Momentum and Impulse

Two snooker balls click together and fly apart, two cars crunch and crumple, a rocket shoves gas one way and leaps the other. In every one of these the same hidden quantity is being carefully passed around and never lost. That quantity is momentum, and once you can track it you can predict the result of almost any collision without knowing a single thing about the messy forces inside it.

What momentum really is

Momentum is just a measure of how hard something is to stop. A slow truck and a fast cricket ball can be equally tricky to halt, because momentum depends on both the mass and the velocity.

• Momentum is mass times velocity, $p = mv$. Double the speed and you double the momentum; double the mass and you double it too.
• It is a vector, so direction matters. A ball going left has the opposite sign to a ball going right, and you must keep track of that sign in every calculation.

A heavy object moving slowly can carry the same momentum as a light object moving fast. That is why a gently rolling truck and a whizzing tennis ball can each be a handful to stop.

Impulse: how momentum gets changed

To change something’s momentum you have to push on it, and the longer you push the bigger the change. That push over time is called the impulse.

Impulse is the force multiplied by how long it acts, $I = F\,\Delta t$, and it equals the change in momentum, $\Delta p$. This is just Newton’s second law turned inside out. A small force over a long time can deliver the same impulse as a big force over a short time, which is exactly why follow through matters in sport and why soft landings save your knees.

When objects collide

Picture two carts on a track. Before they meet, each carries its own momentum. They bump, push on each other, and move off again. Add up the momentum of both carts before the bump and it equals the total after the bump, every time, as long as nothing outside the carts pushes on them.

The blue cart A starts with all the momentum and grey cart B is still. After they lock together they share that momentum as one combined object, shown by the red arrow. Nothing is added and nothing is lost, so the total before equals the total after.

Elastic or inelastic

Momentum survives every collision, but kinetic energy does not. Whether energy is kept tells you what kind of collision you are watching.

• An elastic collision keeps the total kinetic energy. The objects bounce cleanly off each other, like ideal snooker balls.
• An inelastic collision loses kinetic energy to heat, sound and bending. When the objects stick together it is called perfectly inelastic, the most lossy case of all.

In both kinds the momentum still balances perfectly. Only kinetic energy is allowed to go missing, and it goes missing whenever the objects deform or stick.

See it for yourself

Set the masses and velocities of two balls, choose how bouncy the collision is, and watch the momentum bars stay balanced before and after. Slide the elasticity down to zero to make the balls stick together, and notice the total momentum holds steady even as kinetic energy is lost.

How to actually solve one

Collision questions follow one reliable recipe.

1. Choose a positive direction and write down the velocity of everything, using a minus sign for anything going the other way.
2. Write the total momentum before, $\sum p_{\text{before}} = m_1 u_1 + m_2 u_2$.
3. Write the total momentum after. If the objects stick together they share one velocity $v$, so $\sum p_{\text{after}} = (m_1 + m_2)\,v$.
4. Set before equal to after and solve for the unknown.

For an impulse question instead, use $I = \Delta p = m(v_f - v_i)$, and remember the rebound velocity is negative. Watch the signs, because a reversed velocity is where most marks are lost.

See the recipe in action in the Worked Examples tab, then test yourself in Try It.

Lock it in with active recall

Cover the answer and say each one out loud before you flip. Rate yourself honestly — the cards you find hard come back sooner, the ones you know are spaced further out.