Momentum: Concepts, Conservation and Collisions | 动量：概念、守恒与碰撞

📚 Momentum: Concepts, Conservation and Collisions | 动量：概念、守恒与碰撞

Momentum is a fundamental concept in mechanics that links mass and velocity. For CIE A-Level Physics, the topic covers the definition of linear momentum, impulse, Newton’s second law in terms of momentum, the principle of conservation of momentum, and applications to collisions, explosions and two-dimensional problems. Understanding momentum allows you to analyse interactions without needing to know the detailed forces involved.

动量是力学中联系质量与速度的基本概念。CIE A-Level 物理中，本主题涵盖线动量的定义、冲量、用动量表述的牛顿第二定律、动量守恒定律及其在碰撞、爆炸和二维问题中的应用。理解动量可以帮助你在不需要知道具体作用力细节的情况下分析相互作用。

1. Defining Momentum | 动量的定义

Linear momentum p of an object is the product of its mass m and velocity v. Since mass is a scalar and velocity is a vector, momentum is a vector quantity. Its SI unit is kg m s⁻¹, which is equivalent to N s.

线动量 p 是物体的质量 m 与速度 v 的乘积。质量是标量、速度是矢量，因此动量是矢量。它的国际单位是 kg m s⁻¹，也等价于 N s。

p = mv

A large truck moving slowly can have the same momentum as a small car moving quickly, because momentum depends on both mass and velocity. This is why a heavy but slow object can be just as difficult to stop as a light but fast object.

缓慢行驶的大卡车可能与快速行驶的小汽车具有相同的动量，因为动量取决于质量和速度两者。这就是为什么重而慢的物体与轻而快的物体一样难以停下。

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2. Momentum as a Vector | 动量是矢量

Momentum has both magnitude and direction; its direction is the same as the velocity of the object. In one-dimensional problems, choose a positive direction and represent momentum with positive or negative signs. For example, if velocity is 5.0 m s⁻¹ to the right and mass is 2.0 kg, momentum is +10 kg m s⁻¹; if the velocity is 5.0 m s⁻¹ to the left, momentum is −10 kg m s⁻¹.

动量既有大小又有方向；方向与物体的速度方向相同。在一维问题中，先选正方向，并用正负号表示动量。例如，若速度为向右 5.0 m s⁻¹、质量为 2.0 kg，则动量为 +10 kg m s⁻¹；若速度为向左 5.0 m s⁻¹，则动量为 −10 kg m s⁻¹。

In two-dimensional situations, two momenta of the same magnitude can have different effects if their directions differ. You must add momenta by vector addition or resolve them into perpendicular components before applying conservation laws.

在二维情形中，大小相同的两个动量若方向不同，会产生不同的效果。必须通过矢量加法相加，或先将动量分解为垂直分量，再应用守恒定律。

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3. Newton’s Second Law in Terms of Momentum | 用动量表述牛顿第二定律

Newton’s second law can be written in terms of momentum. The resultant force F on an object equals the rate of change of its momentum:

牛顿第二定律可以用动量来表述。作用在物体上的合外力 F 等于其动量的变化率：

F = Δp / Δt

Here Δp is the change in momentum over a time interval Δt. For constant mass, Δp = mΔv, so F = mΔv/Δt = ma. The momentum form is more general and also applies when mass changes, such as a rocket ejecting exhaust gases.

这里 Δp 是经过时间间隔 Δt 的动量变化。质量不变时，Δp = mΔv，因此 F = mΔv/Δt = ma。动量形式更普遍，也适用于质量变化的情况，例如火箭喷出燃气。

This statement explains why a large force changes momentum quickly, while a smaller force takes longer to produce the same change in momentum.

这一表述解释了为什么大力会快速改变动量，而较小的力需要更长的时间才能产生相同的动量变化。

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4. Impulse and Force-Time Graphs | 冲量与力-时间图像

Impulse J is defined as the product of a constant force F and the time interval Δt for which it acts. Its unit is N s, which is identical to kg m s⁻¹. Impulse equals the change in momentum of the object:

冲量 J 定义为恒力 F 与作用时间间隔 Δt 的乘积。其单位是 N s，与 kg m s⁻¹ 相同。冲量等于物体动量的变化量：

J = FΔt

J = Δp = mv − mu

When the force varies, impulse is the area under a force-time graph. This area gives the total change in momentum, even if the force is not constant.

当力变化时，冲量等于力-时间图像下的面积。即使力不是恒定的，该面积也能给出总动量变化。

• Increasing contact time reduces the peak force for the same impulse, as in airbags and crumple zones. 增加接触时间可在冲量相同的情况下减小峰值力，例如安全气囊和溃缩区。
• A short contact time produces a larger force for the same impulse, as in a hammer blow. 接触时间短则同样冲量会产生更大的力，例如锤击。

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5. The Principle of Conservation of Momentum | 动量守恒定律

The principle of conservation of momentum states that, for a system upon which no net external force acts, the total momentum before an interaction equals the total momentum after the interaction. The system is then called isolated or closed.

动量守恒定律指出，若系统不受合外力作用，则相互作用前的总动量等于相互作用后的总动量。这样的系统称为孤立系统或封闭系统。

For two objects colliding in one dimension, if u₁ and u₂ are the initial velocities and v₁ and v₂ are the final velocities, then:

对于一维碰撞的两个物体，若 u₁、u₂ 为初速度，v₁、v₂ 为末速度，则：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Conservation of momentum follows from Newton’s third law. The forces between two interacting objects are equal and opposite, so the impulses on them are equal and opposite, producing equal and opposite momentum changes. Since the changes cancel, total momentum remains constant.

动量守恒源自牛顿第三定律。两个相互作用物体之间的力大小相等、方向相反，因此它们所受的冲量也大小相等、方向相反，产生等大反向的动量变化。由于变化相互抵消，总动量保持不变。

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6. Collisions in One Dimension | 一维碰撞

In one-dimensional collision problems, define a positive direction, write the total momentum before and after, and equate them. If two objects stick together, they move with a common final velocity v, so m₁u₁ + m₂u₂ = (m₁ + m₂)v.

在一维碰撞问题中，先规定正方向，写出碰撞前后的总动量并令其相等。若两个物体粘在一起，它们以共同速度 v 运动，因此 m₁u₁ + m₂u₂ = (m₁ + m₂)v。

Example: A 2.0 kg trolley moving at 3.0 m s⁻¹ collides with a stationary 1.0 kg trolley and they stick together. The common final velocity is v = (2.0 × 3.0 + 1.0 × 0) / (2.0 + 1.0) = 2.0 m s⁻¹ in the original direction.

例子：一辆 2.0 kg 小车以 3.0 m s⁻¹ 与静止的 1.0 kg 小车碰撞并粘在一起。共同末速度为 v = (2.0 × 3.0 + 1.0 × 0) / (2.0 + 1.0) = 2.0 m s⁻¹，方向与原方向相同。

Always check the signs of velocities: a velocity opposite to the positive direction must be entered as a negative value.

始终要检查速度的正负号：与正方向相反的速度必须代入负值。

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7. Elastic and Inelastic Collisions | 弹性与非弹性碰撞

In an elastic collision, both momentum and total kinetic energy are conserved. In an inelastic collision, momentum is conserved but total kinetic energy is not; some kinetic energy is converted into heat, sound or deformation. A perfectly inelastic collision is one in which the colliding bodies stick together and the loss of kinetic energy is a maximum for a given initial momentum.

弹性碰撞中，动量和总动能都守恒。非弹性碰撞中，动量守恒但总动能不守恒；部分动能转化为热、声或形变能。完全非弹性碰撞中，碰撞体粘在一起，对于给定初始动量，动能损失最大。

Collision type 碰撞类型	Momentum conserved 动量守恒	Kinetic energy conserved 动能守恒	Feature 特征
Elastic 弹性	Yes 是	Yes 是	Bodies separate and no permanent deformation 物体分开且无形变
Inelastic 非弹性	Yes 是	No 否	Some kinetic energy is dissipated 部分动能被耗散
Perfectly inelastic 完全非弹性	Yes 是	No, maximum loss 否，损失最大	Bodies stick together 物体粘在一起

Even when kinetic energy is not conserved, total energy, including heat and sound, is always conserved. Distinguish carefully between momentum conservation, which holds for isolated systems in all collisions, and kinetic energy conservation, which holds only for elastic collisions.

即使动能不守恒，包括热和声在内的总能量始终守恒。要仔细区分动量守恒与动能守恒：孤立系统在任何碰撞中动量都守恒，而动能只在弹性碰撞中守恒。

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8. Explosions and Recoil | 爆炸与反冲

An explosion is the reverse of a perfectly inelastic collision. A single object splits into two or more parts; total momentum remains zero if the object was initially at rest. The fragments move apart with equal and opposite momenta:

爆炸是完全非弹性碰撞的逆过程。一个物体分裂成两个或多个部分；如果物体最初静止，则总动量保持为零。碎片以大小相等、方向相反的动量分开：

m₁v₁ + m₂v₂ = 0

Kinetic energy is not conserved in an explosion because stored chemical or internal energy is converted into kinetic energy. The total kinetic energy after the explosion can be greater than before.

爆炸中动能不守恒，因为储存的化学能或内能转化为动能。爆炸后的总动能可能大于爆炸前。

A gun recoil is a common example. When a bullet is fired forward, the gun moves backward with momentum equal in magnitude and opposite in direction to the bullet. This is why a heavier gun recoils more slowly than a lighter one for the same bullet momentum.

枪械反冲是一个常见例子。子弹向前射出时，枪身向后运动，其动量与子弹动量大小相等、方向相反。这就是为什么在子弹动量相同的情况下，较重的枪比轻枪反冲更慢。

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9. Momentum in Two Dimensions | 二维动量问题

In two-dimensional collisions, momentum is conserved independently along each perpendicular axis. Resolve all velocities into components, then apply conservation of momentum separately to the x-direction and y-direction:

在二维碰撞中，动量沿每个垂直坐标轴分别守恒。将所有速度分解为分量，然后对 x 方向和 y 方向分别应用动量守恒：

m₁u₁x + m₂u₂x = m₁v₁x + m₂v₂x

m₁u₁y + m₂u₂y = m₁v₁y + m₂v₂y

This method is important in billiard-ball collisions and particle scattering problems. Even when the collision is not head-on, the total momentum vector before the collision equals the total momentum vector after the collision.

这种方法在台球碰撞和粒子散射问题中很重要。

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