Momentum and Collisions – Combined Mechanics 2.18 | 动量与碰撞——综合力学 2.18

📚 Momentum and Collisions – Combined Mechanics 2.18 | 动量与碰撞——综合力学 2.18

Momentum is one of the most powerful concepts in mechanics, linking mass, velocity and force in a single vector quantity. In Edexcel A‑Level Physics, topic 2.18 brings together momentum, impulse and the conservation law to explain everything from car crashes to rocket launches. This combined study will give you the tools to analyse collisions, explosions and recoil events with confidence.

动量是力学中最重要的概念之一，它将质量、速度和力统一在一个矢量量中。在爱德思 A‑Level 物理中，专题 2.18 将动量、冲量与守恒定律结合在一起，用以解释从汽车碰撞到火箭发射的各种现象。本综合学习将为你提供分析碰撞、爆炸与反冲事件所需的工具与信心。

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1. What Is Momentum? | 什么是动量？

Linear momentum is defined as the product of an object’s mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction. The symbol for momentum is p, and its unit is kg m s⁻¹ or N s.

线动量定义为物体的质量与速度的乘积。它是一个矢量，既有大小也有方向。动量的符号是 p，单位是 kg m s⁻¹ 或 N s。

p = m v

Because velocity depends on the frame of reference, momentum is also frame‑dependent. A passenger sitting on a moving bus has zero momentum relative to the bus but has significant momentum relative to the ground.

由于速度依赖于参考系，动量也随参考系变化。坐在行进公交车上的一名乘客相对于公交车动量为零，但相对于地面却有很大的动量。

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2. Momentum as a Vector | 动量是矢量

In one‑dimensional problems we often choose a positive direction (e.g. to the right) and assign signs to velocities accordingly. The direction of momentum is the same as the direction of velocity.

在一维问题中，我们通常选定正方向（例如向右），并据此为速度赋予正负号。动量的方向与速度方向相同。

For example, Ball A of mass 0.50 kg moving at 4.0 m s⁻¹ to the right has momentum +2.0 kg m s⁻¹. Ball B of the same mass moving at 3.0 m s⁻¹ to the left has momentum −1.5 kg m s⁻¹.

例如，质量为 0.50 kg 的球 A 以 4.0 m s⁻¹ 向右运动，其动量为 +2.0 kg m s⁻¹。质量相同的球 B 以 3.0 m s⁻¹ 向左运动，其动量为 −1.5 kg m s⁻¹。

This vector nature becomes crucial when we add or subtract momentum in collision calculations.

这种矢量特性在碰撞计算中进行动量加减时至关重要。

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3. Impulse and the Rate of Change of Momentum | 冲量与动量变化率

Newton’s second law can be expressed in terms of momentum: the net force acting on an object equals the rate of change of its momentum.

牛顿第二定律可用动量表述：作用在物体上的合力等于其动量随时间的变化率。

F = Δp / Δt

If the force is constant, we can define impulse J as the product of the net force and the time interval during which it acts. Impulse equals the change in momentum:

如果力恒定，我们可以将冲量 J 定义为合力与其作用时间的乘积。冲量等于动量的变化量：

J = F Δt = Δp = m v − m u

This relationship is extremely useful: a large force applied for a short time (like hitting a squash ball) can produce the same momentum change as a smaller force applied for a longer time (like catching an egg). The area under a force–time graph gives the impulse.

这一关系非常有用：短时间内施加的大力（如击打壁球）与较长时间施加的小力（如接住鸡蛋）可以产生相同的动量变化。力–时间图下的面积即为冲量。

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4. Newton’s Laws and Momentum | 牛顿定律与动量

Newton’s third law states that when two objects interact, the forces they exert on each other are equal in magnitude and opposite in direction. If we consider the time of interaction to be the same, the impulses on the two objects are equal and opposite, so their changes in momentum are also equal and opposite.

牛顿第三定律指出，当两个物体相互作用时，彼此施加的力大小相等、方向相反。若考虑相互作用时间相同，则两物体受到的冲量大小相等、方向相反，因此它们的动量变化也大小相等、方向相反。

This is the physical root of the principle of conservation of linear momentum: in a closed system, the total momentum before an interaction equals the total momentum after.

这正是线动量守恒定律的物理根源：在封闭系统中，相互作用前的总动量等于相互作用后的总动量。

Isaac Newton himself recognised momentum as “quantity of motion”, a concept that brilliantly unites his three laws.

艾萨克·牛顿本人将动量视为“运动的量”，这一概念巧妙地统一了他的三大定律。

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5. Principle of Conservation of Linear Momentum | 线动量守恒原理

For any system upon which no net external force acts, the total linear momentum remains constant.

对于没有净外力作用的任何系统，总线动量保持恒定。

Total momentum before = Total momentum after

Mathematically, for a two‑body collision along a straight line:

数学上，对于沿直线发生的两体碰撞：

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

Here u₁ and u₂ are initial velocities, v₁ and v₂ are final velocities. Signs must be assigned carefully according to the chosen positive direction.

其中 u₁ 和 u₂ 为初速度，v₁ 和 v₂ 为末速度。必须根据选定的正方向小心赋予符号。

This principle is universal – it holds in all collisions and explosions, provided external forces (like friction) are negligible or balanced.

该原理具有普适性——只要外力（如摩擦力）可忽略或相互平衡，它适用于所有碰撞与爆炸过程。

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6. Conditions for Conservation | 守恒条件

Conservation of momentum is exact only when the net external force on the system is zero. In practice:

• If the interaction time is very short, impulsive forces dominate, and the effect of external forces is negligible.
• For explosions, internal forces are often huge compared with gravity or friction, so momentum is conserved just before and just after the explosion.

为了动量精确守恒，系统所受净外力必须为零。实际应用中：

• 如果相互作用时间极短，冲力占主导地位，外力的影响可以忽略不计。
• 对于爆炸，内力通常远大于重力或摩擦力，因此在爆炸前后的瞬间动量守恒。

In collision problems, we usually assume the system (e.g. two colliding trolleys) is isolated during the short interaction, making momentum conservation a reliable tool.

在碰撞问题中，我们通常假设系统（如两辆碰撞的小车）在短暂的相互作用期间是孤立的，从而使动量守恒成为可靠的分析工具。

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

A collision is called elastic if kinetic energy, in addition to momentum, is conserved. No kinetic energy is converted into other forms such as heat or sound.

除了动量守恒外，如果动能也守恒，则该碰撞称为弹性碰撞。没有动能转化为热或声等其他形式。

½ m₁ u₁² + ½ m₂ u₂² = ½ m₁ v₁² + ½ m₂ v₂²

In contrast, an inelastic collision loses some kinetic energy. Most everyday collisions are inelastic to some degree.

与此相反，非弹性碰撞会损失一部分动能。大多数日常碰撞或多或少都是非弹性的。

A special case is a perfectly inelastic collision, where the objects stick together after impact and move with a common velocity. Kinetic energy is reduced to a minimum, but momentum is still conserved.

一个特殊情况是完全非弹性碰撞，此时物体在碰撞后粘在一起并以共同速度运动。动能减至最小，但动量依然守恒。

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8. Analysing Perfectly Inelastic Collisions | 分析完全非弹性碰撞

In a perfectly inelastic collision, the objects coalesce. The conservation equation becomes:

在完全非弹性碰撞中，物体合为一体。守恒方程变为：

m₁ u₁ + m₂ u₂ = (m₁ + m₂) v

If we know the initial velocities, we can find the common final velocity v. The kinetic energy lost can be calculated as:

若已知初速度，即可求出共同末速度 v。损失的动能可计算为：

ΔEₖ = (½ m₁ u₁² + ½ m₂ u₂²) − ½ (m₁ + m₂) v²

This energy is dissipated as heat, sound or deformation. Investigating the energy loss helps engineers design safer cars with crumple zones that extend collision time, reducing forces.

这部分能量以热、声或形变的形式耗散。研究能量损失有助于工程师设计更安全的汽车，利用碰撞吸能区延长碰撞时间以减小冲击力。

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9. Explosions and Recoil | 爆炸与反冲

The conservation of momentum also governs explosions and recoil, where objects start from rest or a common state and move apart.

动量守恒同样适用于爆炸与反冲——物体从静止或共同状态开始分离运动。

If a stationary object splits into two fragments, the fragments must move in opposite directions so that the total momentum remains zero:

如果一个静止物体分裂成两个碎片，碎片必须朝相反方向运动，以使总动量保持为零：

0 = m₁ v₁ + m₂ v₂

This explains why a gun recoils when a bullet is fired, or why a rocket accelerates by ejecting exhaust gases backwards. In rocket propulsion, the momentum gained by the rocket is equal and opposite to the momentum carried away by the expelled fuel.

这解释了为什么枪发射子弹时会反冲，或者火箭为何通过向后喷射燃气而加速。在火箭推进中，火箭获得的动量与被排出的燃料带走的动量大小相等、方向相反。

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10. Two‑Dimensional Collisions – A Glimpse | 二维碰撞简介

When collisions occur at an angle, momentum must be conserved independently in perpendicular directions, usually chosen as the x‑ and y‑axes. Vector resolution of momentum becomes essential.

当碰撞发生在有角度的情况下，动量需在相互垂直的方向（通常取 x 轴和 y 轴）上分别守恒。此时动量的矢量分解变得必不可少。

For example, snooker balls colliding obliquely can be modelled by applying conservation of momentum to each axis:

例如，斜碰的桌球可以通过对每个轴应用动量守恒来建立模型：

Σ pₓ before = Σ pₓ after
Σ p_y before = Σ p_y after

Kinetic energy may or may not be conserved depending on the nature of the collision. Two‑dimensional analysis sharpens your understanding of vectors and prepares you for advanced mechanics in further study.

动能是否守恒取决于碰撞的性质。二维分析能加深你对矢量的理解，并为更高阶的力学学习做好准备。

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