的国际单位是千克米每秒，也等价于牛顿秒。
p = m v
The symbol p represents momentum, m is mass and v is velocity. Since velocity changes sign when direction changes, momentum also changes sign. In collision problems, choosing a positive direction and assigning positive or negative signs to all velocities is an essential first step.
符号 p 表示动量，m 表示质量，v 表示速度。由于速度在方向改变时会改变符号，动量也会改变符号。在碰撞问题中，选定正方向并为所有速度赋予正号或负号是必不可少的第一步。
3. Conservation of Linear Momentum | 线动量守恒
For any system with no net external force, total linear momentum remains constant. During a collision, the internal forces between colliding bodies are equal and opposite by Newton’s third law, so they cannot change the total momentum of the system. If external impulse is negligible, the total momentum immediately before the collision equals the total momentum immediately after.
对于任何没有净外力的系统，总线动量保持不变。在碰撞过程中，碰撞物体之间的内力根据牛顿第三定律大小相等、方向相反，因此它们无法改变系统的总动量。如果外力冲量可以忽略，则碰撞前的总动量等于碰撞后的总动量。
For two bodies of masses m₁ and m₂, with initial velocities u₁ and u₂ and final velocities v₁ and v₂, conservation of momentum is written as:
对于质量为 m₁ 和 m₂ 的两个物体，初速度为 u₁ 和 u₂，末速度为 v₁ 和 v₂，动量守恒可写为：
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
This single vector equation applies to all collisions, whether elastic or inelastic. In one dimension it is a scalar equation with signs determined by the chosen positive direction. In two dimensions, it must be applied separately to perpendicular components.
这一矢量方程适用于所有碰撞，无论弹性还是非弹性。在一维情况下，它是一个由选定正方向决定符号的标量方程。在二维情况下，必须分别对垂直分量应用该方程。
4. Impulse and the Force-Time Graph | 冲量与力-时间图像
Impulse is the product of the average force acting on an object and the time for which it acts. It equals the change in momentum of the object. This relationship is derived from Newton’s second law in its most general form.
冲量是作用在物体上的平均力与该力作用时间的乘积。它等于物体动量的变化量。这一关系来源于牛顿第二定律最一般的形式。
J = F Δt = Δp = m v − m u
Here J is impulse, F is the average net force, Δt is the contact time, u is the initial velocity and v is the final velocity. Since impulse equals change in momentum, it has the same unit as momentum: N s or kg m s⁻¹.
这里 J 是冲量，F 是平均净力，Δt 是接触时间，u 是初速度，v 是末速度。由于冲量等于动量的变化量，它的单位与动量相同：N s 或 kg m s⁻¹。
In many collision problems, the force is not constant. The area under a force-time graph gives the impulse delivered to the body. The shape of the graph can therefore be used to estimate the maximum force, the contact time or the total change in momentum.
在许多碰撞问题中，力并不是恒定的。力-时间图像下方的面积给出了传递给物体的冲量。因此，可以利用图像形状来估计最大力、接触时间或总动量变化。
5. Newton’s Law of Restitution | 牛顿恢复定律
The coefficient of restitution, e, measures how bouncy a collision is. It is defined as the ratio of the relative speed of separation after the collision to the relative speed of approach before the collision.
恢复系数 e 衡量碰撞的弹性程度。它被定义为碰撞后分离的相对速度与碰撞前接近的相对速度之比。
e = relative speed of separation / relative speed of approach
For a head-on collision between two objects moving along the same straight line, taking the positive direction from object 1 towards object 2 and assuming object 1 is initially behind object 2, this becomes:
对于两个沿同一直线运动的物体发生正碰，取从物体 1 指向物体 2 为正方向，并假设物体 1 最初在物体 2 后面，则上式可写为：
v₂ − v₁ = e (u₁ − u₂)
The value of e is dimensionless and normally lies between 0 and 1. It depends on the materials involved: glass and steel may have e close to 1, while clay or putty has e close to 0. The coefficient of restitution is assumed constant for given materials and low impact speeds in the standard A-level model.
e 是无量纲量，通常介于 0 和 1 之间。它取决于碰撞材料：玻璃和钢的 e 值可能接近 1，而黏土或油灰的 e 值接近 0。在标准 A-level 模型中，假设在给定材料和较低碰撞速度下，恢复系数保持不变。
6. Elastic, Inelastic and Perfectly Inelastic Collisions | 弹性、非弹性与完全非弹性碰撞
Collisions are classified by whether kinetic energy is conserved. In a perfectly elastic collision, both momentum and total kinetic energy are conserved. In an inelastic collision, momentum is conserved but some kinetic energy is dissipated as heat, sound or permanent deformation. In a perfectly inelastic collision, the two bodies stick together after impact and the loss of kinetic energy is maximised for the given initial conditions.
碰撞根据动能是否守恒来分类。在完全弹性碰撞中，动量和总动能都守恒。在非弹性碰撞中，动量守恒，但部分动能以热、声或永久形变的形式被耗散。在完全非弹性碰撞中，两个物体在碰撞后粘在一起，对于给定的初始条件，动能损失达到最大。
| Collision type | Coefficient e | Kinetic energy | Behaviour |
|---|---|---|---|
| Perfectly elastic | e = 1 | Conserved | Bodies bounce apart |
| Inelastic | 0 < e < 1 | Some lost | Bodies deform, warm up, produce sound |
| Perfectly inelastic | e = 0 | Maximum loss | Bodies stick together |
Kinetic energy is a scalar, so energy calculations in one dimension are straightforward. However, total energy is always conserved in the wider sense: the ‘lost’ kinetic energy is transferred to internal energy of the bodies and the surroundings.
动能是标量，因此一维能量计算比较直接。然而，从更广泛的意义上说，总能量始终守恒：所谓“损失”的动能被转移到物体和周围环境的内能中。
7. Explosions as Reverse Collisions | 作为逆碰撞的爆炸
An explosion can be modelled as a collision run backwards in time. Initially, two or more parts move together as one object; after the explosion, they move apart with separate velocities. Momentum is still conserved because the internal forces from the explosion are equal and opposite.
爆炸可以建模为时间倒流的碰撞。最初，两个或多个部分作为一个整体一起运动；爆炸后，它们以不同的速度分开。动量仍然守恒，因为爆炸产生的内力大小相等、方向相反。
For a stationary object of mass M that splits into two fragments of masses m₁ and m₂, the conservation equation is:
对于质量为 M 的静止物体分裂为质量 m₁ 和 m₂ 的两个碎片，动量守恒方程为：
0 = m₁v₁ + m₂v₂
This implies that the two fragments move in opposite directions along the same line. Unlike an inelastic collision, an explosion increases the total kinetic energy of the system because stored chemical or elastic potential energy is converted into kinetic energy.
这意味着两个碎片沿同一直线向相反方向运动。与非弹性碰撞不同，爆炸会增加系统的总动能，因为储存的化学能或弹性势能转化为动能。
8. One-Dimensional Collision Calculations | 一维碰撞计算
A standard one-dimensional collision problem can be solved by using two equations: conservation of momentum and the restitution equation. If the collision is perfectly elastic, set e = 1. If the bodies stick together, set e = 0 and write v₁ = v₂ = v.
标准的一维碰撞问题可以用两个方程求解：动量守恒方程和恢复系数方程。如果碰撞完全弹性，令 e = 1。如果两物体粘在一起，令 e = 0，并写出 v₁ = v₂ = v。
The recommended method is to draw a clear labelled diagram, choose a positive direction, write all known velocities with correct signs, and then substitute into the two equations. Solving simultaneous equations gives the unknown velocities.
推荐的方法是画出清晰的标注图，选定正方向，写出所有已知速度及其正确符号，然后代入两个方程。求解联立方程即可得到未知速度。
For example, for a perfectly elastic head-on collision between two equal masses where one is initially at rest, the moving mass stops and the stationary mass moves off with the original velocity. This is a well-known result that can be verified from the equations.
例如，两个质量相等的物体发生完全弹性正碰，其中一个最初静止，则运动物体停止，静止物体以原来的速度运动。这是一个可以从方程中验证的著名结果。
9. Two-Dimensional Collision Models | 二维碰撞模型
When two bodies collide obliquely, their final velocities are not necessarily along the original line of motion. Momentum must then be conserved in two perpendicular directions, usually chosen as the x-axis and y-axis.
当两个物体斜碰时，它们的末速度不一定沿着原来的运动方向。此时动量必须在两个垂直方向上守恒，通常选择 x 轴和 y 轴。
The two momentum equations are:
两个动量方程为：
m₁u₁ₓ + m₂u₂ₓ = m₁v₁ₓ + m₂v₂ₓ
m₁u₁_y + m₂u₂_y = m₁v₁_y + m₂v₂_y
In oblique impacts, the coefficient of restitution is applied only along the line of impact, which is the line joining the centres of the two bodies at the instant of contact. The components perpendicular to the line of impact are unaffected if the surfaces are smooth.
在斜碰中，恢复系数仅用于碰撞线方向，即两物体接触瞬间球心连线的方向。如果表面光滑，垂直于碰撞线的分量不受影响。
This model is used in billiard-ball problems and in analysing particles scattered after hitting a target. It shows the importance of resolving velocities into components before applying conservation laws.
该模型用于台球问题以及粒子击中靶后散射的分析。它说明在应用守恒定律之前，将速度分解为分量是非常重要的。
10. Energy Accounting and Common Approximations | 能量核算与常见近似
The kinetic energy change in a collision is found by comparing the total kinetic energy before and after. The difference represents energy transformed into thermal energy, sound and permanent deformation. For a perfectly elastic collision, this difference is zero.
碰撞中动能的变化通过比较碰撞前后的总动能来确定。差值代表转化为热能、声能和永久形变的能量。对于完全弹性碰撞，该差值为零。
ΔKE = KE_after − KE_before
In a perfectly inelastic collision, the two masses move with a common final velocity. The common velocity is found from momentum conservation, and the kinetic energy loss can then be calculated directly.
在完全非弹性碰撞中，两个质量以共同末速度运动。共同速度由动量
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