Linear Momentum

What momentum is

Momentum is the quantity of motion a body possesses. It depends on two things: how much mass the body has, and how fast it is moving.

$\vec{p} = m\vec{v}$

It is a vector — it has the same direction as the velocity.

Why momentum matters more than speed alone

Consider two situations:

• A tennis ball at 100 m/s
• A truck at 10 m/s

The tennis ball is faster, but the truck is far harder to stop. The truck has far more momentum because its mass dominates.

Momentum captures this combined effect of mass and velocity.

Units

$[\vec{p}] = \text{kg} \cdot \text{m/s} = \text{N} \cdot \text{s}$

Both units are equivalent: $\text{kg} \cdot \text{m/s} = \text{kg} \cdot \frac{\text{m}}{\text{s}} = \frac{\text{kg} \cdot \text{m}}{\text{s}^2} \cdot \text{s} = \text{N} \cdot \text{s}$

Momentum and Newton's Second Law

Newton originally stated his Second Law as:

$\vec{F} = \frac{d\vec{p}}{dt}$

Force is the rate of change of momentum. For constant mass:

$\vec{F} = \frac{d(m\vec{v})}{dt} = m\frac{d\vec{v}}{dt} = m\vec{a}$

This shows that $\vec{p} = m\vec{v}$ is not just a definition — it is the quantity that force directly changes.

Momentum is a vector

Since $\vec{p} = m\vec{v}$, momentum has both magnitude and direction.

• Magnitude: $p = mv$
• Direction: same as $\vec{v}$

When dealing with multiple bodies or multiple dimensions, momenta must be added as vectors.

Example: Two cars of equal mass $m$ moving at equal speed $v$, one east and one north. Their momenta are:

$\vec{p}_1 = mv\hat{i}, \quad \vec{p}_2 = mv\hat{j}$

Total momentum:

$\vec{p}_{total} = mv\hat{i} + mv\hat{j}$

Magnitude: $mv\sqrt{2}$, direction: northeast. Not $2mv$ — vector addition, not scalar.

Change in momentum

When a force acts on a body, its momentum changes:

$\Delta\vec{p} = \vec{p}_{final} - \vec{p}_{initial} = m\vec{v}_f - m\vec{v}_i = m(\vec{v}_f - \vec{v}_i)$

The direction of $\Delta\vec{p}$ is the direction of the net force — which may be very different from the direction of motion.

Example: A ball bouncing off a wall. Before: momentum $+mv$ (toward wall). After: momentum $-mv$ (away from wall). Change in momentum: $-2mv$. The wall exerted a force of $-2mv/\Delta t$ on the ball during contact.