Momentum in One Dimension | 一维动量

📚 Momentum in One Dimension | 一维动量

Momentum is one of the most fundamental concepts in physics, linking mass and velocity to describe how objects move and interact. In the Edexcel A-Level Physics specification, the study of momentum in one dimension simplifies real-world collisions and explosions into straight-line problems, allowing students to master the core principles of impulse, conservation laws and energy transfers. Understanding one-dimensional momentum is essential not only for solving examination questions but also for grasping how forces and motion are interconnected in everything from car crashes to rocket launches.

动量是物理学中最基本的概念之一，它将质量和速度联系起来，描述物体如何运动以及如何相互作用。在爱德思 A-Level 物理考纲中，一维动量将现实世界中的碰撞和爆炸简化为直线问题，使学生能够掌握冲量、守恒定律和能量转换的核心原理。理解一维动量不仅对解答考试题目至关重要，而且对于把握从车祸到火箭发射中力与运动的相互联系也必不可少。

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 measure of how difficult it is to stop a moving object. In equation form, momentum p is given by:

动量定义为物体质量与速度的乘积。它衡量了让一个运动物体停下来有多困难。动量 p 的公式为：

p = m × v

where m is the mass in kilograms (kg) and v is the velocity in metres per second (m s⁻¹). The SI unit of momentum is kg m s⁻¹, which is equivalent to N s. Since velocity is a vector, momentum is also a vector quantity, possessing both magnitude and direction. In one-dimensional problems, direction is indicated by a positive or negative sign along a chosen axis.

其中 m 是质量（千克），v 是速度（米每秒）。动量的国际单位是 kg m s⁻¹，等同于 N s。由于速度是矢量，动量也是矢量，既有大小又有方向。在一维问题中，方向通过沿选定轴的正负号来表示。

2. Momentum as a Vector in One Dimension | 一维动量作为矢量

In one-dimensional motion, the direction of momentum is simply given by the sign of the velocity. If we define the positive direction to be to the right, a particle moving to the left will have negative momentum. When adding momenta in a system, vector addition is required: the net momentum of a system is the algebraic sum of the individual momenta, taking signs into account.

在一维运动中，动量的方向简单地由速度的符号给出。如果我们定义向右为正方向，则向左运动的粒子将具有负动量。在将系统中的动量相加时，需要进行矢量加法：系统的净动量是各自动量的代数和，并要考虑符号。

For a two-body system, the total momentum in one dimension can be written as:

对于二体系统，一维总动量可写为：

p_total = m₁v₁ + m₂v₂

where m₁, m₂ are the masses and v₁, v₂ are the velocities with appropriate signs. This simple sign convention is vital for correctly applying conservation laws.

其中 m₁、m₂ 是质量，v₁、v₂ 是带有适当符号的速度。这一简单的符号约定对于正确应用守恒定律至关重要。

3. Impulse and the Impulse-Momentum Theorem | 冲量与冲量-动量定理

When a force acts on a body for a period of time, the product of the force and the time interval is called impulse. Impulse J is defined as:

当一个力在一段时间内作用于物体时，力与时间间隔的乘积称为冲量。冲量 J 定义为：

J = F × Δt

for a constant force, where F is the average force and Δt is the time interval. The impulse-momentum theorem states that the impulse exerted on an object equals the change in its momentum:

对于恒力，其中 F 为平均力，Δt 为时间间隔。冲量-动量定理指出，作用在物体上的冲量等于其动量的变化：

FΔt = Δp = p_final − p_initial

. In one dimension, the force applied in the positive direction increases the momentum in that direction; a force opposing motion reduces momentum. The area under a force–time graph represents the impulse delivered.

该关系式直接由牛顿第二定律 F = Δp/Δt 导出。在一维中，沿正方向作用的力会增加该方向的动量；而阻碍运动的力会减少动量。力-时间图下方的面积代表所施加的冲量。

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

The principle of conservation of linear momentum states that in a closed system (no external forces acting), the total linear momentum remains constant. In one dimension, this means:

线动量守恒原理指出，在一个封闭系统（无外力作用）中，总动量保持不变。在一维中，这意味着：

Total momentum before an event = Total momentum after the event

事件前的总动量 = 事件后的总动量

This conservation arises from Newton’s third law: internal forces between bodies are equal and opposite, so their impulses cancel, leaving the total momentum unchanged. When analysing collisions or explosions, we apply the conservation law along the same straight line. External forces such as friction must be negligible for the law to hold strictly, but in many A-Level problems, we assume ideal conditions.

这种守恒源于牛顿第三定律：物体间的内力大小相等、方向相反，因此它们的冲量相互抵消，总动量保持不变。在分析碰撞或爆炸时，我们沿着同一直线应用守恒定律。要使该定律严格成立，像摩擦力这样的外力必须可以忽略，但在许多 A-Level 题目中，我们假定理想条件。

5. Elastic Collisions in One Dimension | 一维弹性碰撞

An elastic collision is one in which both momentum and kinetic energy are conserved. For two colliding bodies of masses m₁ and m₂, with initial velocities u₁ and u₂ and final velocities v₁ and v₂ (all measured along the same straight line), the conservation laws give two equations:

弹性碰撞是指动量和动能都守恒的碰撞。对于质量分别为 m₁ 和 m₂ 的两个物体，初速度为 u₁、u₂，末速度为 v₁、v₂（均沿同一直线测量），守恒定律得出两个方程：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂²

A useful derived relationship for elastic collisions in one dimension is that the relative speed of approach equals the relative speed of separation:

对于一维弹性碰撞，一个有用的导出关系是趋近相对速度等于分离相对速度：

u₁ − u₂ = −(v₁ − v₂) or v₂ − v₁ = u₁ − u₂

This equation often simplifies problem-solving. Students must be able to solve simultaneous equations to find unknown velocities or masses. Typical exam questions involve a moving object striking a stationary one, or two objects approaching each other.

这个方程通常能简化问题求解。学生必须能够求解联立方程以找到未知的速度或质量。典型的考题涉及运动物体撞击静止物体，或两个物体相互靠近的情形。

6. Inelastic Collisions | 非弹性碰撞

In an inelastic collision, kinetic energy is not conserved, although momentum is still conserved. Some of the initial kinetic energy is transformed into other forms such as internal energy (heat), sound, or permanent deformation of the bodies. In one dimension, only the momentum conservation equation holds, so we cannot use the kinetic energy conservation equation directly.

在非弹性碰撞中，动能不守恒，但动量仍然守恒。一部分初始动能转化为其他形式，如内能（热）、声音或物体的永久变形。在一维情形中，只有动量守恒方程成立，因此我们不能直接使用动能守恒方程。

To solve such problems, additional information is often given, such as the coefficient of restitution e, or the final velocity of one object. The loss in kinetic energy can be calculated as:

要解决此类问题，通常会给出额外信息，例如恢复系数 e，或其中一个物体的末速度。动能损失可按下式计算：

ΔKE = KE_initial − KE_final

Recognising that momentum is always conserved while energy may be dissipated is a key skill. Many everyday collisions are inelastic, such as a ball bouncing off a floor without reaching its original height.

认识到动量始终守恒而能量可能会耗散是一项关键技能。许多日常碰撞都是非弹性的，例如球从地板上弹起却未能回到初始高度。

7. Perfectly Inelastic Collisions | 完全非弹性碰撞

A perfectly inelastic collision is a special case where the colliding bodies stick together after impact and move with a common final velocity, v. In one dimension, momentum conservation gives:

完全非弹性碰撞是碰撞后物体粘在一起并以共同末速度 v 运动的特殊情况。在一维中，动量守恒给出：

m₁u₁ + m₂u₂ = (m₁ + m₂)v

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Kinetic energy is not conserved and the maximum possible energy loss occurs for the given initial conditions. Common examples include a bullet embedding itself in a block (ballistic pendulum) or two railway carriages coupling together. The final kinetic energy can be compared with the initial kinetic energy to find the energy dissipated.

动能不守恒，对于给定的初始条件，会发生最大可能的能量损失。常见例子包括子弹嵌入木块（弹道摆），或两节火车车厢挂接在一起。可以将末动能与初动能进行比较，以求得耗散的能量。

In exam problems, you may be asked to find the final velocity, the impulse experienced by one body, or the loss in kinetic energy. Always remember that momentum direction is crucial: a head-on collision in one dimension means the bodies approach from opposite directions, so signs must be assigned correctly.

在考试问题中，你可能会被要求求出末速度、一个物体所受的冲量或动能的损失。永远记住动量方向至关重要：一维中的迎头碰撞意味着物体从相反方向靠近，因此必须正确赋予符号。

8. Explosions and Recoil | 爆炸与反冲

The principle of conservation of momentum can be applied in reverse: in an explosion or recoil scenario, the total momentum before the event is zero (if initially at rest), and after the separation, the momenta of the fragments must sum to zero. For two fragments:

动量守恒原理也可以反过来应用：在爆炸或反冲场景中，事件前的总动量为零（如果初始静止），分离后各碎片的动量之和必须为零。对于两个碎片：

0 = m₁v₁ + m₂v₂ → m₁v₁ = −m₂v₂

This shows that the fragments move apart with momenta that are equal in magnitude but opposite in direction. The heavier fragment receives a smaller speed. Recoil of a gun after firing a bullet is a classic example; rocket propulsion in one dimension can be analysed by considering the backward ejection of exhaust gases.

这表明碎片以大小相等、方向相反的动量分开。较重的碎片获得较小的速率。开枪后枪的反冲就是一个经典例子；一维火箭推进可以通过考虑废气向后喷射来进行分析。

Energy in explosions comes from chemical or other stored energy, so kinetic energy after an explosion is greater than before. Students must avoid assuming kinetic energy conservation in such scenarios.

爆炸中的能量来自化学能或其他储存能量，因此爆炸后的动能大于爆炸前。学生必须避免在此类场景中假设动能守恒。

9. Force–Time Graphs and Impulse | 力-时间图像与冲量

In many real situations, the force acting on a body is not constant. The impulse delivered can still be found by calculating the area under a force–time graph. For a variable force, the impulse is:

在许多实际情况中，作用在物体上的力不是恒定的。仍可通过计算力-时间图像下的面积来求出传递的冲量。对于变力，冲量为：

Impulse = ∫ F dt = Area under F–t curve

often provide a graph of force vs. time and ask for the change in momentum, the average force, or the maximum force during a collision. For a triangle-shaped graph, the area is ½ × base × height. For a trapezium, the area formula applies.

在一维情境中，面积的正负号表示冲量相对于所选正方向的方向。爱德思考题通常会给出力-时间图，要求求出动量的变化、平均力或碰撞过程中的最大力。对于三角形图像，面积为 ½ × 底 × 高。对于梯形，适用梯形面积公式。

Understanding that the impulse equals change in momentum allows you to link the graph directly to velocities without needing to integrate acceleration over time.

理解冲量等于动量的变化，可以使你直接将图像与速度联系起来，而无需对加速度进行时间积分。

10. Newton’s Third Law and Momentum Conservation | 牛顿第三定律与动量守恒

The link between Newton’s third law and momentum conservation is fundamental. During a collision, object A exerts a force on object B, and B exerts an equal and opposite force on A for the same time interval. Therefore, the impulse on A is equal and opposite to the impulse on B:

牛顿第三定律与动量守恒之间的联系是根本性的。在碰撞过程中，物体 A 对物体 B 施加一个力，同时 B 对 A 施加一个大小相等、方向相反的力，且作用时间相同。因此，A 上的冲量与 B 上的冲量大小相等、方向相反：

J_{A on B} = −J_{B on A}

Since impulse is the change in momentum, the change in momentum of A is Δp_A and of B is Δp_B, and we have Δp_A + Δp_B unchanged. This reasoning holds perfectly in one dimension and underpins all collision and explosion calculations.

由于冲量是动量的变化，A 的动量变化为 Δp_A，B 的动量变化为 Δp_B，并且有 Δp_A + Δp_B = 0。因此总动量保持不变。这一推理在一维中完全成立，并为所有碰撞和爆炸计算奠定了基础。

When drawing free-body diagrams or analysing forces during impact, remember that the forces come in pairs; this pairing is what makes momentum such a powerful tool in multi-body systems.

在画隔离体受力图或分析碰撞过程中的力时，请记住力是成对出现的；正是这种成对性使动量成为多体系统中强大的工具。

11. Solving One-Dimensional Momentum Problems | 一维动量问题求解

To solve typical Edexcel problems, follow a systematic approach:

要解决典型的爱德思问题，可遵循系统方法：

• Sketch the situation and define the positive direction. Label masses, initial velocities and final velocities with appropriate signs.

  画出情景简图并定义正方向。用适当的正负号标注质量、初速度和末速度。

• Write the conservation of momentum equation for the system. If the collision is elastic, also write the kinetic energy conservation equation or the relative speed equation.

  写出系统的动量守恒方程。如果碰撞是弹性的，还要写出动能守恒方程或相对速度方程。

• Solve the equations for unknown quantities. Be careful with signs when substituting numbers; a velocity of 3 m s⁻¹ to the left should be entered as −3 m s⁻¹ if right is positive.

  求解未知量的方程。代入数值时注意符号；如果向右为正，则以 3 m s⁻¹ 向左运动的速度应输入为 −3 m s⁻¹。

• Check that your answers make physical sense: for example, after an elastic head-on collision between objects of equal mass, they exchange velocities.

  检查答案在物理上是否合理：例如，质量相等的物体迎头弹性碰撞后，它们会交换速度。

(continued) If an external impulse is applied, use the impulse-momentum theorem for the individual object. For explosions, set initial total momentum to zero if the system starts from rest.

（续）如果施加了外部冲量，则对单个物体使用冲量-动量定理。对于爆炸，如果系统从静止开始，则将初始总动量设为零。

ation divided by speed of approach: e = (v₂ − v₁)/(u₁ − u₂). For perfectly elastic collisions, e = 1; for perfectly inelastic, e = 0.

考试题目经常将动量与能量概念交织在一起。一种常见的风格是要求求出恢复系数 e，它等于分离速度除以接近速度：e = (v₂ − v₁)/(u₁ − u₂)。对于完全弹性碰撞，e = 1；对于完全非弹性碰撞，e = 0。

12. Common Misconceptions and Exam Tips | 常见误区与考试技巧

Many students mistakenly believe that momentum and kinetic energy are both always conserved. Remember: momentum is conserved in the absence of external forces; kinetic energy is only conserved in elastic collisions. Confusing mass and weight, or treating momentum as a scalar, are other frequent errors. Always use SI units: kg for mass, m s⁻¹ for velocity.

许多学生错误地认为动量和动能总是同时守恒。请记住：在没有外力的情况下动量守恒；动能仅在弹性碰撞中守恒。混淆质量与重量，或将动量当作标量处理，是其他常见错误。务必使用国际单位：质量用 kg，速度用 m s⁻¹。

In one-dimensional problems, a clear sign convention and well-labelled diagrams prevent lost marks. When a question asks for 'magnitude of impulse' or 'speed', give a positive value. If it asks for velocity, include the direction or sign. Practice past paper questions involving graphs, oblique direction change simplified to one dimension, and multi-step analysis to build confidence.

在一维问题中，清晰的符号约定和标注良好的示意图可防止失分。当问题要求“冲量的大小”或“速率”时，给出正值。如果要求速度，则要包括方向或符号。练习包含图像、简化为直线方向的斜向变化以及多步分析的历年真题，以建立信心。

Finally, always read the question carefully to identify whether a collision is described as elastic, inelastic, or perfectly inelastic—this tells you which equations apply. With disciplined practice, momentum in one dimension becomes a predictable and rewarding topic on the Edexcel Physics paper.

最后，务必仔细审题，判断碰撞被描述为弹性、非弹性还是完全非弹性——这告诉你哪些方程适用。通过有规律的练习，一维动量将成为爱德思物理试卷上一个可预测且易得分的主题。

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