Momentum and Impulse | 动量与冲量考点精讲

📚 Momentum and Impulse | 动量与冲量考点精讲

Momentum and impulse are fundamental concepts in mechanics that appear in the IGCSE Mathematics (Higher Tier) syllabus. Understanding these principles helps solve problems involving collisions, explosions, and varying forces. This article breaks down key definitions, formulas, conservation laws, graph interpretations, and typical exam questions, providing a clear bilingual guide for effective revision.

动量和冲量是力学中的基本概念，出现在 IGCSE 数学（高阶）大纲中。理解这些原理有助于解决涉及碰撞、爆炸和变力的问题。本文拆解了关键定义、公式、守恒定律、图像解读以及典型考题，提供了清晰的中英双语复习指南，助力高效备考。

1. What is Momentum? | 动量是什么？

Momentum is a measure of how difficult it is to stop a moving object. It is defined as the product of an object’s mass and its velocity. Because velocity is a vector, momentum is also a vector quantity — it has both magnitude and direction.

动量是衡量一个运动物体停下来难易程度的物理量。它被定义为物体质量与速度的乘积。由于速度是矢量，动量也是一个矢量——具有大小和方向。

p = m v

The symbol p represents momentum, m is mass (in kg), and v is velocity (in m s⁻¹). The SI unit of momentum is kg m s⁻¹, which is equivalent to N s (newton second).

符号 p 表示动量，m 为质量（单位 kg），v 为速度（单位 m s⁻¹）。动量的国际单位是 kg m s⁻¹，它等同于 N s（牛·秒）。

2. Momentum as a Vector | 动量作为矢量

Direction matters when working with momentum. In one-dimensional problems, you must assign a positive direction. An object moving in the opposite direction has negative momentum. Adding momentum values without considering direction will lead to incorrect results.

处理动量时方向至关重要。在一维问题中，必须规定一个正方向。反向运动的物体具有负动量。不考虑方向直接相加动量数值会得出错误结果。

For example, a 2 kg trolley moving right at 3 m s⁻¹ has momentum +6 kg m s⁻¹, while an identical trolley moving left at 3 m s⁻¹ has momentum –6 kg m s⁻¹. The total momentum of the system is zero.

例如，一辆 2 kg 的小车以 3 m s⁻¹ 的速度向右运动，动量为 +6 kg m s⁻¹；而另一辆同样的小车以 3 m s⁻¹ 的速度向左运动，动量为 –6 kg m s⁻¹。该系统的总动量为零。

3. Calculating Momentum | 动量计算

To find the momentum of an object, simply multiply its mass by its instantaneous velocity. Always ensure mass is in kilograms and velocity is in metres per second. The conversion of units may be necessary before calculation.

要计算物体的动量，只需将质量与瞬时速度相乘。务必确保质量以千克为单位，速度以米每秒为单位。计算前有时需要进行单位换算。

For a car of mass 1200 kg moving at 15 m s⁻¹, its momentum is 1200 × 15 = 18 000 kg m s⁻¹. When velocity changes, the momentum changes proportionally.

一辆质量为 1200 kg、以 15 m s⁻¹ 行驶的汽车，其动量为 1200 × 15 = 18 000 kg m s⁻¹。当速度变化时，动量也成比例变化。

4. Conservation of Momentum | 动量守恒

The principle of conservation of momentum states that in a closed system (no external forces act), the total momentum before an interaction is equal to the total momentum after the interaction. This law is fundamental for solving collision and explosion problems.

动量守恒原理指出，在一个封闭系统（无外力作用）中，相互作用前的总动量等于相互作用后的总动量。这一定律是解决碰撞和爆炸问题的基础。

Total momentum before = Total momentum after
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

相互作用前总动量 = 相互作用后总动量
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Here u₁, u₂ are initial velocities and v₁, v₂ are final velocities. In IGCSE maths, you will typically apply this to two-object systems moving along the same straight line.

此处 u₁、u₂ 为初速度，v₁、v₂ 为末速度。在 IGCSE 数学中，通常会将此原理应用于沿同一直线运动的两物体系统。

5. Impulse Defined | 冲量的定义

Impulse is the effect of a force acting over a period of time. It is defined as the product of the average force and the time interval during which the force acts. Impulse is also a vector quantity, having the same direction as the force.

冲量是力在一段时间内作用的效应。它被定义为平均力与力作用时间的乘积。冲量也是矢量，方向与力的方向相同。

Impulse = F × t

The unit of impulse is newton second (N s), which is identical to the unit of momentum (kg m s⁻¹). This identity is crucial for linking impulse to momentum change.

冲量的单位是牛·秒（N s），与动量的单位（kg m s⁻¹）一致。这一等同关系对理解冲量与动量变化的联系至关重要。

6. Impulse-Momentum Theorem | 冲量-动量定理

The impulse-momentum theorem states that the impulse applied to an object equals the change in its momentum. This can be derived from Newton’s second law and is written as:

冲量-动量定理指出，施加在物体上的冲量等于其动量的变化。这可由牛顿第二定律推导得出，写为：

Impulse = Δp = m(v – u)

Where u is the initial velocity and v is the final velocity. This equation allows you to calculate the average force if the contact time is known, or to deduce velocity changes from a known impulse.

其中 u 是初速度，v 是末速度。通过该方程，若已知接触时间便可求出平均力，或根据已知冲量推断速度变化。

7. Impulse from Force-Time Graphs | 力-时间图像的面积——冲量

When a force varies with time, the impulse can be found by calculating the area under a force-time graph. For a constant force, the area is a rectangle; for a linearly varying force, it may be a triangle or trapezium.

当力随时间变化时，可以通过计算力-时间图像下的面积求得冲量。若为恒力，面积为矩形；若力线性变化，面积可能是三角形或梯形。

The relationship is: Impulse = area under F–t graph. This is especially useful for problems where a graph is given and you need to determine the average force or velocity change.

关系式为：冲量 = F–t 图下的面积。这在给定图像并要求确定平均力或速度变化的问题中特别有用。

8. Collisions and Explosions | 碰撞与爆炸

Collisions: When two objects collide, total momentum is conserved. In IGCSE mathematics, collisions are often treated as inelastic (objects move together after impact) or as perfectly elastic (kinetic energy is conserved; although momentum is always the focus). Always identify the masses and velocities before and after, assign a positive direction, and form an equation using the conservation principle.

碰撞：两物体相撞时，总动量守恒。在 IGCSE 数学中，碰撞常被视为完全非弹性（碰后粘在一起运动）或完全弹性（动能守恒；但始终以动量为重点）。解题时要确定碰前、碰后的质量和速度，规定正方向，利用守恒原理建立方程。

Explosions: An explosion separates a single object into fragments. The total momentum before the explosion is zero (if initially stationary), so the total momentum after is also zero. The fragments move apart with momenta that sum to zero vectorially.

爆炸：爆炸使一个物体分离为碎片。爆炸前的总动量为零（若最初静止），因此爆炸后总动量也为零。碎片分离时的动量矢量之和为零。

9. Worked Example 1: Collision | 例题精讲：碰撞

Question: A 3 kg trolley moving at 4 m s⁻¹ to the right catches up a 2 kg trolley moving at 1 m s⁻¹ to the right. After collision, they stick together. Find their common velocity.

题目：一辆 3 kg 小车以 4 m s⁻¹ 的速度向右运动，追上一辆以 1 m s⁻¹ 向右运动的 2 kg 小车。碰撞后两车粘在一起，求它们的共同速度。

Let the positive direction be to the right. Total momentum before = (3×4) + (2×1) = 12 + 2 = 14 kg m s⁻¹. After collision, combined mass = 5 kg, let common velocity be v. Conserving momentum: 5v = 14, so v = 2.8 m s⁻¹ (to the right).

规定向右为正方向。碰前总动量 = (3×4) + (2×1) = 12 + 2 = 14 kg m s⁻¹。碰后总质量 = 5 kg，设共同速度为 v。动量守恒：5v = 14，解得 v = 2.8 m s⁻¹（向右）。

10. Worked Example 2: Impulse | 例题精讲：冲量

Question: A football of mass 0.42 kg hits a wall at 15 m s⁻¹ and rebounds at 12 m s⁻¹. The contact time is 0.08 s. Calculate the average force exerted on the ball by the wall.

题目：一个质量为 0.42 kg 的足球以 15 m s⁻¹ 的速度撞墙，并以 12 m s⁻¹ 反弹。接触时间为 0.08 s。求墙壁对球的平均作用力。

Take the initial direction as positive. Initial momentum = 0.42 × 15 = +6.3 kg m s⁻¹. Final velocity is negative, so final momentum = 0.42 × (–12) = –5.04 kg m s⁻¹. Change in momentum Δp = –5.04 – 6.3 = –11.34 kg m s⁻¹. Impulse = average force × time, so F × 0.08 = –11.34, giving F = –141.75 N. The negative sign means the force acts opposite to the initial direction. Average force ≈ 142 N (away from the wall).

取初速度方向为正。初动量 = 0.42 × 15 = +6.3 kg m s⁻¹。末速度为负，故末动量 = 0.42 × (–12) = –5.04 kg m s⁻¹。动量变化 Δp = –5.04 – 6.3 = –11.34 kg m s⁻¹。冲量 = 平均力 × 时间，即 F × 0.08 = –11.34，得 F = –141.75 N。负号表示力的方向与初速度方向相反。平均力约为 142 N（方向远离墙壁）。

11. Common Mistakes | 常见错误

Forgetting direction: Not assigning a positive direction or forgetting to attach negative signs to velocities in the opposite direction will give an incorrect total momentum.

忘记方向：不规定正方向或忘记给反向的速度加负号，会导致总动量计算错误。

Mixing units: Using grams instead of kilograms, or cm s⁻¹ instead of m s⁻¹. Always convert to SI units before calculations.

混淆单位：用克而不是千克，或用 cm s⁻¹ 而不是 m s⁻¹。计算前务必转换为国际单位。

Wrong sign for velocity after rebound: When an object bounces, its final velocity has the opposite sign. Students often compute change in momentum as m(v – u) without considering the sign change, resulting in an impulse that is too small.

反弹后速度的符号错误：物体反弹时末速度符号应相反。学生经常用 m(v – u) 计算动量变化时不考虑符号变化，导致冲量数值偏小。

12. Exam Tips | 考试技巧

State the law explicitly: When using conservation of momentum, write: “By the principle of conservation of momentum, total momentum before = total momentum after.” This helps secure method marks.

明确写出定律：使用动量守恒时，先写出：“根据动量守恒原理，碰前总动量 = 碰后总动量。”这有助于获得方法分。

Draw a diagram: Sketch the scenario with arrows showing direction and label masses and velocities. This reduces sign errors and clarifies the problem.

画图示意：用箭头标出方向，画出简图，并标注质量和速度。这能减少符号错误，清晰呈现题目信息。

Check units: Give final answers with correct units (kg m s⁻¹ for momentum, N s for impulse, N for force) and round appropriately if instructed.

检查单位：给出正确答案的单位（动量用 kg m s⁻¹，冲量用 N s，力用 N），若题目要求则适当保留有效数字。

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Momentum and Impulse | 动量与冲量 考点精讲