Impulse.

Step 1 of 7

Force times time

To change an object’s momentum takes a force. By Newton’s second law, $F = ma$⁠, and the acceleration is the change in velocity each second, $a = \dfrac{\Delta v}{\Delta t}$$a =$$\dfrac{\Delta v}{\Delta t}$⁠. Substitute and multiply by $\Delta t$⁠:

$F\,\Delta t = m\,\Delta v = \Delta p$$F\,\Delta t =$$m\,\Delta v =$$\Delta p$⁠

The force times the time it acts for is called the impulse, $J = F\,\Delta t$⁠, and it equals the change of momentum. Its unit is the newton second, N·s, exactly the same unit as kg·m/s. A small force acting for a long time can change the momentum as much as a large force acting for a moment.

Step 2 of 7

What B received

In a collision, all the change in a cart’s momentum happens in the short time the bumpers touch. Cart B starts at rest, so all the momentum it leaves with is the impulse A gave it.

Step 3 of 7

Equal and opposite pushes

While the bumpers touch, A pushes on B and B pushes on A. Change Mass ratio B : A and watch on the board how A’s bar shrinks by exactly as much as B’s grows, at the moment of the hit.

Step 4 of 7

The impulse on A

Now B is twice as heavy, and the collision is perfectly bouncy, so A comes back. To work out the impulse on A, subtract its starting momentum from its final momentum, with the signs: right is positive.

Step 5 of 7

Bounce or stick

Equal carts stick together. In a stick the bumpers only stop the carts closing in, and the two roll on together. Predict what happens to the impulse on B when the ends are perfectly bouncy, then watch.

Step 6 of 7

How hard

The model in the Scene treats the collision as instantaneous, but real bumpers squash for a few thousandths of a second. If you know how long the contact lasts, you can work out the average force from the impulse: $F = \dfrac{\Delta p}{\Delta t}$$F =$$\dfrac{\Delta p}{\Delta t}$⁠.

Step 7 of 7

A soft landing

The same change of momentum, spread over a longer time, takes a smaller force. That is the whole idea behind crumple zones in cars, a mat under a high jump and knees that bend on landing.