📚 Momentum and Impulse in GCSE WJEC Mathematics | GCSE WJEC 数学：动量与冲量 考点精讲
In GCSE WJEC Mathematics, you may be asked to solve problems involving momentum and impulse. These concepts come from physics, but the maths paper expects you to interpret a brief, use given formulas correctly, handle units carefully, and apply algebra to find unknown quantities. This revision guide walks you through every key skill you need.
在 GCSE WJEC 数学考试中，你可能会遇到涉及动量和冲量的应用题。这些概念源自物理，但数学卷子要求你读懂题目情境，准确使用给定的公式，仔细处理单位，并运用代数求解未知量。本复习指南将带你逐一掌握所有必要的核心技能。
1. What Is Momentum? | 什么是动量？
Momentum (p) is a measure of 'how much motion' an object has. It depends on two quantities: the mass of the object and its velocity. The formula you will use is:
动量 (p) 是衡量物体“运动量”的物理量。它取决于两个量：物体的质量和速度。你将使用的公式是：
p = m × v
where p is momentum in kilogram metres per second (kg m s⁻¹), m is mass in kilograms (kg), and v is velocity in metres per second (m s⁻¹). In many exam questions, this formula will be provided, but you must recognise what each symbol stands for.
其中 p 是动量，单位为千克米每秒 (kg m s⁻¹)；m 是质量，单位为千克 (kg)；v 是速度，单位为米每秒 (m s⁻¹)。在大多数考题中，此公式会给出，但你必须清楚每个符号的含义。
2. Units and Conversions | 单位与换算
Correct unit conversion is one of the most common sources of error in momentum questions. You must always work with mass in kilograms and velocity in metres per second. Often you will need to convert:
正确的单位换算是动量问题中最常见的错误来源之一。你必须始终使用以千克为单位的质量和以米每秒为单位的速度进行计算。通常你需要进行以下换算：
- Grams to kilograms: divide by 1000. 1 kg = 1000 g, so 250 g = 0.25 kg. | 克转千克：除以 1000。1 kg = 1000 g，因此 250 g = 0.25 kg。
- Kilometres per hour to metres per second: divide by 3.6. 1 m s⁻¹ = 3.6 km h⁻¹. So 72 km h⁻¹ = 72 ÷ 3.6 = 20 m s⁻¹. | 公里每小时转米每秒：除以 3.6。1 m s⁻¹ = 3.6 km h⁻¹。因此 72 km h⁻¹ = 72 ÷ 3.6 = 20 m s⁻¹。
v (m s⁻¹) = v (km h⁻¹) ÷ 3.6
Always check your units before substituting numbers into a formula. If you forget, your final answer will be wrong by a factor of 1000 or more.
代入数字前一定要检查单位。如果忘记换算，你的最终答案可能会因 1000 倍甚至更大的误差而出错。
3. Momentum As a Vector | 动量是矢量
Momentum is a vector quantity—it has both size and direction. In one‑dimensional problems, you can use positive and negative signs to indicate direction. For example, you might take motion to the right as positive, and motion to the left as negative.
动量是矢量——既有大小也有方向。在一维问题中，你可以用正号和负号表示方向。例如，你可以设定向右运动为正，向左运动为负。
This becomes crucial when an object rebounds or changes direction. A ball moving at +4 m s⁻¹ that hits a wall and bounces back at −2 m s⁻¹ has experienced a change in momentum that depends on the sign of each velocity. If you ignore signs, your calculation of impulse or change in momentum will be incorrect.
当物体反弹或改变方向时，这一点至关重要。以 +4 m s⁻¹ 运动的球撞击墙壁后以 −2 m s⁻¹ 弹回，它所经历的动量变化取决于每个速度的符号。如果忽略正负号，你计算出的冲量或动量变化将是错误的。
4. What Is Impulse? | 什么是冲量？
Impulse (I) measures the effect of a force acting over a period of time. The formula for impulse is:
冲量 (I) 衡量力在一段时间内作用所产生的效果。冲量的公式是：
I = F × t
where F is the average force in newtons (N) and t is the time in seconds (s). Impulse has the same units as momentum: N s, which is equivalent to kg m s⁻¹. In WJEC mathematics questions, you may be told that a constant force acts for a given time and asked to find the resulting change in motion.
其中 F 是平均力，单位为牛顿 (N)；t 是时间，单位为秒 (s)。冲量与动量的单位相同：N s，等同于 kg m s⁻¹。在 WJEC 数学试题中，可能会告诉你一个恒力作用了多长时间，然后要求你求出由此产生的运动变化。
5. The Impulse–Momentum Theorem | 冲量–动量定理
The link between impulse and momentum is one of the most important relationships in these problems. The impulse–momentum theorem states that the impulse applied to an object equals its change in momentum:
冲量与动量之间的联系是这类问题中最重要的关系之一。冲量–动量定理指出，作用于物体的冲量等于其动量的变化量：
F t = m v − m u
or more compactly, F t = m (v − u). Here u is the initial velocity and v is the final velocity. This formula allows you to find an unknown force, time, mass, or velocity change when the other quantities are known. Remember to give u and v the correct signs relative to your chosen positive direction.
或者更简洁地表示为 F t = m (v − u)。这里 u 是初速度，v 是末速度。该公式使你能在其他量已知的情况下求出未知的力、时间、质量或速度变化量。请务必根据你选择的正方向为 u 和 v 赋予正确的符号。
6. Conservation of Momentum in Collisions | 碰撞中的动量守恒
In a closed system with no external forces, the total momentum before a collision equals the total momentum after the collision. For two objects A and B, this is written as:
在没有外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量。对于两个物体 A 和 B，可以写成：
m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂
where m₁ and m₂ are the masses, u₁ and u₂ are initial velocities, and v₁ and v₂ are final velocities. In GCSE WJEC mathematics, you may be given a diagram of a collision and some of these values, and asked to find a missing velocity or mass.
其中 m₁ 和 m₂ 为质量，u₁ 和 u₂ 为初速度，v₁ 和 v₂ 为末速度。在 GCSE WJEC 数学中，你可能会看到一幅碰撞示意图以及其中部分数值，被要求求出缺失的速度或质量。
Be especially careful with direction signs when using this equation. If one object moves in the opposite direction, its velocity must be entered with a negative sign. This is a common trick in exam questions.
使用该公式时，要特别注意方向符号。如果某个物体朝相反方向运动，其速度必须用负号代入。这是考试题目中常见的陷阱。
7. Step‑by‑Step Problem Solving | 分步解题策略
Following a clear method will help you avoid mistakes. Use these steps every time:
遵循清晰的解题方法有助于避免错误。每次做题都请按以下步骤操作：
- Identify the knowns and the unknown. Write down what mass, velocities, time or force are given, and what you need to find. | 明确已知量和未知量。写下给出的质量、速度、时间或力，以及你需要求什么。
- Convert all units to SI. Ensure mass is in kg, velocity in m s⁻¹, time in s, force in N. | 将所有单位转换为国际单位。确保质量以 kg 计，速度以 m s⁻¹ 计，时间以 s 计，力以 N 计。
- Choose a positive direction and assign signs. Draw a simple arrow diagram if it helps. | 选定正方向并赋予符号。如有必要，画一个简单的箭头图辅助。
- Select the appropriate formula. Write down p = mv, I = Ft, Ft = m(v − u), or the conservation equation. | 选择合适的公式。写下 p = mv、I = Ft、Ft = m(v − u) 或动量守恒方程。
- Substitute carefully and solve. Show each step of your algebra clearly. | 仔细代入并求解。清晰地展示每一步代数运算。
- Check your answer’s units and direction. Make sure the magnitude is realistic and the sign matches your chosen positive direction. | 检查答案的单位和方向。确保数值大小合理，且符号与你选择的正方向一致。
8. Typical WJEC Style Questions | WJEC 典型考题
Example 1 – Rebound and change in momentum
例 1 – 反弹与动量变化
A tennis ball of mass 0.06 kg hits a wall at 18 m s⁻¹ and rebounds at 12 m s⁻¹ in the opposite direction. Calculate the change in momentum.
一个质量为 0.06 kg 的网球以 18 m s⁻¹ 的速度撞击墙壁，并以 12 m s⁻¹ 的速度沿相反方向弹回。计算动量的变化量。
Solution:
Take the initial direction as positive. Then u = +18 m s⁻¹, v = −12 m s⁻¹.
Initial momentum = m u = 0.06 × 18 = 1.08 kg m s⁻¹.
Final momentum = m v = 0.06 × (−12) = −0.72 kg m s⁻¹.
tude is 1.80 kg m s⁻¹, and the negative sign shows the change is opposite to the initial motion.
解：
设初速度方向为正。则 u = +18 m s⁻¹，v = −12 m s⁻¹。
初动量 = m u = 0.06 × 18 = 1.08 kg m s⁻¹。
末动量 = m v = 0.06 × (−12) = −0.72 kg m s⁻¹。
动量变化 = 末 − 初 = −0.72 − 1.08 = −1.80 kg m s⁻¹。变化量为 1.80 kg m s⁻¹，负号表示变化方向与初始运动方向相反。
Example 2 – Impulse and final velocity
例 2 – 冲量与末速度
A constant force of 400 N acts on a stationary 80 kg go‑kart for 3 seconds. Find its final velocity.
一个 400 N 的恒力作用在一辆静止的 80 kg 卡丁车上，持续 3 秒。求末速度。
Solution:
Using F t = m v − m u, with u = 0: 400 × 3 = 80 × v ⇒ 1200 = 80 v ⇒ v = 15 m s⁻¹ in the direction of the force.
解：
使用 F t = m v − m u，其中 u = 0：400 × 3 = 80 × v ⇒ 1200 = 80 v ⇒ v = 15 m s⁻¹，方向与力的方向相同。
Example 3 – Conservation of momentum
例 3 – 动量守恒
A 2 kg trolley moving at 5 m s⁻¹ collides with a stationary 3 kg trolley. After the collision, they lock together. Find their common speed.
一辆 2 kg 的小车以 5 m s⁻¹ 的速度运动，与一辆静止的 3 kg 小车碰撞。碰撞后两者锁在一起。求它们的共同速度。
Solution:
Total momentum before = (2 × 5) + (3 × 0) = 10 kg m s⁻¹.
After, total mass = 5 kg. Let common velocity be V. Then 5 V = 10 ⇒ V = 2 m s⁻¹ in the original direction.
解：
碰撞前总动量 = (2 × 5) + (3 × 0) = 10 kg m s⁻¹。
碰撞后总质量 = 5 kg。设共同速度为 V。则 5 V = 10 ⇒ V = 2 m s⁻¹，方向与原方向相同。
9. Common Mistakes and How to Avoid Them | 常见错误与规避方法
| Mistake / 错误 | How to avoid it / 如何避免 |
|---|---|
| Using grams instead of kilograms | Always convert mass to kg before substituting into any formula. |
| 使用克而不是千克 | 代入公式前，务必先将质量转换为千克。 |
| Forgetting that velocity is a vector – ignoring direction signs | Define a positive direction and stick to it. Mark velocities with + or − accordingly. |
| 忘记速度是矢量——忽略方向符号 | 规定正方向，并始终遵循。给相应的速度标记 + 或 −。 |
| Confusing 'change in momentum' with 'final momentum' | Change in momentum = final momentum − initial momentum. Read the question carefully. |
| 混淆“动量变化”与“末动量” | 动量变化 = 末动量 − 初动量。仔细读题。 |
| Mixing up force and momentum units | Force is in N, momentum in kg m s⁻¹. Impulse bridges them: 1 N s = 1 kg m s⁻¹. |
| 混淆力的单位与动量单位 | 力用 N，动量用 kg m s⁻¹。冲量将两者联系起来：1 N s = 1 kg m s⁻¹。 |
10. Practice Calculations | 计算练习
Question 1: A car of mass 1500 kg accelerates from 8 m s⁻¹ to 22 m s⁻¹ in 7 seconds. Calculate the average force acting on it.
问题 1：一辆质量为 1500 kg 的汽车在 7 秒内从 8 m s⁻¹ 加速到 22 m s⁻¹。计算作用在汽车上的平均力。
Solution: F t = m(v−u) ⇒ F × 7 = 1500 × (22−8) = 1500 × 14 = 21000 ⇒ F = 3000 N.
解：F t = m(v−u) ⇒ F × 7 = 1500 × (22−8) = 1500 × 14 = 21000 ⇒ F = 3000 N。
Question 2: A bullet of mass 15 g is fired at 320 m s⁻¹ into a 1.5 kg wooden block resting on a smooth surface. The bullet becomes embedded. Find the speed of the block immediately after impact.
问题 2：一颗质量为 15 g 的子弹以 320 m s⁻¹ 的速度射入一个静止在光滑表面上的 1.5 kg 的木块。子弹嵌入其中。求木块在撞击后的瞬时速度。
Solution: Convert 15 g = 0.015 kg. Total momentum before = 0.015×320 + 1.5×0 = 4.8 kg m s⁻¹.
After, total mass = 1.515 kg. Let V be common speed: 1.515 V = 4.8 ⇒ V ≈ 3.17 m s⁻¹ (to 3 s.f.).
解：换算 15 g = 0.015 kg。碰撞前总动量 = 0.015×320 + 1.5×0 = 4.8 kg m s⁻¹。
碰撞后总质量 = 1.515 kg。设共同速度为 V：1.515 V = 4.8 ⇒ V ≈ 3.17 m s⁻¹（保留三位有效数字）。
11. Using the Formula Sheet Effectively | 有效使用公式表
requires them will provide the relevant formula in the text. You will see expressions like "use the formula p = m v" or "impulse = force × time" ant to work faster; focus on interpreting the data and following the steps outlined above.
在 WJEC GCSE 数学考试中，公式表通常不会列出动量方程，但每道涉及动量的题目都会在题干中给出相关公式。你会看到诸如“使用公式 p = m v”或“冲量 = 力 × 时间”这样的表述。你的任务是识别该用哪个公式，并正确代入数字。无须专门记忆公式，除非你想提高答题速度；重点应放在理解数据和遵循上述解题步骤上。
12. Summary and Key Takeaways | 总结与要点
pulse equals change in momentum; and total momentum is conserved in collisions when no external force acts. Practise converting units and assigning signs until it becomes automatic. By mastering these skills, you will be able to tackle any momentum problem the exam throws at you with confidence.
WJEC GCSE 数学中的动量和冲量问题考察你运用科学公式、处理单位以及考虑方向的能力。请始终牢记：动量 = 质量 × 速度；冲量 = 力 × 时间；冲量等于动量的变化；在没有外力作用的碰撞中，总动量守恒。多加练习单位换算和符号赋值，直至变成一种习惯。掌握了这些技能，你就能自信地应对考试中出现的任何动量问题。
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