GCSE Maths: Momentum and Impulse | GCSE 数学：动量与冲量考点精讲

📚 GCSE Maths: Momentum and Impulse | GCSE 数学：动量与冲量考点精讲

Momentum and impulse are not only fundamental concepts in physics but also provide excellent practice for applying mathematical skills such as multiplication, rearranging equations, interpreting graphs, and solving simultaneous equations. In GCSE Maths, you may encounter questions that require you to calculate momentum, use the impulse-momentum relationship, or analyse a force-time graph. This revision guide covers all the key mathematical techniques you need.

动量和冲量不仅是物理学的基本概念，也是练习数学技能（如乘法、方程变换、图表解读和联立方程求解）的绝佳主题。在 GCSE 数学中，你可能会遇到要求计算动量、运用冲量-动量关系或分析力-时间图的题目。本复习指南涵盖你所需的所有关键数学技巧。

1. What is Momentum? | 动量的定义

Momentum is a measure of how difficult it is to stop a moving object. It depends on two quantities: mass and velocity.

动量是衡量运动物体难以停止的程度的量。它取决于两个量：质量和速度。

The greater an object’s mass or the faster it moves, the larger its momentum. In mathematical terms, momentum is directly proportional to both mass and velocity.

物体的质量越大或运动速度越快，其动量就越大。用数学术语来说，动量与质量和速度都成正比。

When solving problems, always start by identifying the moving object’s mass (in kg) and its velocity (in m s⁻¹). Momentum is a vector, meaning direction matters.

解题时，首先要确定运动物体的质量（单位 kg）及其速度（单位 m s⁻¹）。动量是矢量，意味着方向很重要。

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

The momentum of an object is calculated using the formula:

物体的动量使用以下公式计算：

p = m × v

where p is momentum, m is mass, and v is velocity. The SI unit of momentum is kilogram metre per second, written as kg m s⁻¹.

其中 p 是动量，m 是质量，v 是速度。动量的国际单位是千克米每秒，记作 kg m s⁻¹。

You can rearrange the equation to find mass or velocity: m = p / v or v = p / m. These are simple algebraic manipulations often required in exam questions.

你可以变换方程来求质量或速度：m = p / v 或 v = p / m。考试中常要求这些简单的代数变换。

Note that the units must be consistent: mass in kilograms, velocity in metres per second. If mass is given in grams, convert to kilograms by dividing by 1000.

注意单位必须一致：质量用千克，速度用米每秒。如果质量以克为单位，除以 1000 转换为千克。

3. Momentum as a Vector in Calculations | 计算中的动量矢量性

Because velocity is a vector, momentum also has direction. In one-dimensional problems, you can use positive and negative signs to represent opposite directions.

由于速度是矢量，动量也有方向。在一维问题中，你可以用正负号表示相反的方向。

For example, take the right as positive. A car moving right at 20 m s⁻¹ has momentum +m × 20, while one moving left at the same speed has momentum −m × 20.

例如，规定向右为正。一辆以 20 m s⁻¹ 向右行驶的汽车动量为 +m × 20，而以相同速度向左行驶的汽车动量为 −m × 20。

This sign convention is essential when applying the conservation of momentum or impulse-momentum theorem. Neglecting the sign can lead to completely wrong answers.

在应用动量守恒定律或冲量-动量定理时，这种符号规定至关重要。忽略符号会导致完全错误的答案。

Always state your positive direction clearly before writing any equation. Use it consistently throughout the calculation.

在写任何方程之前，务必清楚说明正方向，并在整个计算过程中一致使用。

4. What is Impulse? | 什么是冲量？

Impulse quantifies the effect of a force acting over a time interval. Mathematically, impulse (I) is the product of the average force (F) and the time interval (t) during which the force acts.

冲量量化了力在一段时间间隔内作用的效果。数学上，冲量 (I) 是平均力 (F) 与作用时间间隔 (t) 的乘积。

I = F × t

The unit of impulse is newton second (N s). Because 1 N = 1 kg m s⁻², a newton second is equivalent to 1 kg m s⁻¹, the same unit as momentum.

冲量的单位是牛顿秒 (N s)。因为 1 N = 1 kg m s⁻²，所以 1 N s 等价于 1 kg m s⁻¹，与动量的单位相同。

Impulse is also a vector quantity; it acts in the direction of the force. In calculations, always consider whether the force is positive or negative relative to your chosen axis.

冲量也是矢量；它的方向与力的方向一致。在计算中，务必根据所选坐标轴考虑力是正还是负。

5. The Impulse-Momentum Theorem | 冲量-动量定理

The impulse-momentum theorem directly links the impulse experienced by an object to its change in momentum. It is derived from Newton’s second law.

冲量-动量定理直接将物体受到的冲量与其动量变化联系起来。它由牛顿第二定律推导而来。

Starting with F = m × a and substitution a = (v − u) / t gives F = m × (v − u) / t. Multiplying both sides by t yields:

从 F = m × a 出发，代入 a = (v − u) / t 得到 F = m × (v − u) / t。两边同乘以 t 得到：

F × t = m × (v − u)

In words, impulse = final momentum − initial momentum, i.e. I = Δp. This is the key equation you will use in most exam problems.

用文字表述就是：冲量 = 末动量 − 初动量，即 I = Δp。这是你在大多数考题中会用到的关键方程。

You can use this theorem to find any one of average force, time interval, mass, or change in velocity, provided the other quantities are known. Rearranging is a core mathematical skill.

在已知其他量的情况下，你可以利用此定理来求平均力、时间间隔、质量或速度变化量。方程变换是一项核心数学技能。

6. Interpreting Force-Time Graphs | 解读力-时间图

In GCSE Maths, you may be given a force-time graph and asked to find the impulse. The impulse equals the area under the force-time graph.

在 GCSE 数学中，你可能会遇到力-时间图，并被要求求冲量。冲量等于力-时间图下的面积。

For a constant force, the graph is a rectangle. The impulse is simply F × Δt, which is the area of the rectangle.

对于恒力，图形是一个矩形。冲量就是 F × Δt，即矩形的面积。

If the force varies linearly from zero to a maximum, the area is a triangle: impulse = ½ × base × height = ½ × t × F_max. For a trapezium, use A = ½ (a + b) × h.

如果力从零线性增加到最大值，面积是一个三角形：冲量 = ½ × 底 × 高 = ½ × t × F_max。对于梯形，使用 A = ½ (a + b) × h。

The mathematical skill here is finding areas of simple shapes. Remember to check the scale on each axis carefully, and express your answer in N s.

这里的数学技巧是求简单图形的面积。记住仔细检查每个轴的刻度，并用 N s 表示答案。

Once you have the impulse, you can equate it to m(v − u) to find the change in velocity or the mass. This links graphical analysis to algebraic problem-solving.

求出冲量后，你可以用它等于 m(v − u) 来求速度变化量或质量。这将图形分析与代数问题求解结合起来。

7. Conservation of Momentum | 动量守恒

In a closed system with no external forces, the total momentum before an event (such as a collision or explosion) equals the total momentum after the event.

在没有外力的封闭系统中，事件（如碰撞或爆炸）前的总动量等于事件后的总动量。

Mathematically, for two objects colliding, you write:

数学上，对于两个物体碰撞，可写为：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

where u represents initial velocities and v represents final velocities. All velocities must include a sign to indicate direction.

其中 u 代表初速度，v 代表末速度。所有速度都必须包含表示方向的符号。

This principle allows you to set up linear equations and solve for unknown velocities. It is widely used in maths problems involving collisions and recoils.

利用这个原理，你可以建立线性方程并求解未知速度。它广泛应用于涉及碰撞和反冲的数学问题中。

8. Solving Momentum Conservation Equations | 求解动量守恒方程

When using the conservation equation, first assign a positive direction. Then substitute the known masses and velocities with their signs into the equation.

使用守恒方程时，首先规定正方向。然后将已知的质量和带有符号的速度代入方程。

You will often need to isolate the unknown velocity. For example, if two objects stick together after collision, v₁ = v₂ = v (common velocity). The equation simplifies to: m₁u₁ + m₂u₂ = (m₁ + m₂)v.

你通常需要隔离未知速度。例如，如果碰撞后两个物体粘在一起，则 v₁ = v₂ = v（共同速度）。方程化简为：m₁u₁ + m₂u₂ = (m₁ + m₂)v。

Then solve for v by dividing both sides by (m₁ + m₂). This is classic algebraic manipulation: collecting terms, simplifying, and dividing.

然后两边除以 (m₁ + m₂) 求解 v。这是经典的代数操作：合并同类项、化简并相除。

If the bodies separate and move in opposite directions, make sure the signs reflect that. A common mistake is forgetting to put a negative sign for a velocity opposite to the chosen positive direction.

如果物体分离并朝相反方向运动，确保符号反映出这一点。一个常见错误是忘记给与选定正方向相反的速度加上负号。

9. Worked Example: Collision Analysis | 实例分析：碰撞问题

Example 1 (stick together): Trolley A (0.2 kg) moves right at 1.5 m s⁻¹ and hits stationary trolley B (0.3 kg). After the collision, they couple together. Find the combined speed.

示例 1（粘在一起）： 小车 A（0.2 kg）以 1.5 m s⁻¹ 向右运动，撞上静止的小车 B（0.3 kg）。碰撞后它们连在一起。求二者的共同速度。

Take right as positive. Initial total momentum = (0.2)(1.5) + (0.3)(0) = 0.3 kg m s⁻¹. Final momentum = (0.2 + 0.3)v = 0.5v. Equating: 0.5v = 0.3 ⇒ v = 0.6 m s⁻¹ to the right.

取向右为正。初始总动量 = (0.2)(1.5) + (0.3)(0) = 0.3 kg m s⁻¹。末动量 = (0.2+0.3)v = 0.5v。方程：0.5v = 0.3 ⇒ v = 0.6 m s⁻¹，方向向右。

Example 2 (rebound): Ball X (0.5 kg) at 2 m s⁻¹ right hits stationary ball Y (0.5 kg). After impact, X moves left at 0.5 m s⁻¹. Find Y’s velocity.

示例 2（反弹）： 球 X（0.5 kg）以 2 m s⁻¹ 向右运动，撞击静止的球 Y（0.5 kg）。碰撞后，X 以 0.5 m s⁻¹ 向左运动。求 Y 的速度。

Right positive. Initial momentum: (0.5)(2) + (0.5)(0) = 1 kg m s⁻¹. Final: (0.5)(−0.5) + (0.5)v_Y = −0.25 + 0.5v_Y. Set equal: 1 = −0.25 + 0.5v_Y ⇒ 1.25 = 0.5v_Y ⇒ v_Y = 2.5 m s⁻¹ to the right.

向右为正。初始动量：(0.5)(2)+(0.5)(0)=1 kg m s⁻¹。末动量：(0.5)(−0.5)+(0.5)v_Y = −0.25+0.5v_Y。列等式：1=−0.25+0.5v_Y ⇒ 1.25=0.5v_Y ⇒ v_Y = 2.5 m s⁻¹，方向向右。

These examples show how careful sign use and algebraic steps lead to the correct numerical answer. Always check that your answer makes physical sense.

这些例子表明，谨慎使用符号和代数步骤可以得到正确的数值答案。一定要检查答案是否在物理意义上合理。

10. Common Mistakes and Exam Advice | 常见错误与考试建议

Forgetting to convert mass to kilograms. Always use SI units: mass in kg, velocity in m s⁻¹, time in s. 忘记将质量转换为千克。始终使用国际单位：质量用 kg，速度用 m s⁻¹，时间用 s。
Treating momentum as a scalar. Include direction with + or − signs in all calculations. 将动量当作标量。在所有计算中用 + 或 − 表示方向。
Confusing impulse and momentum. Impulse equals change in momentum, not momentum itself. 混淆冲量和动量。冲量等于动量的变化，而不是动量本身。
Using the wrong area formula for a force-time graph. Revise area formulas for rectangle, triangle and trapezium. 使用错误的力-时间图面积公式。复习矩形、三角形和梯形的面积公式。
Not checking whether objects stick together or separate. This changes the form of the momentum equation. 没有检查物体是粘在一起还是分离。这会影响动量方程的形式。
Algebra slips when rearranging equations. Practise changing the subject of a formula thoroughly. 方程变换时出现代数错误。充分练习公式变形。

During the exam, read the question carefully to identify whether you need momentum, impulse, or both. Show all working steps, especially the initial equation and sign conventions, to gain method marks.

考试时要仔细读题，确定是需要动量、冲量还是两者都需要。展示所有解题步骤，特别是初始方程和符号规定，以获得方法分。

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GCSE Maths: Momentum and Impulse | GCSE 数学：动量与冲量 考点精讲