Momentum & Impulse, Conservation of Momentum | 动量定理与动量守恒

📚 Momentum & Impulse, Conservation of Momentum | 动量定理与动量守恒

Momentum is one of the most powerful concepts in physics. It allows us to describe and predict the outcomes of collisions and explosions without needing to know the intricate forces involved. In this article, we will explore the definition of momentum, the impulse-momentum theorem, and the principle of conservation of momentum, all tailored to the Edexcel IGCSE Physics syllabus.

动量是物理学中最重要的概念之一。它使我们能够描述和预测碰撞与爆炸的结果，而无需知道过程中复杂的力。在本文中，我们将深入探讨动量的定义、冲量-动量定理以及动量守恒定律，内容完全针对 Edexcel IGCSE 物理考纲要求。

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1. What is Momentum? | 什么是动量？

Momentum is defined as the product of an object’s mass and its velocity. It is a vector quantity, which means it has both magnitude and direction. The direction of momentum is always the same as the direction of the object’s velocity.

动量的定义是物体的质量与速度的乘积。它是一个矢量，这意味着它既有大小也有方向。动量的方向始终与物体的速度方向相同。

Momentum = mass × velocity

p = m × v

The SI unit of momentum is kilogram metre per second (kg·m/s). When an object is at rest, its momentum is zero, because its velocity is zero.

动量的国际单位是千克·米每秒（kg·m/s）。当物体静止时，它的动量为零，因为速度为零。

• The greater the mass, the greater the momentum for a given velocity. | 在速度一定时，质量越大，动量越大。
• The greater the velocity, the greater the momentum for a given mass. | 在质量一定时，速度越大，动量越大。
• Momentum is a vector — direction matters. | 动量是矢量，方向很重要。

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2. The Impulse-Momentum Theorem | 冲量-动量定理

When a resultant force acts on an object, it changes the object’s momentum. The change in momentum is equal to the force multiplied by the time for which the force acts. This product is called the impulse.

当合外力作用在物体上时，它会改变物体的动量。动量的变化量等于力乘以力作用的时间。这个乘积称为冲量。

Impulse = Force × Time

I = F × t

According to Newton’s second law, force equals the rate of change of momentum. Therefore, the impulse is also equal to the change in momentum:

根据牛顿第二定律，力等于动量的变化率。因此，冲量也等于动量的变化量：

F × t = Δp = m × v − m × u

Where u is the initial velocity and v is the final velocity. This equation is known as the impulse-momentum theorem. It tells us that to change an object’s momentum, we can either apply a large force for a short time, or a small force over a longer time.

其中 u 是初速度，v 是末速度。这个方程称为冲量-动量定理。它告诉我们，要改变物体的动量，既可以施加短时间的大力，也可以施加较长时间的小力。

The unit of impulse is Newton-second (N·s), which is equivalent to kg·m/s.

冲量的单位是牛顿·秒（N·s），它与 kg·m/s 是等价的。

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3. Understanding the Equation F = ma via Momentum | 通过动量理解 F = ma

The impulse-momentum theorem is actually a more general form of Newton’s second law. Let us see how they connect.

冲量-动量定理实际上是牛顿第二定律的更普遍形式。让我们看看它们之间是如何联系的。

Starting from F × t = m × v − m × u, we can rearrange:

从 F × t = m × v − m × u 出发，我们可以整理得到：

F = (m × v − m × u) / t = m × (v − u) / t

Since acceleration a = (v − u) / t, we obtain F = m × a. This shows that Newton’s second law F = ma is a specific case of the momentum version of the second law, valid when the mass remains constant.

由于加速度 a = (v − u) / t，我们得到 F = m × a。这表明牛顿第二定律 F = ma 是动量形式第二定律的一个特例，适用于质量保持不变的情况。

When the mass of an object changes (such as a rocket burning fuel), the more general form F = Δp / t must be used.

当物体的质量发生变化时（如火箭燃烧燃料），就必须使用更普遍的形式 F = Δp / t。

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4. Increasing Time to Reduce Force: Crumple Zones | 延长作用时间以减小力：缓冲区的应用

The impulse-momentum theorem has very important real-world applications, especially in car safety design. In a collision, the change in momentum is fixed — the car and passengers must come to rest. To reduce the harmful force, the time over which the collision occurs must be increased.

冲量-动量定理在现实生活中有非常重要的应用，尤其在汽车安全设计中。在碰撞中，动量的变化量是固定的——汽车和乘客必须停下来。为了减小伤害力，就必须增加碰撞发生的时间。

• Crumple zones (缓冲区): They deform on impact, extending the collision time and reducing the force. | 缓冲区：在撞击时发生形变，延长碰撞时间，从而减小力。
• Airbags (安全气囊): They inflate and gradually deflate, increasing the time for the passenger’s momentum to reach zero. | 安全气囊：充气并逐渐放气，增加乘客动量归零所需的时间。
• Seat belts (安全带): They stretch slightly during a collision, again increasing the stopping time. | 安全带：在碰撞时略微拉伸，同样增加了停止时间。
• Foam packaging (泡沫包装): Protects fragile items by increasing the time of impact. | 泡沫包装：通过增加撞击时间来保护易碎物品。

In all these examples, the goal is not to stop the object — the stopping is inevitable. The goal is to spread the change in momentum over a longer time so that the force experienced is smaller.

在所有这些例子中，目标不是阻止物体——停止是不可避免的。目标是将动量的变化分摊到更长的时间上，从而使物体受到的力更小。

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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 when two or more objects interact (in a closed system with no external forces), the total momentum before the interaction is equal to the total momentum after the interaction.

动量守恒定律指出：当两个或多个物体在不受外力的封闭系统中相互作用时，相互作用前的总动量等于相互作用后的总动量。

Total momentum before = Total momentum after

相互作用前的总动量 = 相互作用后的总动量

This principle is universal. It applies to all types of collisions, explosions, and any form of interaction between objects. It holds true whether the objects are billiard balls, cars, or subatomic particles.

这个定律是普遍适用的。它适用于所有类型的碰撞、爆炸以及物体间的任何形式的相互作用。无论是台球、汽车还是亚原子粒子，它都成立。

The key condition required: there must be no external resultant force acting on the system. External forces such as friction or air resistance may need to be ignored or accounted for.

关键条件是：系统不能受到外部合力的作用。摩擦力和空气阻力等外力可能需要被忽略或加以考虑。

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6. Elastic vs Inelastic Collisions | 弹性碰撞与非弹性碰撞

In physics, collisions are classified as either elastic or inelastic. The difference lies in whether kinetic energy is conserved.

在物理学中，碰撞分为弹性碰撞和非弹性碰撞。区别在于动能是否守恒。

Feature | 特征	Elastic Collision 弹性碰撞	Inelastic Collision 非弹性碰撞
Momentum conserved | 动量守恒	Yes | 是	Yes | 是
Kinetic energy conserved | 动能守恒	Yes | 是	No (some converted to heat/sound) | 否（部分转化为热和声音）
Objects bounce apart | 物体弹开	Yes | 是	May stick together | 可能粘在一起

In a perfectly elastic collision, objects separate after impact and total kinetic energy is conserved. In practice, perfectly elastic collisions only occur between very hard objects, like billiard balls or gas molecules.

在完全弹性碰撞中，物体在撞击后分开，总动能守恒。实际上，完全弹性碰撞只发生在非常坚硬的物体之间，比如台球或气体分子。

In an inelastic collision, some kinetic energy is transferred to other forms of energy, such as heat, sound, and deformation energy. Remember, momentum is always conserved, but kinetic energy is not necessarily conserved.

在非弹性碰撞中，部分动能转化为其他形式的能量，如热能、声能和形变能。记住，动量总是守恒的，但动能不一定守恒。

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7. Solving Momentum Questions: The Step-by-Step Approach | 解决动量问题的分步方法

To solve momentum conservation problems, follow these systematic steps:

要解决动量守恒问题，请遵循以下系统步骤：

• Step 1: Draw a diagram showing the objects before and after the interaction. Assign positive and negative directions. | 第一步：画出物体在相互作用前后的示意图。指定正方向和负方向。
• Step 2: Write down the known quantities: masses, velocities, and directions. | 第二步：写下已知量：质量、速度和方向。
• Step 3: Apply the conservation of momentum equation. | 第三步：应用动量守恒方程。
• Step 4: Substitute values into the equation carefully, including signs. | 第四步：仔细代入数值，注意符号。
• Step 5: Solve for the unknown quantity and specify its direction. | 第五步：解出未知量并指明其方向。

Remember: velocities in opposite directions must have opposite signs. If the final answer is positive, the direction is the positive direction; if negative, it is opposite.

记住：反方向的速度必须乘以相反的符号。如果最终答案为正，方向就是正方向；如果为负，则是反方向。

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8. Worked Example: Sticky Collision | 例题精讲：粘性碰撞

A trolley of mass 2 kg moving at 3 m/s collides with a stationary trolley of mass 1 kg. The two trolleys stick together. Calculate the final velocity of the combined trolleys.

一个质量为 2 kg 的小车以 3 m/s 的速度运动，撞上一个质量为 1 kg 的静止小车。两个小车粘在一起运动。计算两个小车结合后的最终速度。

Step 1: Identify the momentum before the collision:

第一步：确定碰撞前的动量：

Total momentum before = (2 × 3) + (1 × 0) = 6 kg·m/s

Step 2: After the collision, the total mass is 2 + 1 = 3 kg. Let the final velocity be v.

第二步：碰撞后，总质量为 2 + 1 = 3 kg。设最终速度为 v。

Total momentum after = 3 × v

Step 3: Apply conservation of momentum:

第三步：应用动量守恒：

6 = 3 × v → v = 2 m/s

Therefore, the combined trolleys move at 2 m/s in the same direction as the initial motion. This is an inelastic collision — kinetic energy is not conserved because the trolleys stick together.

因此，结合后的小车以 2 m/s 的速度沿初始运动方向前进。这是一个非弹性碰撞——动能不守恒，因为小车粘在一起了。

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9. Worked Example: Explosions | 例题精讲：爆炸与反冲

Explosions are the reverse of collisions. Before an explosion, an object is at rest with zero total momentum. After the explosion, the fragments fly apart, but their total momentum must still be zero.

爆炸是碰撞的反过程。爆炸前，物体静止，总动量为零。爆炸后，碎片向各方向飞出，但它们的总动量必须仍然为零。

Classic example: A stationary gun of mass 2 kg fires a bullet of mass 0.01 kg at 300 m/s. What is the recoil velocity of the gun?

经典例题：一支质量为 2 kg 的静止枪支发射一颗质量为 0.01 kg 的子弹，子弹速度为 300 m/s。枪的反冲速度是多少？

Initial total momentum = 0. Therefore, final total momentum = 0:

初始总动量 = 0。因此，最终总动量 = 0：

0 = (0.01 × 300) + (2 × v)

0 = 3 + 2v → v = −1.5 m/s

The negative sign indicates the gun recoils in the opposite direction to the bullet. The magnitude of the recoil velocity is 1.5 m/s.

负号表示枪向子弹相反的方向反冲。反冲速度的大小为 1.5 m/s。

This is how Newton’s third law and the conservation of momentum are connected: the force from the gun on the bullet is equal and opposite to the force from the bullet on the gun.

这就是牛顿第三定律与动量守恒之间的联系：枪对子弹的作用力与子弹对枪的作用力大小相等、方向相反。

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10. Newton’s Cradle and the Conservation of Momentum | 牛顿摆与动量守恒

Newton’s cradle is a classic demonstration of conservation of momentum. When one ball on the left is lifted and released, it strikes the row of balls. A single ball on the right pops out with the same velocity.

牛顿摆是对动量守恒的经典演示。当左侧的一个球被抬起并释放时，它撞击一排气球。右侧会弹出一个具有相同速度的球。

Why doesn’t two balls pop out? If two balls were ejected at half the speed, momentum would be conserved. However, kinetic energy would not be conserved. The ejection of one ball at the same speed conserves both momentum and kinetic energy, matching the nearly elastic nature of the steel balls.

为什么不会弹出两个球？如果弹出两个速度减半的球，动量可以守恒。但动能不会守恒。只有一个球以相同的速度弹出，才能同时守恒动量和动能，这符合钢球近乎弹性的特性。

This demonstrates that for analysing collisions, both momentum and kinetic energy must be considered together — together they completely determine the outcome.

这说明在分析碰撞时，必须同时考虑动量和动能——两者共同完全决定了碰撞的结果。

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11. Momentum in Two Dimensions | 二维动量分析

In the IGCSE Edexcel course, most momentum problems are solved in one dimension. However, you should know that momentum is a vector and must be analysed separately in perpendicular directions for two-dimensional problems.

在 IGCSE Edexcel 课程中，大多数动量题都是在一维中解决的。然而，你应该知道动量是矢量，在二维问题中必须分别分析垂直方向的分量。

For example, when two cars collide at a right-angle intersection, momentum in the x-direction is conserved separately, and momentum in the y-direction is conserved separately:

例如，当两辆车在十字路口成直角碰撞时，x 方向动量分别守恒，y 方向动量分别守恒：

Total pₓ before = Total pₓ after

Total pᵧ before = Total pᵧ after

You can then find the resultant momentum using vector addition, typically using Pythagoras’ theorem. This is an advanced application for examination questions requiring more complex analysis.

然后你可以使用矢量叠加来求合动量，通常使用勾股定理。这是一个高级应用，用于需要更复杂分析的考试题目。

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12. Examination Focus: Common Mistakes and Tips | 考试重点：常见错误与技巧

There are several common mistakes that IGCSE students make when tackling momentum questions. Being aware of these will help you avoid them and gain marks.

IGCSE 学生在做动量题时经常会犯一些错误。意识到这些问题可以帮助你避免它们并获得分数。

• Forgetting momentum is a vector. Always assign directions and signs. | 忘记动量是矢量。始终设定方向和符号。
• Incorrectly applying the formula. Check what mass and velocity must be used before and after the collision. | 公式应用错误。检查碰撞前后应该使用什么质量和速度。
• Forgetting to convert units. Ensure mass is in kg and velocity in m/s. | 忘记单位换算。确保质量单位是 kg，速度单位是 m/s。
• Conservation of momentum only holds without external forces. In real problems, friction and air resistance may affect results. | 动量守恒仅在无外力时成立。在真实问题中，摩擦力和空气阻力可能影响结果。
• Confusing momentum with kinetic energy. Momentum is not conserved in all cases where energy is — and vice versa. | 混淆动量与动能。动量不是在所有能量守恒的情况下都守恒——反之亦然。

Key equations to memorise for the exam:

考试中必须记住的关键方程：

p = m × v

F × t = Δp = m × v − m × u

Total p before = Total p after

Also, be ready to interpret velocity-time graphs and force-time graphs. The area under a force-time graph represents the impulse, which equals the change in momentum. This graphical insight is frequently tested in the Edexcel IGCSE exam.

此外，要准备好解读速度-时间图和力-时间图。力-时间图下的面积代表冲量，等于动量的变化量。这种图形分析的能力在 Edexcel IGCSE 考试中经常被考查。

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Summary | 总结

Momentum is the product of mass and velocity. The impulse-momentum theorem connects force, time, and change in momentum. The principle of conservation of momentum states that in a closed system, total momentum remains constant during collisions and explosions. Understanding these concepts, along with their applications to cars, guns, and everyday life, is essential for success in IGCSE Physics.

动量是质量与速度的乘积。冲量-动量定理将力、时间和动量变化联系起来。动量守恒定律指出在封闭系统中，碰撞和爆炸期间总动量保持不变。理解这些概念及其在汽车、枪支和日常生活中的应用，对在 IGCSE 物理中取得成功至关重要。

Master these principles, practise plenty of past paper questions, and momentum will become one of your strongest topics.

掌握这些原理，大量练习历年真题，动量将成为你最擅长的考点之一。

Published by TutorHao | Physics Revision Series | aleveler.com

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