Conservation of Momentum
- State the conservation of momentum for an isolated system
- Apply it to recoil and explosion problems
- Solve for an unknown velocity after an interaction
Isolated systems conserve momentum
If no net external force acts on a system, its total momentum stays constant. Internal forces — like two objects pushing on each other — come in equal-and-opposite Newton’s-third-law pairs, so they cancel within the system and cannot change the total. This holds through collisions, explosions, and recoil, and it does not care whether energy is conserved, which makes it enormously powerful.
Recoil and explosions from rest
When something starts at rest and then flies apart — a gun firing, a skater throwing a ball, a rocket expelling gas — the total momentum must stay zero. So the momenta of the pieces are equal in size and opposite in direction: the lighter piece flies off fast, the heavier piece recoils slowly. That is why a cannon lurches back only a little while the ball rockets forward.
A 200 kg cannon, initially at rest, fires a 4 kg cannonball forward at 50 m/s. Find the cannon’s recoil speed.
- 1.Before firing, total momentum is zero (everything at rest).
- 2.After firing it must still be zero: m_ball·v_ball + m_cannon·v_cannon = 0.
- 3.So 4 × 50 + 200 × v_cannon = 0 → 200 + 200·v_cannon = 0.
- 4.Solve: v_cannon = −200 ÷ 200 = −1 m/s (backward).
Momentum is a vector — objects moving in opposite directions have opposite-sign momenta. Adding their magnitudes instead of accounting for direction is the classic conservation-of-momentum mistake.
A 60 kg astronaut, floating at rest, throws a 3 kg wrench at 8 m/s. What is the astronaut’s recoil speed?
Two skaters stand at rest and push off each other. Skater A (40 kg) glides away at 3 m/s. If skater B has a mass of 60 kg, what is B’s speed?
For any "at rest, then flies apart" problem, set total momentum equal to zero. The pieces carry equal and opposite momenta, so the mass ratio is the inverse of the speed ratio — heavier means slower.
Answer the 2 checkpoints as you read.
Sign in to save your progress