Newton's third law and conservation of momentum
For every action there is an equal and opposite reaction, acting on a different body. So the total momentum of a system stays the same when no outside force acts.
Newton's third law in everyday life
- Statement: for every action, there is an equal and opposite reaction. Action and reaction act on two different bodies, so they never cancel each other.
- Walking: your foot pushes the ground backward; the ground pushes you forward.
- Swimming: a swimmer pushes the water backward; the water pushes the swimmer forward.
- A bird flying: the wings push air downward; the air pushes the bird upward.
- Gun recoil: the gun pushes the bullet forward; the bullet pushes the gun backward, so the gun recoils (kicks back).
- Jumping off a boat: as you jump forward to the bank, you push the boat backward.
Proving the law of conservation of linear momentum
**Law**: the total linear momentum of a system of bodies stays **constant** if **no net external force** acts on it. We prove it using the second and third laws.
- 1Two balls of masses m₁ and m₂ move in a straight line with velocities u₁ and u₂ (u₁ > u₂). They collide for a short time t, then move with velocities v₁ and v₂.
- 2Force on ball 2 by ball 1 (action): F₂ = m₂(v₂ − u₂)/t.
- 3Force on ball 1 by ball 2 (reaction): F₁ = m₁(v₁ − u₁)/t.
- 4By the third law, F₁ = −F₂. So m₁(v₁ − u₁)/t = −m₂(v₂ − u₂)/t.
- 5★Cancel t and rearrange: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Total momentum before collision = total momentum after collision.
★ key step
Rocket propulsion
Rocket propulsion works on **Newton's third law** and the **conservation of linear momentum**.
- 1Before launch the rocket is at rest, so its total momentum is zero.
- 2The fuel (propellant) burns and makes hot gases.
- 3The gases rush out of the nozzle at the back at very high speed. They carry momentum backward.
- 4★To keep the total momentum unchanged, the rocket gains an equal forward momentum and rises.
- 5As the fuel burns, the mass of the rocket decreases, so its velocity keeps increasing. It can reach the escape velocity (11.2 km s⁻¹) needed to escape the Earth's pull.
★ key step
How to draw the collision diagram
- Draw a straight horizontal line for the path, and three sketches side by side.
- Before: two circles labelled m₁ and m₂, with arrows u₁ and u₂ pointing the same way (draw the u₁ arrow longer).
- During: the two balls touching, with an arrow F₁ on m₁ pointing backward and an arrow F₂ on m₂ pointing forward.
- After: the two balls apart, with arrows v₁ and v₂.
- Write Before, During and After under the three sketches.
Recoil of a gun
A gun of mass 5 kg fires a bullet of mass 0.01 kg with a velocity of 250 m s⁻¹. Find the recoil velocity of the gun.
- Given: gun m₁ = 5 kg, bullet m₂ = 0.01 kg, bullet velocity v₂ = 250 m s⁻¹. Both are at rest before firing.
- Momentum before firing = 0.
- Conservation of momentum: m₁v₁ + m₂v₂ = 0.
- 5 × v₁ + 0.01 × 250 = 0, so 5v₁ = −2.5.
- v₁ = −2.5 ÷ 5 = −0.5 m s⁻¹.
The gun recoils at 0.5 m s⁻¹. The minus sign shows that it moves opposite to the bullet.
When you jump from a country boat on to the bank of a river, the boat moves back. ISRO rockets launched from Sriharikota rise the same way, by pushing hot gases downward at great speed.
2 marks: “State Newton's third law of motion. Give two examples.” For every action there is an equal and opposite reaction, acting on different bodies. Examples: walking, gun recoil, swimming.
4 marks: “State and prove the law of conservation of linear momentum.” Statement; two bodies m₁, m₂ collide; F₂ = m₂(v₂ − u₂)/t, F₁ = m₁(v₁ − u₁)/t; F₁ = −F₂ by the third law; so m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
7 marks: “State the law of conservation of momentum and explain rocket propulsion. A 4 kg gun fires a 0.02 kg bullet at 200 m s⁻¹. Find the recoil velocity.” Statement and proof; rocket: gases go backward, rocket goes forward, mass falls so speed rises; recoil: 4v + 0.02 × 200 = 0, so v = −1 m s⁻¹ (1 m s⁻¹ backward).
Action and reaction are twins on different bodies: equal, opposite, and they never cancel.