Lesson 3 of 4

Conservation of momentum.

About 30 minutes · 6 practice questions

Read the full article: Momentum and collisions

The total no collision can change: the total momentum before and after, carts that stick, momentum handed over, a cart that bounces back, and carts heading towards each other.

Step 1 of 6 · What stays the same

Step 1 of 6

What stays the same

In the last Lesson you saw that while they touch, A and B push on each other with equal and opposite forces, by Newton’s third law, so their impulses are equal and opposite too. Whatever momentum B gains, A loses.

If nothing outside pushes the carts along the track, the total for the two cannot change. This is the conservation of momentum:

$m_A v_A + m_B v_B = m_A v_A' + m_B v_B'$ ⁠

The prime marks the velocities after the collision. Here B is three times as heavy as A, and the collision is half bouncy. Predict, and then Look inside will open at Momentum and energy.

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Step 2 of 6

Sticking together

When the carts stick, after the hit they are one body of mass $m_A + m_B$ ⁠, moving at one velocity, $v'$ ⁠. Conservation of momentum gives it straight away:

$v' = \frac{m_A v_A + m_B v_B}{m_A + m_B}$ ⁠

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Step 3 of 6

Momentum handed over

Bounciness, which physicists call the coefficient of restitution, is the speed at which the carts move apart after the hit divided by the speed at which they closed in before it:

$e = \frac{v_B' - v_A'}{v_A - v_B}$ ⁠

When they stick, $e = 0$ ⁠: they do not move apart at all. Perfectly bouncy, $e = 1$ ⁠: they move apart as fast as they met. Conservation of momentum and a known bounciness are two equations, and they give both velocities after the hit.

Now the carts are equal and stick. Predict what happens when they are perfectly bouncy.

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Step 4 of 6

Bouncing back

When B is heavier than A and the collision is bouncy enough, B can take more momentum than A brought. Then A must come out of the hit with negative momentum, so that the total stays as it was.

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Step 5 of 6

Stopping A dead

B is twice as heavy as A. With little bounce A rolls on forward after the hit, and perfectly bouncy it comes back. In between there is one bounciness at which it stops dead. Change Bounciness, and watch the arrow over A after the hit.

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Step 6 of 6

Towards each other

Conservation of momentum works when both carts are moving before the collision too. When they head towards each other, one of them has negative momentum, and it takes away from the total. If the total comes out positive the pair carries on to the right; if it comes out zero, both stop.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Predict, then look

Cart A arrives carrying 0.6 kg·m/s into cart B, three times as heavy and standing still. The collision is half bouncy, and some of the kinetic energy becomes heat and sound. What will the total momentum of the two carts be after the hit?

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Practice · Calculate

Cart A, 1 kg, arrives at 0.6 m/s at cart B, 3 kg, and sticks to it. At what speed do the two roll on together?

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Practice · Predict, then look

Equal carts, and they stick: after the hit both roll on at 0.3 m/s. What will happen when the ends are perfectly bouncy?

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Practice · Calculate

Cart B is four times as heavy, 4 kg, and the collision is perfectly bouncy. After the hit B moves off to the right at 0.24 m/s. What is the velocity of cart A, 1 kg, after the hit? Right is positive.

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Practice · Reach the target

Cart B is twice as heavy. Change Bounciness until cart A stops dead after the hit and hands all its momentum to B, then press Check.

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Practice · Calculate

Two carts roll towards each other: a 2 kg cart to the right at 0.5 m/s, and a 1 kg cart to the left at 0.4 m/s. They stick. What is their common velocity after the hit? Right is positive.