Momentum in One Dimension | 一维动量

📚 Momentum in One Dimension | 一维动量

Momentum is a fundamental quantity in mechanics that combines mass and velocity. In the Edexcel A-Level Physics specification, you need to define linear momentum, apply the impulse-momentum relationship, and use the principle of conservation of momentum to solve one-dimensional collision and explosion problems. This article covers the key concepts, equations, worked examples and common pitfalls to help you master the topic.

动量是力学中结合质量与速度的基本物理量。在 Edexcel A-Level 物理考试中，你需要定义线性动量、应用冲量-动量关系，并利用动量守恒定律求解一维碰撞与爆炸问题。本文涵盖关键概念、公式、例题和常见错误，帮助你掌握这一主题。

1. Linear Momentum | 线性动量

Linear momentum is defined as the product of an object’s mass and its velocity. It is a vector quantity, so in one dimension its direction is indicated by a positive or negative sign according to a chosen coordinate system.

线性动量定义为物体质量与速度的乘积。它是一个矢量，因此在一维问题中，其方向根据所选坐标系用正号或负号表示。

The defining equation is:

定义方程为：

p = m v

Here p is momentum, m is mass and v is velocity. The SI unit of momentum is kg m s⁻¹, which is equivalent to N s because 1 N = 1 kg m s⁻².

这里 p 表示动量，m 表示质量，v 表示速度。动量的国际单位是 kg m s⁻¹，也等价于 N s，因为 1 N = 1 kg m s⁻²。

For example, a car of mass 1200 kg moving at 15 m s⁻¹ due east has momentum 18 000 kg m s⁻¹ east. If east is taken as positive, then p = +18 000 kg m s⁻¹. The same car moving west at the same speed would have p = -18 000 kg m s⁻¹.

例如，一辆质量为 1200 kg 的汽车以 15 m s⁻¹ 向东行驶，其动量为 18 000 kg m s⁻¹ 向东。若取向东为正，则 p = +18 000 kg m s⁻¹。同一辆车以相同速率向西行驶时，p = -18 000 kg m s⁻¹。

2. Impulse and Change in Momentum | 冲量与动量变化

Impulse is the product of the resultant force acting on an object and the time for which it acts. It is equal to the change in momentum of the object, which follows directly from Newton’s second law.

冲量是作用在物体上的合力与作用时间的乘积。它等于物体动量的变化，这直接由牛顿第二定律得出。

The impulse-momentum relationship is:

冲量-动量关系为：

J = F Δt = Δp = m v – m u

Here J is impulse, F is the average resultant force, Δt is the time interval, u is the initial velocity and v is the final velocity. Impulse is also a vector quantity, so its direction must be considered in one-dimensional problems.

这里 J 表示冲量，F 表示平均合力，Δt 表示时间间隔，u 表示初速度，v 表示末速度。冲量也是矢量，因此在一维问题中必须考虑其方向。

On a force-time graph, the impulse is represented by the area under the graph. If the force is not constant, you can use the average force multiplied by the time interval, or find the area by counting squares or using geometric shapes.

在力-时间图像中，冲量由图像下的面积表示。如果力不是恒定的，可以用平均力乘以时间间隔，或者通过数方格或使用几何形状求面积。

The units of impulse are N s, which are identical to kg m s⁻¹. This equivalence is important when converting between force and momentum quantities in exam questions.

冲量的单位是 N s，与 kg m s⁻¹ 完全相同。在考试题目中，在力与动量之间进行转换时，这一等价关系非常重要。

3. Principle of Conservation of Momentum | 动量守恒定律

The principle of conservation of momentum states that in a closed system with no external resultant force, the total momentum before an interaction is equal to the total momentum after the interaction.

动量守恒定律指出，在没有外合力的封闭系统中，相互作用前的总动量等于相互作用后的总动量。

In one dimension, this is written as:

在一维情况下，可写为：

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

The subscripts 1 and 2 refer to two objects, u represents initial velocities and v represents final velocities. All velocities must be measured in the same direction convention, and negative signs must be included for objects moving in the opposite direction.

下标 1 和 2 表示两个物体，u 表示初速度，v 表示末速度。所有速度必须采用同一方向约定，并且对于沿相反方向运动的物体必须包含负号。

Conservation of momentum arises from Newton’s third law. When two objects collide, they exert equal and opposite forces on each other for the same time interval. The impulses on the two objects are therefore equal and opposite, so the total change in momentum of the system is zero.

动量守恒源于牛顿第三定律。当两个物体碰撞时，它们在相同时间间隔内对彼此施加大小相等、方向相反的力。因此两个物体受到的冲量大小相等、方向相反，系统的总动量变化为零。

This law applies to all types of collisions and explosions, regardless of whether kinetic energy is conserved, provided there is no external resultant force such as friction.

只要没有摩擦等外合力，该定律适用于所有类型的碰撞和爆炸，无论动能是否守恒。

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

In an elastic collision, both momentum and kinetic energy are conserved. This is an idealised situation, but it closely describes collisions between hard objects such as billiard balls or gas molecules.

在弹性碰撞中，动量和动能都守恒。这是一种理想化情况，但它近似描述了台球或气体分子等硬物之间的碰撞。

For an elastic collision in one dimension, the total kinetic energy before equals the total kinetic energy after:

对于一维弹性碰撞，碰撞前的总动能等于碰撞后的总动能：

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

In an inelastic collision, momentum is conserved but kinetic energy is not. Some kinetic energy is converted into heat, sound or permanent deformation. A completely inelastic collision is one in which the two objects stick together and move with a common final velocity.

在非弹性碰撞中，动量守恒但动能不守恒。部分动能转化为热量、声音或永久形变。完全非弹性碰撞是指两个物体粘在一起并以共同末速度运动的碰撞。

For a completely inelastic collision, the conservation equation becomes:

对于完全非弹性碰撞，守恒方程变为：

m₁ u₁ + m₂ u₂ = (m₁ + m₂) v

Here v is the common final velocity of the combined objects. Remember that momentum is always conserved, but kinetic energy is only conserved in elastic collisions.

这里 v 是两个物体合并后的共同末速度。请记住，动量总是守恒的，但动能仅在弹性碰撞中守恒。

5. Explosions and Recoil | 爆炸与反冲

An explosion is an interaction in which a single object breaks apart into two or more fragments. If the object is initially at rest, its total initial momentum is zero. By conservation of momentum, the total final momentum of all fragments must also be zero.

爆炸是一个物体分裂成两个或多个碎片的相互作用。如果物体最初静止，其初始总动量为零。根据动量守恒，所有碎片的总末动量也必须为零。

This means the fragments must move in opposite directions with momenta that cancel each other out. In one dimension, if two fragments are produced, we can write:

这意味着碎片必须沿相反方向运动，且动量相互抵消。在一维情况下，如果产生两个碎片，可以写成：

0 = m₁ v₁ + m₂ v₂

Therefore m₁ v₁ = -m₂ v₂. The lighter fragment moves faster, and the velocities are inversely proportional to the masses.

因此 m₁ v₁ = -m₂ v₂。较轻的碎片运动更快，速度与质量成反比。

A common example is the recoil of a gun. When a bullet is fired forward, the gun recoils backward. If the bullet has mass m and velocity v_bullet, and the gun has mass M and recoil velocity V_gun, then:

一个常见的例子是枪的反冲。当子弹向前发射时，枪向后反冲。如果子弹质量为 m，速度为 v_bullet，枪的质量为 M，反冲速度为 V_gun，则：

0 = m v_bullet + M V_gun

This gives M V_gun = -m v_bullet, so the gun recoils in the opposite direction with a smaller speed if M is much larger than m.

由此得到 M V_gun = -m v_bullet，因此枪沿相反方向反冲，如果 M 远大于 m，枪的反冲速度较小。

6. Solving One-Dimensional Momentum Problems | 一维动量问题解法

To solve a one-dimensional momentum problem successfully, follow a clear step-by-step method. First, choose a positive direction and state it clearly. Then write down all given masses and velocities with appropriate signs.

要成功解决一维动量问题，请按照清晰的步骤进行。首先，选择正方向并明确说明。然后写下所有给定的质量和速度，并加上适当的正负号。

Next, identify whether the event is a collision or an explosion, and decide whether kinetic energy is relevant. For collisions, write the conservation of momentum equation. For elastic collisions, also write the conservation of kinetic energy equation if needed.

接下来，确定事件是碰撞还是爆炸，并判断动能是否相关。对于碰撞，写出动量守恒方程。对于弹性碰撞，如需还写出动能守恒方程。

Then substitute the known values and solve for the unknown quantity. Finally, check that the sign of your answer is consistent with the chosen positive direction, and state the direction in your final answer.

然后代入已知值并求解未知量。最后，检查答案的符号是否与所选正方向一致，并在最终答案中说明方向。

Always ask yourself whether the answer is physically reasonable. A final speed significantly larger than the initial speeds may indicate a sign error or a mistake in the conservation equation.

始终问自己答案是否物理上合理。末速度明显大于初速度可能表明符号错误或守恒方程有误。

7. Worked Example – Collision | 例题：碰撞

A trolley A of mass 2.0 kg moves to the right at 3.0 m s⁻¹. It collides with a stationary trolley B of mass 1.0 kg. After the collision, trolley A continues to the right at 1.0 m s⁻¹. Calculate the velocity of trolley B after the collision.

一个质量为 2.0 kg 的小车 A 以 3.0 m s⁻¹ 向右运动。它与一个质量为 1.0 kg 的静止小车 B 碰撞。碰撞后，小车 A 以 1.0 m s⁻¹ 继续向右运动。计算碰撞后小车 B 的速度。

Take right as positive. The given quantities are: m₁ = 2.0 kg, u₁ = +3.0 m s⁻¹, v₁ = +1.0 m s⁻¹, m₂ = 1.0 kg, u₂ = 0 m s⁻¹. Let v₂ be the unknown final velocity of trolley B.

取向右为正。已知量为：m₁ = 2.0 kg，u₁ = +3.0 m s⁻¹，v₁ = +1.0 m s⁻¹，m₂ = 1.0 kg，u₂ = 0 m s⁻¹。设 v₂ 为小车 B 的未知末速度。

Using conservation of momentum:

使用动量守恒：

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

2.0 × 3.0 + 1.0 × 0 = 2.0 × 1.0 + 1.0 × v₂

6.0 = 2.0 + v₂

v₂ = 4.0 m s⁻¹

The positive sign means trolley B moves to the right at 4.0 m s⁻¹. This is physically reasonable because trolley A transferred part of its momentum to trolley B.

正号表示小车 B 以 4.0 m s⁻¹ 的速度向右运动。这在物理上是合理的，因为小车 A 将部分动量传递给了小车 B。

8. Worked Example – Explosion / Recoil | 例题：爆炸/反冲

A rifle of mass 3.0 kg fires a bullet of mass 0.020 kg. The bullet leaves the rifle with a speed of 400 m s⁻¹ horizontally to the right. Calculate the recoil velocity of the rifle.

一支质量为 3.0 kg 的步枪发射一颗质量为 0.020 kg 的子弹。子弹以 400 m s⁻¹ 的水平速度向右离开步枪。计算步枪的反冲速度。

Take right as positive. Before firing, the total momentum of the rifle and bullet system is zero because both are at rest. After firing, the bullet has velocity +400 m s⁻¹ and the rifle has unknown recoil velocity V.

取向右为正。发射前，步枪和子弹系统的总动量为零，因为二者均静止。发射后，子弹的速度为 +400 m s⁻¹，步枪具有未知的反冲速度 V。

By conservation of momentum:

根据动量守恒：

0 = m_bullet v_bullet + m_rifle V

0 = 0.020 × 400 + 3.0 × V

0 = 8.0 + 3.0 V

3.0 V = -8.0

V = -2.7 m s⁻¹

The negative sign means the rifle recoils to the left at 2.7 m s⁻¹. The rifle is much heavier than the bullet, so its recoil speed is much smaller than the bullet’s speed.

负号表示步枪以 2.7 m s⁻¹ 的速度向左反冲。步枪比子弹重得多，因此其反冲速度远小于子弹的速度。

9. Common Mistakes and How to Avoid Them | 常见错误与规避

One of the most common mistakes in momentum questions is forgetting that momentum is a vector. In one dimension, you must assign signs to velocities. A positive direction must be chosen, and any object moving in the opposite direction must have a negative velocity.

动量问题中最常见的错误之一是忘记动量是矢量。在一维问题中，必须给速度赋予正负号。必须选择正方向，任何沿相反方向运动的物体都必须具有负速度。

Another common error is assuming that kinetic energy is always conserved. Kinetic energy is only conserved in elastic collisions. In most real collisions and explosions, kinetic energy is not conserved, but momentum is always conserved if no external resultant force acts.

另一个常见错误是假设动能始终守恒。动能仅在弹性碰撞中守恒。在大多数真实碰撞和爆炸中，动能不守恒，但如果没有外合力作用，动量始终守恒。

Students sometimes confuse mass and weight, or use incorrect units. Mass must be in kg and velocity in m s⁻¹ to get momentum in kg m s⁻¹. Check that you have converted grams to kilograms before substituting into equations.

学生有时会混淆质量和重量，或使用错误的单位。质量必须以 kg 为单位，速度以 m s⁻¹ 为单位，才能得到以 kg m s⁻¹ 为单位的动量。代入方程前检查是否已将克换算为千克。

In explosion problems, a common mistake is to write the total initial momentum as something other than zero when the object is initially at rest. Always start with total initial momentum equal to zero for a stationary object.

在爆炸问题中，一个常见错误是当物体最初静止时，把初始总动量写成非零值。对于静止物体，始终从初始总动量为零开始。

10. Exam Tips Summary | 考试技巧总结

Before starting any momentum problem, write down the key equation and define your positive direction. This simple step prevents many sign errors and helps you organise the given information clearly.

在开始任何动量问题之前，写下关键方程并定义正方向。这一简单步骤可以避免许多符号错误，并帮助你清晰地组织已知信息。

Remember the two key equations for one-dimensional momentum:

记住一维动量的两个关键方程：

J = F Δt = Δp = m v – m u

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

Use the impulse equation when a force acts over a known time interval, and use the conservation equation for collisions and explosions. For elastic collisions, also apply the kinetic energy conservation equation if the question asks for two unknown velocities.

当已知力作用的时间间隔时使用冲量方程，对于碰撞和爆炸使用守恒方程。对于弹性碰撞，如果题目要求求解两个未知速度，还需应用动能守恒方程。

In your final answer, always state both the magnitude and the direction of the velocity or momentum. If the question asks for speed, give only the magnitude without the sign.

在最终答案中，始终说明速度或动量的大小和方向。如果题目要求速率，只给出大小，不带符号。

Finally, practise a range of questions including collisions, explosions and force-time graphs. This will build your confidence in applying the correct equation and checking your answers.

最后，练习各种类型的题目，包括碰撞、爆炸和力-时间图像。这将增强你应用正确方程并检查答案的信心。

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Momentum in One Dimension | 一维动量