The principle
- The principle of conservation of momentum states that, when no outside force acts on a group of interacting objects, the total momentum of the group before the interaction equals the total momentum after the interaction
- "Interaction" in this context can mean:
- a collision between two objects that come together
- an explosion where one object breaks into two or more pieces
- Because momentum is a vector, the signs of velocities must be tracked carefully when totals are taken, because opposite-direction velocities can partially or fully cancel
Applying conservation to a collision
- Pick a positive direction for the whole problem at the start
- Write the total momentum before the collision:
p_before = m₁u₁ + m₂u₂
- Write the total momentum after the collision:
p_after = m₁v₁ + m₂v₂
- Set the two equal (the heart of conservation) and solve for the unknown velocity:
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- If two objects start moving towards each other at equal and opposite momenta, the system's total momentum is zero, and the same zero must come out after the collision (e.g. the two objects might end up at rest, or they might fly off back-to-back at matched momenta)
Example — a 1500 kg car travelling at 12 m/s collides with a stationary 2400 kg lorry. The car slows to 3.0 m/s after the impact (still moving forward). Calculate the lorry's velocity straight after the collision.
- Take "forward" as positive
- Total momentum before = (1500 × 12) + (2400 × 0) = 18 000 kg m/s
- Total momentum after = (1500 × 3.0) + (2400 × v_lorry) = 4500 + 2400 v_lorry
- Set them equal: 18 000 = 4500 + 2400 v_lorry
- Solve: 2400 v_lorry = 18 000 − 4500 = 13 500
- v_lorry = 13 500 / 2400 = 5.625 m/s forward