Conservation of momentum

Two skaters at rest

Two skaters stand still: total momentum zero. They push apart and glide off in opposite directions. One has momentum $+p$, the other $-p$. The total is still zero.

Why the total stays the same

In a push or a collision, the third law gives each object an equal and opposite force for the same time. So each gains as much momentum as the other loses:

If no outside force acts, the total momentum of a group of objects does not change.

You should find that the total-after bar always equals the total-before bar. With B moving towards A, the totals can be small or even negative.

Watch out. Speeds in opposite directions get opposite signs. And the rule needs no outside force: friction from the ground must be small during the collision.

A worked example

A $3\ \text{kg}$ trolley at $4\ \text{m/s}$ runs into a $1\ \text{kg}$ trolley at rest, and they stick together. How fast do they move?

1. Before. $3\ \text{kg} \times 4\ \text{m/s} + 1\ \text{kg} \times 0\ \text{m/s} = 12\ \text{kg m/s}$.
2. After. $(3\ \text{kg} + 1\ \text{kg})\,v = 12\ \text{kg m/s}$, so $v = 3\ \text{m/s}$.
3. Check. $4\ \text{kg} \times 3\ \text{m/s} = 12\ \text{kg m/s}$: the same total.

Practise with new numbers

Try it in Python

The program works out the trolleys.

Remember these

• With no outside force, total momentum before $=$ total momentum after.
• Use signs for opposite directions.
• Recoil: starting at rest, the two momenta afterwards are equal and opposite.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A $50\ \text{kg}$ boy jumps from a $100\ \text{kg}$ boat at $2\ \text{m/s}$. How fast does the boat move back?

2. Two equal trolleys move towards each other at the same speed and stick. What happens?

3. Why does a gun kick back when it fires?

4. A $2\ \text{kg}$ trolley at $5\ \text{m/s}$ hits a $3\ \text{kg}$ trolley at rest and stops. How fast does the $3\ \text{kg}$ trolley move?