3.6 Momentum

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Main Content

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1. Conservation of Linear Momentum

Momentum is the "quantity of motion" an object has. It depends on both mass and velocity:

$$\text{Momentum} = m \times v$$

where $m$ is mass (in kg) and $v$ is velocity (in m s⁻¹). Momentum is measured in kg m s⁻¹. It is a vector — it has both size and direction, so direction matters.

The Principle of Conservation of Linear Momentum states:

In plain English: the total "amount of motion" in the system does not change during a collision.

For two objects with masses $m_1$ and $m_2$, moving with velocities $u_1$ and $u_2$ before the collision and $v_1$ and $v_2$ after the collision:

$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

This equation is your starting point for almost every collision problem.

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2. Newton's Experimental Law and the Coefficient of Restitution

Conservation of momentum alone is not enough to solve most collision problems — it gives you one equation but you often have two unknowns ($v_1$ and $v_2$). You need a second equation. This is where Newton's Experimental Law comes in.

Newton's Experimental Law (also called the Law of Restitution) describes how "bouncy" a collision is. It says:

Written as a formula:

$$e = \frac{\text{speed of separation}}{\text{speed of approach}}$$

More precisely, using velocities along the line of impact:

$e = \frac{v_2 - v_1}{u_1 - u_2}$

Here:

• $u_1$ and $u_2$ are the velocities of the two objects before the collision (in the positive direction).
• $v_1$ and $v_2$ are the velocities of the two objects after the collision (in the positive direction).
• $e$ is the coefficient of restitution — a number that tells you how elastic (springy) the collision is.

The letter $e$ stands for the coefficient of restitution. Think of it as a "bounciness score."