📚 Momentum | 动量考点精讲
Momentum is one of the most frequently examined topics in CIE A-Level Physics, appearing in questions on collisions, explosions, impulse, and conservation laws. A deep understanding of momentum not only helps you solve quantitative problems but also strengthens your grasp of Newton’s laws and energy concepts.
动量是 CIE A-Level 物理中最常考的专题之一，在碰撞、爆炸、冲量以及守恒定律的题目中反复出现。透彻理解动量不仅能帮你解决定量计算，还能加深你对牛顿定律和能量概念的整体把握。
1. Defining Momentum | 动量的定义
Linear momentum (symbol p) is defined as the product of an object’s mass and its velocity. Because velocity is a vector, momentum is also a vector quantity — it has both size and direction. For an object of mass m travelling with velocity v, the momentum is:
线动量（符号 p）定义为物体质量与速度的乘积。因为速度是矢量，动量也是矢量 —— 既有大小也有方向。对于质量为 m、速度为 v 的物体，其动量为：
p = mv
The SI unit of momentum is kilogram metre per second (kg m s⁻¹). This is equivalent to the newton second (N s), since 1 N = 1 kg m s⁻² and therefore 1 N s = 1 kg m s⁻¹. The momentum vector always points in the same direction as the velocity vector.
动量的国际单位是千克米每秒 (kg m s⁻¹)，等同于牛顿秒 (N s)，因为 1 N = 1 kg m s⁻²，所以 1 N s = 1 kg m s⁻¹。动量矢量的方向始终与速度矢量的方向一致。
Momentum should not be confused with kinetic energy. Kinetic energy is a scalar that depends on speed only (½mv²), whereas momentum depends on velocity and thus carries directional information. Two identical cars moving at the same speed in opposite directions have equal kinetic energy but opposite momentum, so their total momentum is zero.
动量不可与动能混淆。动能是标量，仅决定于速率 (½mv²)，而动量的定义涉及速度，因此包含方向信息。两辆相同的小车以相同速率沿相反方向运动，动能相等，但动量方向相反，因此总动量为零。
2. Impulse and the Change in Momentum | 冲量与动量变化
Impulse (J) is the product of a constant net force and the time interval over which it acts. Impulse equals the change in momentum produced by that force:
冲量 (J) 是恒定的净力与其作用时间的乘积。冲量等于该力所引起的动量变化：
J = F Δt = Δp = m(v − u)
where u is the initial velocity and v is the final velocity. Like momentum, impulse is a vector. The direction of the impulse is the same as the direction of the net force. If the force is not constant, the impulse is the area under a force–time graph.
其中 u 为初速度，v 为末速度。与动量一样，冲量也是矢量。冲量的方向与净力方向相同。如果力不是恒定的，则冲量等于力–时间图下的面积。
In many exam questions, you are given the force and time and asked to find the change in speed. Rearranging Δp = FΔt to Δv = FΔt / m gives a direct route to the answer. Always pay attention to the signs: if you define one direction as positive, forces and velocities in the opposite direction must be written with a minus sign.
在许多考题中，题目给出力和时间，要求你计算速率的变化。将 Δp = FΔt 移项得 Δv = FΔt / m，可直接得出答案。始终要注意符号：若规定某一方向为正，则相反方向的力和速度必须带上负号。
3. Newton’s Second Law in Momentum Form | 牛顿第二定律的动量形式
Newton’s second law is often written as F = ma, but this is only valid when mass is constant. The more general form, which CIE examiners expect you to recognise, is:
牛顿第二定律常写作 F = ma，但这仅在质量不变时才成立。CIE 考官希望你掌握的更普遍的形式为：
F = Δp / Δt
The net force equals the rate of change of momentum. This version is essential for problems involving rockets, sand falling onto conveyors, or water jets — cases where mass changes over time. In these problems, the change in momentum per second gives the thrust or force exerted.
净力等于动量的变化率。这个版本对于火箭、沙子落到传送带上或水柱等质量随时间变化的问题尤为关键。这时候，每秒的动量变化就是所产生的推力或作用力。
Even in constant‑mass situations, F = Δp/Δt provides a useful link between impulse and force. A very short collision time produces a large force for a given Δp, which explains why crumple zones in cars reduce injury by extending the stopping time.
即使在质量恒定的情况下，F = Δp/Δt 也给出了冲量与力之间的有用联系。对于给定的 Δp，碰撞时间极短会产生巨大的力，这正解释了为什么汽车的溃缩区能通过延长停车时间来减小伤害。
4. Interpreting Force–Time Graphs | 解读力–时间图像
CIE often presents a graph of force against time and asks for the impulse or the average force. The impulse delivered by a varying force is simply the area between the force–time curve and the time axis. For a triangular or trapezoidal graph, you can calculate the area using basic geometry.
CIE 经常给出力随时间变化的图像，并要求你求冲量或平均力。变力提供的冲量就是力–时间曲线与时间轴之间所围的面积。对于三角形或梯形图，你可以用基本几何方法计算面积。
If the question asks for the average force, divide the total impulse by the total time during which the force acts. Remember that the impulse‑momentum theorem, Δp = area under F–t graph, is always true regardless of whether the force is constant.
如果题目要求平均力，就用总冲量除以力作用的总时间。记住，无论力是否恒定，动量-冲量定理 Δp = F–t 图下面积始终成立。
Exam tip: when the graph shows a negative force (e.g. during a bounce), treat the area as negative impulse, corresponding to a reversal of momentum direction. Adding areas with their proper signs gives the net impulse.
考试贴士：当图像显示负的力（例如在反弹期间），将该面积视为负冲量，对应动量方向的反转。将各面积连同正确的符号相加即可得到净冲量。
5. The Principle of Conservation of Momentum | 动量守恒原理
The principle states that in a closed system (no external resultant force acts), the total linear momentum remains constant. For two interacting bodies:
动量守恒原理指出，在一个封闭系统（无外合力作用）中，总线动量保持恒定。对于两个相互作用的物体：
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
where u₁, u₂ are the velocities before the interaction, and v₁, v₂ are the velocities afterwards. This vector equation means direction must be accounted for by assigning positive and negative signs to velocities.
其中 u₁、u₂ 为作用前的速度，v₁、v₂ 为作用后的速度。这个矢量方程意味着必须通过给速度设置正负号来计入方向。
Conservation of momentum is derived from Newton’s third law: the forces two objects exert on each other are equal and opposite, and act for the same time, producing equal and opposite impulses — hence the total momentum change is zero.
动量守恒可由牛顿第三定律推导：两物体间相互施加的力等大反向且作用时间相同，产生的冲量等大反向，因此总动量变化为零。
6. Elastic Collisions | 弹性碰撞
An elastic collision is one in which both momentum and total kinetic energy are conserved. In such collisions, the objects bounce apart without any permanent deformation or generation of heat. The two conservation conditions are:
弹性碰撞是指动量和总动能都守恒的碰撞。在这种碰撞中，两物体弹开，不产生永久形变或生成热量。两个守恒条件为：
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂²
For a head‑on elastic collision between two objects, these equations lead to a simple relationship: the relative speed of approach equals the relative speed of separation: u₁ − u₂ = v₂ − v₁ (assuming u₁ > u₂). In a special case where two equal masses collide elastically, they simply exchange velocities — a classic exam scenario.
对于两个物体的正碰弹性碰撞，这两个方程导向一个简单关系：相对接近速率等于相对分离速率：u₁ − u₂ = v₂ − v₁（假定 u₁ > u₂）。在等质量弹性碰撞的特殊情况下，两物体正好交换速度 —— 这是经典的考试场景。
Real collisions between steel balls or air‑track gliders approximate elastic behaviour, but on a macroscopic scale perfectly elastic collisions are rare. Even so, CIE uses the elastic model frequently both for calculations and for conceptual questions.
钢球之间或气垫导轨上的真实碰撞接近弹性行为，但在宏观尺度上完全弹性碰撞十分罕见。即便如此，CIE 仍然频繁地使用弹性模型进行相关的计算和概念辨析。
7. Inelastic Collisions | 非弹性碰撞
In an inelastic collision, momentum is conserved but kinetic energy is not. Some kinetic energy is transformed into internal energy, sound, or plastic deformation. The total momentum after the collision can still be found using the conservation equation, even though the initial and final speeds cannot both be determined from momentum alone.
在非弹性碰撞中，动量守恒而动能不守恒。一部分动能转变为内能、声音或塑性形变。碰撞后的总动量仍可借助守恒方程求得，尽管仅靠动量不能同时确定初速和末速。
Candidates often mistake ‘lost kinetic energy’ for ‘lost momentum’. Momentum is never lost in a closed system; it is always conserved. The ‘loss’ refers solely to kinetic energy, which is converted into other forms.
考生常将“损失的动能”错误地理解为“损失的动量”。在封闭系统中，动量绝不会丢失，始终守恒。所谓“损失”仅仅针对动能而言，它被转化为其他形式的能量。
8. Perfectly Inelastic Collisions | 完全非弹性碰撞
A perfectly inelastic collision is an extreme case of inelastic collision in which the colliding bodies stick together and move with a common final velocity v. Momentum conservation gives:
完全非弹性碰撞是非弹性碰撞的一种极端情形：碰撞物体粘合在一起，以共同的末速度 v 运动。由动量守恒得：
v =
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