📚 Momentum and Energy Conservation in A-Level Mechanics: Core Concepts and Exam Skills | A-Level 物理力学动量与能量守恒核心考点
Momentum and energy are the two most dependable problem-solving tools in A-Level mechanics. Almost every collision, explosion, braking, rebound or swing question in Papers 1, 2 and 3 can be attacked with one of them, and many can only be solved cleanly when both are used together. This article rebuilds the topic from first principles: the definition and vector nature of momentum, impulse and the impulse-momentum theorem, the conditions for conservation of momentum, the classification of collisions, work and the work-energy theorem, gravitational potential energy, mechanical energy conservation, the coefficient of restitution, power and efficiency, the standard practical work, and finally the exam habits that convert understanding into marks.
动量与能量是 A-Level 力学中最可靠的两件解题工具。试卷一、试卷二、试卷三里的碰撞、爆炸、刹车、回弹、摆动等题目，几乎都能用其中之一攻破，而不少题目只有把两者结合使用才能干净利落地解出。本文将从头重建这一专题：动量的定义与矢量性、冲量与冲量—动量定理、动量守恒的成立条件、碰撞的分类、功与动能定理、重力势能、机械能守恒、恢复系数、功率与效率、标准实验操作，最后是把你理解转化为分数的应试习惯。
1. Why These Two Principles Dominate Mechanics Papers | 为什么这两大原理主导力学试卷
In A-Level physics, forces tell you what happens at an instant, but momentum and energy tell you what happens over an interaction. That is why examiners keep returning to them: a question about a car crash, a trolley collision or a falling ball can be answered without ever knowing the detailed force-time history. Momentum handles the vector side of the interaction; energy handles the scalar side. Strong candidates learn to ask first: "which quantity is conserved here, and under what condition?"
在 A-Level 物理中，力描述的是某一瞬间的处境，而动量与能量描述的是整个相互作用过程的结果。这正是考官反复考它们的原因：一道关于撞车、小车碰撞或落体的问题，不需要知道力随时间变化的细节也能解出。动量负责过程里的矢量一面，能量负责标量一面。优秀考生的第一反应应是先问：这里哪个物理量守恒？在什么条件下守恒？
The two principles are not interchangeable. Momentum is conserved in every collision and explosion as long as the resultant external force is negligible, but kinetic energy is only conserved in elastic interactions. Energy in total is always conserved, yet it leaks into thermal, sound and deformation energy in inelastic events.
这两个原理不能互相替代。只要合外力可忽略，动量在任何碰撞与爆炸中都守恒；但动能只在弹性相互作用中守恒。能量总量永远守恒，只是在非弹性过程中会流向热能、声能与形变能。
- Momentum route — use when masses and velocities before and after an interaction are involved, or when a force acts for a very short time. / 动量路线——涉及相互作用前后的质量与速度，或力作用时间极短时使用。
- Energy route — use when heights, speeds, work done against friction or energy loss are involved. / 能量路线——涉及高度、速率、克服摩擦做功或能量损失时使用。
- Combined route — use in elastic collisions, where both equations must be solved simultaneously. / 组合路线——弹性碰撞中两式需联立求解。
2. Momentum: Definition, Vector Nature and Units | 动量：定义、矢量性与单位
Linear momentum is defined as the product of mass and velocity. It is a vector quantity, so it always carries a direction, and in one-dimensional problems that direction is expressed through a sign convention. The SI unit is the kilogram metre per second, which is also equivalent to the newton second.
线动量定义为质量与速度的乘积。它是矢量，因此永远带有方向；在一维问题中，方向通过正负号约定来表达。国际单位是千克米每秒，它等价于牛顿秒。
p = m v
A 2.0 kg ball moving at 3.0 m s⁻¹ to the right has momentum 6.0 kg m s⁻¹ to the right; the same ball moving left has momentum −6.0 kg m s⁻¹ if right is taken as positive. Choosing the positive direction before writing any numbers is the single most effective habit for avoiding sign errors in the whole topic. Write it down on the page, not just in your head.
一个 2.0 kg 的小球以 3.0 m s⁻¹ 向右运动，动量为 6.0 kg m s⁻¹ 向右；若规定向右为正，则同一个小球向左运动的动量为 −6.0 kg m s⁻¹。在写下任何数字之前先规定正方向，是整个专题中最有效的防错号习惯。请把它写在卷面上，而不是只在脑中默念。
Newton' cause it also holds when mass changes, such as for a rocket burning fuel or a conveyor belt being loaded.
牛顿第二定律可以写成动量形式：合外力等于动量变化率。这一形式比 F = ma 更基本，因为它在质量变化时依然成立，例如火箭燃烧燃料或传送带被装载的情形。
F = Δp / Δt = (m v − m u) / Δt
Do not confuse momentum with kinetic energy. Two bodies can have identical kinetic energy but different momenta, and identical momenta but different kinetic energies if their masses differ. Momentum is a vector proportional to v; kinetic energy is a scalar proportional to v², so doubling the speed quadruples the kinetic energy but only doubles the momentum.
不要把动量与动能混为一谈。两个物体的动能可以相同而动量不同；质量不同时，动量相同而动能也可以不同。动量是与 v 成正比的矢量，动能是与 v² 成正比的标量，所以速率加倍会让动能变为四倍，而动量只变为两倍。
3. Impulse and the Impulse-Momentum Theorem | 冲量与冲量—动量定理
Impulse is the product of a force and the time for which it acts. The impulse-momentum theorem states that the impulse of the resultant force equals the change in momentum. Because impulse is a vector, it points in the same direction as the change in momentum, not necessarily in the direction of motion.
冲量是力与作用时间的乘积。冲量—动量定理指出：合外力的冲量等于动量的变化量。由于冲量是矢量，它的方向与动量变化量方向相同，而不一定与运动方向相同。
J = F Δt = Δp = m v − m u
When the force varies with time, the impulse equals the area under the force-time graph. This is a standard examination skill: candidates are given a curved F-t graph and asked either to estimate the area by counting squares or to interpret why a longer contact time produces a lower peak force for the same impulse.
当力随时间变化时，冲量等于力—时间图像下的面积。这是标准考点：题目给出一条弯曲的 F–t 图像，要求用数格子法估算面积，或解释为什么在冲量相同时接触时间越长峰值力越小。
This is the physics behind every safety device you will be asked about. Crumple zones, airbags, seat belts, crash barriers and packaging all increase the contact time Δt. Since Δp is fixed by the change in momentum, a larger Δt means a smaller average force, which reduces injury and damage.
这正是所有安全装置背后的物理原理。溃缩区、安全气囊、安全带、防撞护栏与包装材料都在增大接触时间 Δt。由于 Δp 由动量变化决定是个定值，Δt 越大平均力就越小，从而减轻伤害与损坏。
- Airbag — increases the time of the impact so the peak force on the passenger is reduced. / 安全气囊——延长撞击时间，从而减小乘客受到的峰值力。
- Follow-through in sport — a longer contact time between bat and ball gives a larger change in momentum and hence a faster ball. / 运动中的随挥——球拍与球接触时间越长，动量变化越大，出球越快。
- Bending knees on landing — lengthens the deceleration time, reducing the average force on the joints. / 落地屈膝——延长减速时间，减小关节受到的平均力。
4. Conservation of Momentum: Conditions and Method | 动量守恒：条件与解题方法
The principle of conservation of momentum states that, for a system of interacting bodies, the total momentum remains constant provided no resultant external force acts on the system. Internal forces between the bodies are equal and opposite by Newton's third law, so they cancel in pairs and cannot change the total momentum of the system; they can only redistribute it.
动量守恒定律指出：对一个相互作用的系统，只要系统不受合外力作用，总动量保持不变。系统内部各物体之间的力按牛顿第三定律大小相等、方向相反，成对抵消，无法改变系统的总动量，只能重新分配动量。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
In practice, the condition is satisfied in two common situations. First, a genuinely isolated system, such as two trolleys on a level air track. Second, a system where external forces such as weight and friction are negligible compared with the very large internal forces during a brief interaction, which is exactly why the principle works so well for collisions and explosions lasting milliseconds.
实践中，条件通常在两类情形下满足。第一类是真正孤立的系统，例如水平气垫导轨上的两辆小车。第二类是相互作用时间极短、内部力极大，使重力、摩擦等外力相比之下可忽略的系统——这正是动量守恒在持续仅毫秒级的碰撞与爆炸中如此好用的原因。
The standard method never changes. Choose and state a positive direction; write down the momentum of each body before the interaction; write the total momentum after; equate the two; solve. If the answer comes out negative, that is not an error but information: the body moves in the direction you defined as negative.
标准解法永远不变：选定并写明正方向；写出相互作用前每个物体的动量；写出相互作用后的总动量；令两者相等；求解。若结果出现负号，那不是错误而是信息：该物体朝你规定为负的方向运动。
- Step 1 — define positive direction and state it in words. / 第一步——规定正方向并用文字写明。
- Step 2 — list masses and velocities before, with correct signs. / 第二步——列出各物体相互作用前的质量与速度，注意正负号。
- Step 3 — list masses and velocities after, using a single unknown where possible. / 第三步——列出相互作用后的量，尽量只保留一个未知数。
- Step 4 — equate total momentum before and after, then solve and check units. / 第四步——令前后总动量相等，求解并检查单位。
5. Elastic, Inelastic and Perfectly Inelastic Collisions | 弹性、非弹性与完全非弹性碰撞
Collisions are classified by what happens to kinetic energy, not by what happens to momentum. Momentum is conserved in all three types; kinetic energy is conserved only in the elastic case. An inelastic collision loses kinetic energy, typically converted into thermal energy, sound and permanent deformation.
碰撞按动能的变化分类，而不是按动量分类。三类碰撞中动量都守恒；只有弹性碰撞中动能守恒。非弹性碰撞会损失动能，通常转化为热能、声能与永久形变。
| Type / 类型 | Momentum / 动量 | Kinetic energy / 动能 | Example / 例子 |
| Elastic / 弹性 | Conserved / 守恒 | Conserved / 守恒 | Gas molecule collisions / 气体分子碰撞 |
| Inelastic / 非弹性 | Conserved / 守恒 | Decreases / 减少 | Car crash / 车祸碰撞 |
| Perfectly inelastic / 完全非弹性 | Conserved / 守恒 | Maximum loss for given p / 给定动量下损失最大 | Two trolleys sticking together / 两车粘在一起 |
In a perfectly inelastic collision the bodies move with a common final velocity, which reduces the momentum equation to (m₁ + m₂) v. This makes the algebra simpler but the energy analysis more interesting: the kinetic energy lost is exactly the amount that must appear as thermal energy in the coupling.
在完全非弹性碰撞中，两物体碰后以共同速度运动，动量方程简化为 (m₁ + m₂) v。这使代数更简单，但能量分析更有意思：损失的动能恰好等于必须出现在连接处的热能。
For elastic collisions, a useful shortcut is that the relative speed of approach equals the relative speed of separation. For a body of mass m₁ hitting a stationary body of mass m₂ head-on elastically, the final velocities are given below — a result worth memorising for multiple-choice speed.
对弹性碰撞有一个有用的捷径：接近时的相对速率等于分离时的相对速率。质量为 m₁ 的物体正碰静止的质量为 m₂ 的物体且完全弹性时，末速度如下——这个结果值得记住，可加快选择题速度。
v₁ = (m₁ − m₂) u₁ / (m₁ + m₂) v₂ = 2 m₁ u₁ / (m₁ + m₂)
6. Explosions, Recoil and Momentum in Two Dimensions | 爆炸、反冲与二维动量
An explosion is the reverse of a perfectly inelastic collision. The total momentum before the event is zero, so the total momentum afterwards must also be zero. Every fragment therefore carries momentum that is cancelled by the momentum of the others, taken as vectors.
爆炸是完全非弹性碰撞的逆过程。爆炸前的总动量为零，所以爆炸后的总动量也必须是零。因此每一块碎片携带的动量，都会被其他碎片以矢量方式抵消掉。
The classic examples are a rifle recoiling when fired, a rocket ejecting exhaust gases, and an unstable nucleus emitting an alpha particle. In each case, writing the initial momentum as zero immediately gives a relationship between the masses and speeds of the two products.
经典例子包括步枪射击时的后坐、火箭喷出燃气，以及不稳定原子核放出 α 粒子。每一情形下，把初动量写为零，立刻就能得到两个产物质量与速率之间的关系。
0 = m₁ v₁ + m₂ v₂ → m₁ v₁ = − m₂ v₂
For two-dimensional problems, resolve momentum into two perpendicular directions and apply conservation separately in each. The vector triangle of momenta is a clean way to see the geometry: the initial momentum vector equals the vector sum of the final momentum vectors. Do not simply add magnitudes, because momentum is a vector and the angle matters.
在二维问题中，把动量分解到两个互相垂直的方向，并在每个方向上分别应用守恒。动量矢量三角形是看清几何关系的干净方法：初动量矢量等于末动量矢量的矢量和。不要简单相加大小，因为动量是矢量，夹角会影响结果。
7. Work, Kinetic Energy and the Work-Energy Theorem | 功、动能与动能定理
Work is done when a force moves its point of application through a distance. Only the component of the force along the displacement does work, which is why the cosine factor appears. Work is a scalar and is measured in joules.
当力使其作用点发生位移时，就做了功。只有沿位移方向的分力做功，这就是余弦因子出现的原因。功是标量，单位为焦耳。
W = F s cos θ
When the force varies with displacement, the work done equals the area under the force-displacement graph. This mirrors the impulse idea: the area under an F-t graph gives impulse, and the area under an F-x graph gives work done.
当力随位移变化时，做功等于力—位移图像下的面积。这与冲量的思路相对应：F–t 图像下的面积是冲量，F–x 图像下的面积是做功。
k done on a body equals its change in kinetic energy. The theorem follows directly from F = ma combined with v² = u² + 2as when the force is constant.
物体的动能定义为其质量与速率平方乘积的一半；动能定理指出，对物体做的合力功等于其动能变化量。当力恒定时，该定理可由 F = ma 结合 v² = u² + 2as 直接推出。
Eₖ = ½ m v² Wnet = ΔEₖ = ½ m v² − ½ m u²
f d = ½ m u² and d = u² / (2 f / m). The result confirms the familiar fact that braking distance is proportional to the square of the speed, not to the speed itself.
这是解决刹车距离问题最稳妥的路线。若质量为 m 的汽车以速率 u 运动直至停下，摩擦力 f 作用距离 d，则 f d = ½ m u²，得 d = u² / (2 f / m)。这一结果印证了熟悉的结论：刹车距离与速率的平方成正比，而非与速率成正比。
8. Gravitational Potential Energy and Mechanical Energy Conservation | 重力势能与机械能守恒
Gravitational potential energy near the Earth's surface is the energy a body possesses because of its height above a reference level. Only changes in potential energy are physically meaningful, so the reference level may be chosen freely; the value of ΔEₚ is the same whatever zero you pick.
地球表面附近的重力势能，是物体因相对某参考高度而具有的能量。只有势能的变化量才有物理意义，所以参考面可自由选取；无论零点选在哪，ΔEₚ 的数值都相同。
ΔEₚ = m g Δh
The principle of conservation of mechanical energy states that, when the only forces doing work are conservative, the sum of kinetic and potential energy stays constant. A swinging pendulum, a roller coaster on a frictionless track and a ball rolling down a smooth ramp are the standard examples.
机械能守恒定律指出：当只有保守力做功时，动能与势能之和保持不变。单摆摆动、无摩擦轨道上的过山车、沿光滑斜面滚下的小球都是标准例子。
½ m u² + m g h₁ = ½ m v² + m g h₂
When friction or air resistance is present, mechanical energy is not conserved, but total energy still is. The missing mechanical energy becomes thermal energy, so the extension of the equation is to add the work done against resistive forces to the final side.
当存在摩擦或空气阻力时，机械能不守恒，但总能量仍然守恒。消失的机械能变成了热能，因此方程的扩展形式是把克服阻力做的功加到末状态一侧。
½ m u² + m g h₁ = ½ m v² + m g h₂ + Wfriction
Notice the difference in how the two principles behave. Momentum is conserved even when mechanical energy is lost, but energy conservation is universal and never fails. Examiners deliberately design questions where using the wrong one leads to a contradiction, so always check the list of forces acting.
请注意两个原理行为方式的差别：即使机械能损失，动量仍可守恒；而能量守恒是普适的，永不失效。考官会有意设计用错原理就会自相矛盾的题目，因此务必先核对受力的完整清单。
9. Energy Accounting in Collisions and the Coefficient of Restitution | 碰撞中的能量核算与恢复系数
After applying conservation of momentum, a colliding system will generally have a different total kinetic energy. The kinetic energy lost is found by subtracting the total kinetic energy after the interaction from that before. Always calculate the total for the system, not for a single body, since one body can gain kinetic energy while the total falls.
在应用动量守恒之后，碰撞系统的总动能通常会发生变化。损失的动能等于相互作用前的总动能减去相互作用后的总动能。务必计算整个系统的总量而非单个物体，因为完全可能出现某个物体动能增加而系统总量下降的情况。
ΔEₖ = (½ m₁u₁² + ½ m₂u₂²) − (½ m₁v₁² + ½ m₂v₂²)
The coefficient of restitution measures the elasticity of a collision. It is defined as the relative speed of separation divided by the relative speed of approach, and its value lies between 0 and 1. A value of 1 describes a perfectly elastic collision; a value of 0 describes a perfectly inelastic one
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