The third law and conservation of momentum
Forces come in pairs
Press your hand against a wall. You push the wall, and you can feel the wall pushing back on your hand.
Two skaters stand face to face on ice. One pushes the other. Both of them slide back, in opposite directions.
Forces always come in pairs. Newton’s third law: to every action, there is always an equal and opposite reaction.
Draw the pair as arrows. The force on B by A, and the force on A by B. Same length, opposite directions: F A B equals minus F B A.
Key point: the two forces act on different bodies. One arrow sits on A, the other on B. That is why they never cancel each other out.
Neither force comes first. Action and reaction happen at the same instant. You can call either one the action.
A falling apple: the Earth pulls the apple down, and the apple pulls the Earth up with exactly the same force.
Why doesn’t the Earth move towards the apple? Same force, but enormous mass. By a equals F by m, the Earth’s acceleration is far too tiny to notice.
Recoil and conservation of momentum
A rifle fires a bullet. The bullet flies forward, and the rifle kicks back into the shooter’s shoulder. This kick is called recoil.
While the gas pushes, the rifle pushes the bullet forward with force F. By the third law, the bullet pushes the rifle backwards with the same force.
Both forces act for the same short time, delta t. So the impulses are equal and opposite, and the changes in momentum are equal and opposite.
Before firing, nothing moves: total momentum is zero. After firing, the bullet’s forward momentum is exactly cancelled by the rifle’s backward momentum.
The bullet is light, so it gets a big speed. The rifle is heavy, so its recoil speed is small. Rifle speed equals bullet mass times bullet speed, divided by rifle mass.
This is the law of conservation of momentum: the total momentum of an isolated system of interacting particles stays constant.
Isolated means no external force acts. Internal forces, like the push between the bullet and the rifle, can never change the total.
The same law explains collisions between snooker balls, a boat moving back as you jump onto the bank, and how a rocket pushes itself forward.
Try it: carts pushed apart
- Release the spring with equal carts first. Both move off at the same speed.
- Now make one cart heavier. The forces on the two carts are still equal and opposite.
- Watch the momentum bars: always equal in size, so the total stays zero.
- Try the Gun and bullet preset: the light cart flies off ten times faster.
Quick check
A 4 kg rifle fires a 20 g bullet with a speed of 400 m/s. What is the recoil speed of the rifle?
- 0.5 m/s
- 2 m/s
- 8 m/s
- 20 m/s
Answer
B. Total momentum is zero before and after firing. Bullet momentum = 0.020 × 400 = 8 kg·m/s forward. So the rifle has 8 kg·m/s backwards: V = 8 / 4 = 2 m/s, opposite to the bullet.
Spot the pair
A book rests on a table. Its weight W acts downwards and the normal force N from the table acts upwards. Are W and N an action–reaction pair?
- Yes, they are equal and opposite
- Yes, because the book is at rest
- No. Both act on the book. W pairs with the book’s pull on the Earth
- No. W is bigger than N
Answer
C. An action–reaction pair always acts on two different bodies. W and N both act on the book; they balance because the book is in equilibrium (first and second laws). The reaction to W is the book pulling the Earth upwards. The reaction to N is the book pushing down on the table.
Summary
- Third law (F_AB = force on A by B):
F_AB = −F_BA - Conservation of momentum:
m₁v₁ + m₂v₂ = constant - Gun recoil:
V = −mv / M
- Third law: to every action there is an equal and opposite reaction. Forces always come in pairs.
- Action and reaction act on different bodies, at the same instant, so they never cancel each other.
- For an isolated system, internal forces cannot change the total momentum. It stays constant.
- In recoil, the light body gets a large speed and the heavy body a small one, but their momenta are equal and opposite.