Momentum and Impulse

The 'quantity of motion'

A bowling ball rolling slowly can knock down pins that a fast-flying ping-pong ball cannot. A heavy lorry is far harder to stop than a bicycle moving at the same speed. Both ideas — how much mass is moving, and how fast — combine into a single quantity physicists call momentum. It is sometimes described as the "quantity of motion" an object carries.

Closely linked is impulse, which explains how momentum changes and why airbags, crumple zones and crash mats save lives. If you have met collisions in momentum and collisions, this lesson sharpens those ideas into equations you can calculate with.

Momentum: p = mv

Momentum is defined as mass multiplied by velocity:

p = m × v

where:

• p is the momentum in kilogram metres per second (kg·m/s),
• m is the mass in kilograms (kg),
• v is the velocity in metres per second (m/s).

Because velocity has direction, momentum is a vector — it also has direction. A 1000 kg car going 20 m/s east has a momentum of 20 000 kg·m/s east.

Worked example. A 0.4 kg football is kicked at 25 m/s. Its momentum is:

p = mv = 0.4 × 25 = 10 kg·m/s

Impulse: changing momentum

To change an object's momentum you must apply a force for some time. The combination of force and time is called impulse:

impulse = force × time = F × t

And the central result, sometimes called the impulse–momentum theorem, is:

F × t = change in momentum = m × v − m × u

This comes straight from Newton's second law. Force equals mass × acceleration, and acceleration is the change in velocity over time — rearranging gives Ft = change in momentum. The units of impulse (N·s) turn out to be identical to those of momentum (kg·m/s).

Why time changes the force

Look closely at F × t = change in momentum. For a given change in momentum, if you make the time t longer, the force F must be smaller. This single idea is behind almost all modern safety design:

• A car crumple zone crushes slowly, stretching the collision time so the force on passengers drops.
• An airbag extends the time your head takes to stop, reducing the force on your skull.
• A crash mat or trampoline lets a falling gymnast stop gradually instead of suddenly.
• Bending your knees when you land spreads the stop over more time, protecting your joints.

In every case the momentum change is the same, but stretching the time slashes the force.

Worked examples

Example 1 — impulse. A 0.15 kg cricket ball arrives at 30 m/s and is caught, stopping in 0.05 s. What force does the catcher's hands feel?

Change in momentum = m × v − m × u = (0.15 × 0) − (0.15 × 30) = −4.5 kg·m/s F = change in momentum ÷ time = −4.5 ÷ 0.05 = −90 N

The minus sign shows the force opposes the ball's motion. Now imagine the catcher pulls their hands back so the ball stops in 0.2 s instead:

F = −4.5 ÷ 0.2 = −22.5 N

Four times the time gives one quarter of the force — exactly why fielders "give" with the catch.

Example 2 — conservation of momentum. Two ice skaters stand still facing each other. A 60 kg skater pushes off, moving at 2 m/s. How fast does the 40 kg skater move the other way?

Total momentum before = 0 (both at rest). Total momentum after must also be 0. So (60 × 2) + (40 × v) = 0 → 120 + 40v = 0 → v = −3 m/s

The lighter skater moves off at 3 m/s in the opposite direction. This is conservation of momentum.

Conservation of momentum

In a closed system — one with no outside forces — the total momentum stays constant. Whatever momentum one object gains, another loses an equal and opposite amount. This is one of the deepest and most reliable laws in physics, used to analyse everything from billiard balls to rocket launches and even subatomic particle collisions. It connects directly to Newton's third law, which you can review in Newton's laws of motion.

Try it yourself! 🧪

Feel impulse with a dropped egg — the classic safe demo.

You need two raw eggs, a baking tray, and two landing surfaces: a hard plate and a thick folded towel or pillow.

1. Hold one egg about 20 cm above the hard plate (over the tray to catch the mess) and let it drop. It breaks — the egg stops almost instantly, so the force is large.
2. Hold the second egg the same height above the thick towel and drop it. The soft towel lets the egg slow down over a longer time, so the force is smaller — it often survives.

Both eggs had the same momentum just before landing, but the towel stretched the stopping time and cut the force. That is impulse in action — the same physics that protects you in a car crash. Do this over a tray or outside, and let an adult help with clean-up.