Conservation of Momentum | 动量守恒

📚 Conservation of Momentum | 动量守恒

Momentum is one of the most useful quantities in mechanics because it allows us to predict the outcome of collisions, explosions, and recoil events without knowing the detailed forces involved. The principle of conservation of momentum is a cornerstone of the Edexcel A-level Physics specification and applies whenever a system experiences no external resultant force.

动量是力学中最有用的物理量之一，因为我们可以利用它来预测碰撞、爆炸和反冲事件的结果，而不必了解其中复杂的力。动量守恒定律是 Edexcel A-level 物理考纲的核心内容，只要系统不受外力合力作用，该定律就适用。

1. Momentum and the Conservation Law | 动量与守恒定律

Momentum p is defined as the product of an object’s mass and its velocity. It is a vector quantity, so its direction is the same as the velocity of the object.

动量 p 定义为物体质量与速度的乘积。它是一个矢量，因此其方向与物体的速度方向相同。

The SI unit of momentum is the kilogram metre per second, written kg m s⁻¹. Since 1 N = 1 kg m s⁻², an equivalent unit is the newton second, N s.

动量的国际单位是千克米每秒，写作 kg m s⁻¹。由于 1 N = 1 kg m s⁻²，等价单位也可用牛秒 N s。

p = m v

The principle of conservation of momentum states that when no external resultant force acts on a system, the total momentum of the system remains constant. This applies before and after collisions, explosions, and other interactions.

动量守恒定律指出：当系统不受外力合力作用时，系统的总动量保持不变。这适用于碰撞、爆炸以及其他相互作用发生之前和之后。

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

2. Connecting to Newton’s Laws | 与牛顿定律的联系

Newton’s second law can be written in terms of momentum: the resultant force is equal to the rate of change of momentum of the object. This is a more general form than F = ma because it also applies when mass changes.

牛顿第二定律可以用动量表述：物体所受的合力等于其动量的变化率。这是比 F = ma 更普遍的表述，因为在质量变化时它同样适用。

F = Δp / Δt

In a collision between two objects, the forces they exert on each other are equal in magnitude and opposite in direction, as required by Newton’s third law. These interaction forces act for the same time interval, so the impulses on the two objects are equal and opposite.

在两个物体碰撞时，它们彼此施加的力大小相等、方向相反，这正是牛顿第三定律的要求。这些相互作用力作用的时间相同，因此两个物体受到的冲量大小相等、方向相反。

3. Deriving Conservation from Newton’s Third Law | 由牛顿第三定律推导守恒

Consider two objects A and B of masses m_A and m_B. Let their initial velocities be u_A and u_B, and their final velocities after collision be v_A and v_B.

考虑质量分别为 m_A 和 m_B 的两个物体 A 和 B。设碰撞前速度分别为 u_A 和 u_B，碰撞后速度分别为 v_A 和 v_B。

During the collision, object A exerts a force F on object B. By Newton’s third law, object B exerts a force -F on object A. The contact time Δt is the same for both objects.

碰撞过程中，物体 A 对物体 B 施加力 F。根据牛顿第三定律，物体 B 对物体 A 施加力 -F。两物体的接触时间 Δt 相同。

Applying the impulse-momentum equation to each object:

对每个物体应用冲量-动量方程：

F Δt = m_B v_B – m_B u_B

-F Δt = m_A v_A – m_A u_A

Adding the two equations cancels the impulses on the left-hand side, giving:

将两式相加，左边的冲量相互抵消，得到：

m_A u_A + m_B u_B = m_A v_A + m_B v_B

4. Elastic and Inelastic Collisions | 弹性碰撞与非弹性碰撞

Collisions are classified according to whether kinetic energy is conserved. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but some kinetic energy is transformed into heat, sound, or deformation.

碰撞根据动能是否守恒进行分类。在弹性碰撞中

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