Newton's third law and conservation of momentum

Forces come in pairs

Push a wall and it pushes back on your hand: you can feel it. Row a boat: you push the water back, and the water pushes the boat forward. A force is never alone. It is one half of an interaction between two bodies.

Newton's third law

If body A exerts a force on body B, then B exerts a force on A that is equal in size and opposite in direction.

Three things to remember about a third-law pair:

• The two forces act on different bodies, so they never cancel each other.
• They are the same kind of force: both contact forces, or both gravity.
• They act at the same time: neither comes first. "Action" and "reaction" are just names.

Watch out. A book on a table: its weight (the Earth pulls the book) and the table's push up (the table pushes the book) are equal and opposite, but they are not a third-law pair. Both act on the book. The partner of the weight is the book's pull on the Earth.

Conservation of momentum

When two bodies interact, the third law makes their forces equal and opposite, for the same time. So the impulses are equal and opposite, and the momentum one gains, the other loses:

The total momentum of a system stays the same when no net external force acts on it.

Recoil and explosions

A gun and its bullet are at rest: total momentum zero. After the shot it is still zero, so the gun moves back: $MV = -mv$.

Rockets work the same way: hot gas is pushed back, and the rocket is pushed forward. No ground or air is needed.

Collisions

In every collision, momentum is conserved (if no outside force acts). Kinetic energy usually is not: some becomes sound, heat and dents. When two bodies stick together, the most energy is lost. Watch the bars.

You should find that the momentum bars always match, while the energy after coupling is less, and the loss is biggest when the wagons have equal masses.

A worked example

A boy of mass $45\ \text{kg}$ runs at $4.0\ \text{m/s}$ and jumps onto a skateboard of mass $3.0\ \text{kg}$ at rest. How fast do they move together?

1. Before. $45\ \text{kg} \times 4.0\ \text{m/s} + 0 = 180\ \text{kg m/s}$.
2. After. $(45\ \text{kg} + 3.0\ \text{kg})\,v = 180\ \text{kg m/s}$, so $v = 3.75\ \text{m/s}$.

Think first. The board is light, so the speed drops only a little. Had the boy jumped onto a $45\ \text{kg}$ trolley, they would move off at $2.0\ \text{m/s}$, half the speed.

Practise with new numbers

Try it in Python

The program couples two railway wagons and checks both momentum and kinetic energy.

Remember these

• Third law: forces come in equal and opposite pairs, acting on different bodies.
• With no net external force, total momentum stays the same.
• Recoil: $MV = mv$. Sticking together: $v = \dfrac{m_1u_1}{m_1 + m_2}$.
• Momentum is conserved in collisions; kinetic energy usually is not.

Check yourself

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

1. A swimmer pushes the water backward. Which force moves her forward?

2. If the Earth pulls an apple with $2\ \text{N}$, how hard does the apple pull the Earth?

3. A $60\ \text{kg}$ skater at rest pushes a $40\ \text{kg}$ skater at rest, who moves off at $3.0\ \text{m/s}$. How fast does the first skater move?

4. Why do the third-law forces between a horse and its cart not cancel, so that nothing could move?