Coaching Centre
Newton’s third law and momentum
Action–reaction force pairs and Newton's third law, why the pair does not cancel, rockets, walking and recoil, momentum as mass × velocity, conservation of momentum in collisions and explosions, and impulse.
A rocket on the launch pad has nothing to push against once it leaves the ground, and in space there is no air at all. So how does it speed up? It throws hot gas backwards at thousands of metres per second, and the gas pushes the rocket forwards. Every force comes with a partner.
Newton's third law
Newton's third law: when object A exerts a force on object B, object B exerts a force on object A that is equal in size and opposite in direction.
The two forces are called an action–reaction pair (or a third-law pair). They always:
- are the same size
- act in opposite directions
- act on two different objects
- are the same type of force (both contact forces, both gravity, and so on)
- happen at the same time: neither comes first, despite the names.
- Walking: your foot pushes backwards on the ground (using friction), and the ground pushes forwards on your foot. That forward push is what accelerates you. On slippery ice, your foot can't push back, so you can't push forward either.
- Rockets and jets: the engine pushes exhaust gases backwards; the gases push the rocket forwards. No air is needed.
- Recoil: a rifle pushes a bullet forwards and the bullet pushes the rifle back into the shooter's shoulder. A garden hose pushes back on your hand for the same reason.
- Swimming and rowing: hands, flippers or oars push water backwards; the water pushes the swimmer or boat forwards.
Why don't the pair cancel out?
If the two forces are equal and opposite, why does anything move? Because they act on different objects. Forces only cancel when they act on the same object. When two skaters push apart, the 60 N on skater A acts on A and the 60 N on skater B acts on B. Each skater has an unbalanced force on them, so each accelerates. By F = ma, the lighter skater has the bigger acceleration.
Don't confuse a third-law pair with balanced forces. A book resting on a table has its weight (Earth pulling the book down) balanced by the support force (table pushing the book up). Those two act on the same object, so they are not a third-law pair. The partner of the book's weight is the book pulling the Earth up; the partner of the support force is the book pushing the table down.
Momentum
A truck and a bicycle moving at the same speed are not equally hard to stop. Momentum (symbol p) measures how much "motion" an object has, combining mass and velocity:
momentum = mass × velocity p = m v (units: kg m/s)
Momentum is a vector: it has the same direction as the velocity. Choose one direction as positive, and momentum in the other direction is negative.
A 0.16 kg cricket ball bowled at 40 m/s: p = 0.16 × 40 = 6.4 kg m/s.
A 1500 kg car at 20 m/s: p = 1500 × 20 = 30 000 kg m/s.
To have the same momentum as the car, a 100 kg cyclist would need v = 30 000 ÷ 100 = 300 m/s, which is impossible on a bike.
Conservation of momentum
When objects collide or push apart, the third law means they exert equal and opposite forces on each other for the same time. So whatever momentum one gains, the other loses. As long as no outside force acts (such as friction), the total momentum stays the same.
Law of conservation of momentum: total momentum before = total momentum after
Before: 2 × 3 + 1 × 0 = 6 kg m/s. After, the joined 3 kg moves at v: 3 v = 6, so v = 2 m/s.
KE before = ½ × 2 × 3² = 9 J. KE after = ½ × 3 × 2² = 6 J. Momentum is conserved, but 3 J of kinetic energy became heat, sound and the work of squashing the Velcro.
A 1 kg and a 3 kg trolley start together at rest, so total momentum = 0. A spring pushes them apart and the 3 kg trolley moves right at 1 m/s (+3 kg m/s). The 1 kg trolley must have −3 kg m/s, so it moves left at 3 m/s.
- Choose Sticky, cart A 2 kg at 3 m/s and cart B 1 kg at rest. Press Go. Check the final velocity is 2 m/s and the total momentum stays 6 kg m/s. What happens to the kinetic energy?
- Choose Bouncy and make both carts 2 kg, with B at rest. What happens? Is kinetic energy conserved too?
- Choose Explosion with A = 1 kg and B = 3 kg. Explain how the total momentum can still be zero after both carts are moving.
- Make the carts head towards each other: A 2 kg at 2 m/s, B 2 kg at −2 m/s, Sticky. Predict the result first.
Force and change in momentum
Newton actually wrote his second law in terms of momentum: the net force equals the rate of change of momentum. Since a = (v − u) ÷ t, F = m a can be rearranged as
F × t = m v − m u (force × time = change in momentum)
F × t is called the impulse. To bring a given momentum to zero, you can use a large force for a short time or a small force for a long time. Bending your knees when you land, "giving" with your hands as you catch a cricket ball, and car airbags all increase the stopping time and so reduce the force. You'll use this idea for car safety in 5I.
Test yourself
A person pushes on a wall with a force of 30 N. The wall pushes on the person with a force of
Third law: equal size, opposite direction.
Which is an action–reaction pair?
Only the first pair acts on two different objects (foot and ground).
The momentum of a 70 kg runner moving at 8 m/s is
p = m v = 70 × 8 = 560 kg m/s.
A 4 kg trolley at 3 m/s collides with a stationary 2 kg trolley and they stick together. Their velocity afterwards is
Momentum before = 12 kg m/s; 12 ÷ 6 kg = 2 m/s.
Two astronauts float at rest. A (100 kg) pushes B (50 kg) with 80 N. Which is true?
Equal forces of 80 N; a = F ÷ m gives 0.8 m/s² for A and 1.6 m/s² for B.
Label the action–reaction forces.
Is each pair of forces an action–reaction pair?
Newton's third law forces are equal in size, opposite in direction and act on objects. Momentum equals mass × and is measured in . In a collision with no outside forces, the total is conserved.
A 0.01 kg bullet is fired at 400 m/s from a 4 kg rifle that was at rest. (a) Use conservation of momentum to calculate the recoil velocity of the rifle. (b) Explain the recoil using Newton's third law. (c) Explain why the shooter is told to hold the rifle firmly against their shoulder.
(a) Before firing, everything is at rest, so the total momentum is zero. After firing, the bullet's momentum is 0.01 × 400 = 4 kg m/s forwards, so the rifle must have 4 kg m/s backwards for the total to stay zero. Recoil velocity = 4 ÷ 4 = 1 m/s backwards. (b) When the gunpowder explodes, the rifle pushes the bullet forwards. By Newton's third law the bullet pushes back on the rifle with a force of equal size in the opposite direction, for exactly the same time, so the rifle gains the same amount of momentum in the opposite direction. (c) If the rifle is held firmly against the shoulder, the shooter and rifle recoil together as one much larger mass, so the recoil velocity is much smaller. If it is held loosely, the rifle can reach its full recoil speed before hitting the shoulder, and is then stopped over a very short time, producing a large, painful force.