# Conservation of momentum

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u3-conservation-of-momentum/

This subtopic explains the principle of conservation of linear momentum, its application to one-dimensional collisions and explosions, and how to distinguish between elastic and inelastic collisions. It is a core topic assessed in both multiple choice and structured questions.

**Prerequisites:** [Newton's laws of motion](https://www.owlsprep.com/study/cie-9702-u3-newtons-laws/); [Linear momentum and impulse](https://www.owlsprep.com/study/cie-9702-u3-momentum-impulse/)

## Learning objectives

- State the principle of conservation of linear momentum
- Apply conservation of momentum to collisions and explosions
- Distinguish between elastic and inelastic collisions
- Solve one-dimensional momentum problems for CIE exams

## The Principle of Conservation of Momentum

**Principle of Conservation of Linear Momentum** — For a closed system of interacting objects, the total linear momentum in any given direction remains constant, provided no net external force acts on the system.

This principle is derived directly from Newton's Third Law. When two objects interact, the force each exerts on the other is equal and opposite, so the total change in momentum of the system is zero.

$$\sum p_{initial} = \sum p_{final} \quad \text{where} \quad p = mv$$

**Worked example:** A 2 kg mass moving at 3 m s⁻¹ to the right collides with a stationary 1 kg mass. What is the total momentum after the collision?

1. Assign right as the positive direction
2. Calculate total initial momentum:
3. $$p_{initial} = m_1 v_1 + m_2 v_2 = (2 \times 3) + (1 \times 0) = 6 \text{ kg m s}^{-1}$$
4. By conservation of momentum, total final momentum equals total initial momentum
5. Final total momentum = 6 kg m s⁻¹

> **Exam tip:** Always state the full principle when answering structured CIE questions to gain full marks.

## Elastic and Inelastic Collisions

All collisions obey conservation of momentum (for closed systems), but they are classified based on whether kinetic energy is conserved.

**Elastic Collision** — A collision where both total momentum and total kinetic energy are conserved

*Example:* Ideal billiard ball collisions, collisions between gas molecules

**Inelastic Collision** — A collision where total momentum is conserved, but kinetic energy is not. Kinetic energy is converted to heat, sound, or deformation energy

*Example:* A car crash, a ball sticking to a surface after impact

**Worked example:** A 0.5 kg ball moving at 4 m s⁻¹ collides head-on with a stationary 0.5 kg ball. After collision, the first ball stops, and the second moves at 4 m s⁻¹. Show the collision is elastic.

1. Check momentum is conserved:
2. $$p_{initial} = (0.5 \times 4) + (0.5 \times 0) = 2 \text{ kg m s}^{-1} \\ p_{final} = (0.5 \times 0) + (0.5 \times 4) = 2 \text{ kg m s}^{-1}$$
3. Calculate initial and final kinetic energy:
4. $$KE_{initial} = \frac{1}{2}(0.5)(4)^2 + 0 = 4 \text{ J} \\ KE_{final} = 0 + \frac{1}{2}(0.5)(4)^2 = 4 \text{ J}$$
5. Kinetic energy is unchanged, so the collision is elastic

> **Exam tip:** CIE almost always asks to distinguish between elastic and inelastic collisions, so memorize both definitions.

## Explosions and Recoil

Explosions are the reverse of collisions: a single stationary object splits into two or more parts. Total initial momentum is zero, so by conservation of momentum, total final momentum must also be zero. The parts move in opposite directions with equal magnitude momentum, which causes recoil of the larger object.

**Worked example:** A stationary cannon of mass 1000 kg fires a 10 kg cannonball at 150 m s⁻¹. Calculate the recoil speed of the cannon.

1. Assign the cannonball's direction as positive. Total initial momentum = 0, since the cannon is stationary.
2. Apply conservation of momentum:
3. $$0 = m_{ball} v_{ball} + m_{cannon} v_{cannon}$$
4. Rearrange to solve for recoil velocity:
5. $$v_{cannon} = - \frac{m_{ball} v_{ball}}{m_{cannon}} = - \frac{10 \times 150}{1000} = -1.5 \text{ m s}^{-1}$$
6. The negative sign indicates the cannon moves opposite to the cannonball. Recoil speed is 1.5 m s⁻¹

> **tip**
>
> Always start with total initial momentum equal to zero for explosions of stationary objects, this simplifies calculations.

## Common pitfalls

- **Wrong:** Ignoring that momentum is a vector, so signs for opposite directions are omitted
  - Why it fails: This leads to incorrect total momentum calculations for objects moving in opposite directions
  - Correct: Assign a positive direction at the start of the problem, and use negative signs for velocity in the opposite direction
- **Wrong:** Claiming kinetic energy is conserved in all collisions
  - Why it fails: Only elastic collisions conserve kinetic energy; nearly all real collisions are inelastic
  - Correct: Only state kinetic energy is conserved if the question explicitly identifies the collision as elastic
- **Wrong:** Applying conservation of momentum to systems with external net forces
  - Why it fails: The principle only holds for closed systems with no net external force
  - Correct: Check for unbalanced external forces (e.g. friction) before applying the principle
- **Wrong:** Confusing mass and velocity when calculating recoil speed after an explosion
  - Why it fails: Recoil speed is inversely proportional to mass for a fixed momentum change
  - Correct: Set total initial momentum to zero, then solve for the unknown speed using equal and opposite momentum

## Cheatsheet

| Concept | Key Rule | Formula |
| --- | --- | --- |
| Conservation of momentum | Closed system, no external force | $\sum p_{initial} = \sum p_{final}$ |
| Elastic collision | Momentum and KE conserved | $KE_{initial} = KE_{final}$ |
| Inelastic collision | Only momentum conserved | $KE_{initial} > KE_{final}$ |
| Stationary explosion | Total initial $p=0$ | $m_1 v_1 = -m_2 v_2$ |

## What's next

Conservation of momentum is a fundamental conservation law that extends to two-dimensional problems and is used across many areas of physics, including nuclear reactions, where it helps calculate the momentum of emitted particles such as alpha particles. It also connects back to Newton's laws of motion and underpins concepts like impulse and force interactions. Mastery of one-dimensional momentum problems is essential for all dynamics questions in the CIE A-Level exam, and prepares you for more complex problems in further topics.

- [Collisions](https://www.owlsprep.com/study/cie-9702-u3-collisions/)
- [Mass and weight](https://www.owlsprep.com/study/cie-9702-u3-mass-and-weight/)
- [Forces, density and pressure](https://www.owlsprep.com/study/cie-9702-u4-overview/)

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