Momentum and Impulse: Key Concepts for KS3 Maths | KS3 数学：动量与冲量考点精讲

📚 Momentum and Impulse: Key Concepts for KS3 Maths | KS3 数学：动量与冲量考点精讲

In KS3 Maths, you may come across problems that link algebra and proportional reasoning with real‑world physics ideas like momentum and impulse. Understanding these topics will sharpen your ability to rearrange formulas, substitute values correctly, and interpret graphs – all essential maths skills.

在 KS3 数学中，你可能会遇到将代数、比例推理与动量、冲量等实际物理概念相结合的题目。掌握这些内容能提升你转化公式、正确代入数值以及解读图像的能力，这些都是重要的数学技能。

1. What Is Momentum? | 什么是动量？

Momentum is a measurement of how much motion an object has. It depends on two things: the object’s mass and its velocity. The greater the mass or the faster the speed, the more momentum the object carries.

动量用来衡量一个物体具有的“运动量”，它取决于两个因素：物体的质量和速度。质量越大或速度越快，物体的动量就越大。

In maths terms, momentum is directly proportional to both mass and velocity when the other quantity is fixed. This relationship can be described by the formula:

从数学角度看，当其中一个量固定时，动量与质量和速度都成正比。这种关系可以用公式表示：

2. Momentum Formula and Units | 动量公式与单位

The standard momentum formula is:

标准的动量公式为：

p = m × v

where p stands for momentum, m is mass in kilograms (kg), and v is velocity in metres per second (m/s). Therefore the unit of momentum is kg m/s. In KS3 maths questions, you may also see mass in grams (g) or velocity in km/h; always convert to standard units first.

其中 p 代表动量，m 是质量（单位千克，kg），v 是速度（单位米每秒，m/s）。因此动量的单位是 kg m/s。在 KS3 数学题中，有时质量会以克（g）或速度以公里每小时（km/h）给出，务必先换算成标准单位。

To find mass or velocity when momentum is known, use inverse operations: m = p ÷ v and v = p ÷ m. This is typical rearrangement practice.

已知动量求质量或速度时，使用逆运算：m = p ÷ v，v = p ÷ m。这是典型的公式变形练习。

3. What Is Impulse? | 什么是冲量？

Impulse measures the overall effect of a force acting over a period of time. When you push or pull something for a certain duration, you deliver an impulse that changes the object’s momentum.

冲量用来衡量一个力在一段时间内产生的总体效果。当你对物体施加推力或拉力并持续一段时间时，你就提供了一个冲量，这个冲量会改变物体的动量。

Mathematically, impulse is linked to the product of force and time. It is a useful concept when a force is not constant, but in KS3 we work with constant forces.

从数学上讲，冲量与力乘以时间有关。当力不是恒力时这个概念更有用，但在 KS3 阶段我们只涉及恒力。

4. Impulse Formula and Units | 冲量公式与单位

The impulse delivered by a constant force is given by:

恒力产生的冲量公式为：

I = F × t

where I is impulse in newton seconds (N s), F is force in newtons (N), and t is time in seconds (s). Notice that 1 N s is equivalent to 1 kg m/s, so impulse and momentum share the same unit type.

其中 I 表示冲量（单位牛秒，N s），F 是力（单位牛顿，N），t 是时间（单位秒，s）。注意 1 N s 等同于 1 kg m/s，因此冲量和动量属于同一类单位。

Again, rearranging gives F = I ÷ t and t = I ÷ F. These algebraic steps are frequently tested in KS3 problem solving.

同样，公式变形可得 F = I ÷ t 和 t = I ÷ F。这些代数步骤在 KS3 解题中经常考查。

5. Relationship Between Momentum and Impulse | 动量与冲量的关系

The key connection is that the impulse applied to an object equals the change in its momentum. This is written as:

关键联系在于：物体受到的冲量等于它动量的变化量。写作：

I = Δp = m × Δv

Here Δ (delta) means ‘change in’. So if an object’s velocity changes from u to v, the change in momentum is m(v – u) and this must equal the impulse F t.

这里 Δ（德尔塔）表示“变化量”。因此如果物体速度从 u 变为 v，动量变化量为 m(v − u)，并且它必须等于冲量 F t。

This relationship allows you to solve problems linking force, time, mass and velocity change, using only one equation I = m(v – u). It’s a powerful example of direct proportion and linear equations.

利用这个关系，你可以用一个方程 I = m(v − u) 解决涉及力、时间、质量和速度变化的问题。这是正比例和线性方程的有力示例。

6. Conservation of Momentum Basics | 动量守恒基础

In isolated systems, total momentum before an event equals total momentum after. For two objects colliding or pushing apart, we can write:

在孤立系统中，事件前的总动量等于事件后的总动量。对于两个碰撞或相互推开的物体，可写为：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

This looks complicated, but KS3 questions often keep one object stationary initially, reducing the equation to something manageable like m₁u₁ = (m₁ + m₂)v for a perfectly inelastic collision. It’s essentially solving for an unknown using balance.

这看起来复杂，但 KS3 题目常令其中一个物体最初静止，从而将方程简化为可控形式，如完全非弹性碰撞中的 m₁u₁ = (m₁ + m₂)v。实质上就是利用等量关系求解未知数。

Understanding conservation of momentum helps you practise equation setting and substitution.

理解动量守恒有助于练习建立方程和代入数值。

7. Velocity–Time Graphs and Impulse | 速度–时间图像与冲量

When a constant force acts, an object accelerates uniformly, producing a straight sloping line on a velocity–time graph. The area under a force–time graph represents impulse, but in KS3 maths we often use velocity–time graphs to find acceleration and then use F = ma to link back to impulse.

当恒力作用时，物体均匀加速，在速度–时间图上形成一条倾斜直线。力–时间图下的面积代表冲量，但在 KS3 数学中，我们经常利用速度–时间图求加速度，再通过 F = ma 与冲量联系起来。

For example, a graph of velocity against time can give Δv over a known interval. Multiplying by mass gives momentum change, which equals impulse. This integrates graph interpretation with formula work.

例如，速度–时间图可以给出已知时间段内的 Δv。乘以质量就得到动量变化，即冲量。这便把图像解读与公式运用结合起来了。

8. Worked Example 1: Car Braking | 计算示例 1：汽车制动

A car of mass 1200 kg reduces its speed from 20 m/s to 5 m/s in 3 seconds. Calculate (a) the change in momentum, and (b) the average braking force.

一辆质量为 1200 kg 的汽车在 3 秒内速度从 20 m/s 减至 5 m/s。计算 (a) 动量变化量，(b) 平均制动力。

(a) Δp = m(v – u) = 1200 × (5 – 20) = 1200 × (–15) = –18 000 kg m/s. The negative sign shows the momentum decreases.

(a) Δp = m(v − u) = 1200 × (5 − 20) = 1200 × (−15) = −18 000 kg m/s。负号表示动量减少。

(b) Using I = F t = Δp, so F = Δp ÷ t = –18 000 ÷ 3 = –6000 N. The force is negative, meaning it acts opposite to motion. The magnitude is 6000 N.

(b) 利用 I = F t = Δp，因此 F = Δp ÷ t = −18 000 ÷ 3 = −6000 N。力为负值表示与运动方向相反，大小为 6000 N。

9. Worked Example 2: Football Kick | 计算示例 2：足球射门

A stationary football of mass 0.43 kg is kicked and flies off at 25 m/s. If the foot is in contact with the ball for 0.08 s, what is the average force applied?

一个静止的足球质量为 0.43 kg，被踢后以 25 m/s 飞出。若脚与球接触时间为 0.08 s，施加的平均力是多少？

Initial momentum = 0. Final momentum = 0.43 × 25 = 10.75 kg m/s. Δp = 10.75 kg m/s. Impulse = F t = 10.75, so F = 10.75 ÷ 0.08 = 134.375 N (approx. 134 N).

初动量 = 0。末动量 = 0.43 × 25 = 10.75 kg m/s。Δp = 10.75 kg m/s。冲量 = F t = 10.75，因此 F = 10.75 ÷ 0.08 = 134.375 N（约 134 N）。

This shows how a short contact time requires a large force to produce a large momentum change.

这说明较短的接触时间需要较大的力才能产生巨大的动量变化。

10. Direct and Inverse Proportion Traps | 正比与反比常见陷阱

Many KS3 questions test whether you recognise that for a given momentum, mass and velocity are inversely proportional. If velocity triples and mass stays the same, momentum triples. But to keep momentum constant while velocity doubles, mass must halve.

许多 KS3 题考查你是否意识到：在动量一定时，质量与速度成反比。如果速度变为 3 倍而质量不变，动量也变为 3 倍。但如果要保持动量不变而速度加倍，质量必须减半。

Similarly, for a fixed impulse, force and time are inversely proportional: doubling the force requires halving the contact time to deliver the same impulse.

同样，冲量固定时，力与时间成反比：力加倍则需要时间减半才能产生同样的冲量。

Always read the question carefully to check which quantities are constant.

始终仔细读题，核实哪些量是不变的。

11. Checking Units and Conversions | 单位检查与换算

A common mistake is mixing grams with kilograms or centimetres with metres. Remember:

1 kg = 1000 g, so to convert g to kg divide by 1000.
1 m/s = 3.6 km/h; to change km/h to m/s divide by 3.6.
Time may be given in milliseconds (ms); 1 s = 1000 ms.

常见错误是混淆克与千克、厘米与米。请记住：

1 kg = 1000 g，所以要把 g 换算成 kg 需除以 1000。
1 m/s = 3.6 km/h；要把 km/h 换成 m/s 需除以 3.6。
时间可能以毫秒 (ms) 给出；1 s = 1000 ms。

Always write the unit next to each numerical answer; unmarked answers lose marks in exams.

每个数值答案旁边都要写明单位；无单位的答案在考试中会失分。

12. Recap and Key Points | 总结与关键点

Concept	Formula	Unit
Momentum	p = m × v	kg m/s
Impulse	I = F × t	N s
Impulse–Momentum	F t = m(v – u)	N s = kg m/s

These three ideas form the foundation for solving momentum and impulse problems in KS3. Practise rearranging formulas and interpreting physical situations as mathematical equations.

这三个概念构成了 KS3 阶段求解动量和冲量问题的基础。多加练习公式变形，并将物理情景转化为数学方程。

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Momentum and Impulse: Key Concepts for KS3 Maths | KS3 数学：动量与冲量 考点精讲