Momentum — p = mv | Stewdent.eu

Momentum is a measure of how difficult it is to stop a moving object. It depends on both mass and velocity. Momentum is always conserved in a closed system — the total momentum before a collision equals the total momentum after.

What is Momentum?

Momentum ( $p$ ) is the product of an object's mass and velocity.

$p = mv$

Symbol	Quantity	Unit
$p$	momentum	kg·m/s
$m$	mass	kg
$v$	velocity	m/s

Note

Momentum is a vector quantity — it has both size and direction. It acts in the same direction as the velocity.

Calculating Momentum

A 60 kg cyclist rides at 5 m/s. What is their momentum?

$p = mv = 60 \times 5 = 300 \text{ kg·m/s}$

Conservation of Momentum

In a closed system (no external forces), the total momentum before a collision or interaction equals the total momentum after.

$p_{\text{before}} = p_{\text{after}}$

$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

Collision Example

A 2 kg ball moving at 3 m/s hits a stationary 1 kg ball. After the collision, the 2 kg ball moves at 1 m/s. Find the speed of the 1 kg ball.

Before: $p = 2 \times 3 + 1 \times 0 = 6$ kg·m/s

After: $p = 2 \times 1 + 1 \times v = 2 + v$

$6 = 2 + v \Rightarrow v = 4$ m/s

Momentum and Force

A force causes a change in momentum. The greater the force or the longer it acts, the greater the change in momentum:

$F = \frac{\Delta p}{\Delta t} = \frac{m \Delta v}{\Delta t}$

This is actually Newton's Second Law in its most general form.

Real world: crumple zones in cars

In a crash, a 1 500 kg car decelerates from 15 m/s to 0. The change in momentum is always $\Delta p = 1\,500 \times 15 = 22\,500$ kg·m/s regardless of crash design. However, $F = \Delta p / \Delta t$ : crumple zones extend the collision time from ~0.05 s to ~0.2 s, reducing the average force from 450 000 N to 112 500 N — four times smaller. This is why modern cars are designed to crumple; the bodywork absorbs the impact to protect the people inside.

Real world: snooker — transferring momentum

In snooker, a cue ball (mass ~0.17 kg) rolls at 2 m/s and strikes a stationary red ball of the same mass head-on. Momentum before: $0.17 \times 2 = 0.34$ kg·m/s. By conservation of momentum, the red ball moves off at 2 m/s and the cue ball stops completely. Professional players use this principle to control where the cue ball ends up after each shot.

Real world: rocket exhaust and jet engines

A jet engine expels mass (exhaust gases) backward at high velocity, giving those gases a large backward momentum. By conservation of momentum, the aircraft gains an equal and opposite forward momentum. The greater the mass flow rate and the higher the exhaust speed, the greater the thrust — which is why engineers work to increase both to improve fuel efficiency and power in aircraft engines.

Tip

Where do you see momentum? Car safety design (crumple zones, airbags), snooker and billiards, rocket and jet propulsion, a goalkeeper catching a fast ball (absorbing momentum gradually by letting hands move back), and ice skating — all involve changing or transferring momentum.

Practice Problems

A 1,500 kg car travels at 20 m/s. Calculate its momentum.

Show answer

p = mv = 1500 × 20 = 30,000 kg·m/s

A 0.5 kg ball has momentum of 10 kg·m/s. What is its speed?

Show answer

v = p/m = 10/0.5 = 20 m/s

A 3 kg trolley moving at 4 m/s collides with a stationary 1 kg trolley and they stick together. What is their combined speed?

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Before: p = 3 × 4 = 12 kg·m/s. After (combined mass = 4 kg): 12 = 4 × v → v = 3 m/s.