4.3 Conservation of Linear Momentum

Momentum is conserved when isolated

For a closed, isolated system (no net external force), total linear momentum is conserved: p_initial = p_final. Internal forces (like collisions or explosions) can transfer momentum between objects but cannot change the system's total momentum. This follows directly from Newton's third law, since internal forces cancel in pairs.

Collisions: elastic vs. inelastic

In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is conserved but kinetic energy is not — some converts to heat, sound, or deformation. Perfectly inelastic collisions have objects sticking together, moving with a common final velocity, maximizing KE loss.

Impulse-momentum theorem connects forces to Δp

Impulse (J) = ∫F dt = Δp. When analyzing systems, set total initial momentum equal to total final momentum along each axis independently, since momentum is a vector. This lets you solve for unknown velocities or masses even without knowing the details of the interaction force.

Key terms

Conservation of momentum

Total momentum of an isolated system (zero net external force) remains constant: p_initial = p_final.

Elastic collision

A collision where both momentum and kinetic energy are conserved.

Inelastic collision

A collision where momentum is conserved but kinetic energy is not; some KE converts to heat/sound/deformation.

Perfectly inelastic collision

A special inelastic collision where the colliding objects stick together and move with a common final velocity.

Why internal forces don't change total momentum

By Newton's third law, internal forces occur in equal and opposite pairs, so they cancel out when summed over the whole system.

Try an AP-style question

A 60 kg person on frictionless ice throws a 2 kg ball at 15 m/s. What is the person's recoil speed?

1. 2 m/s
2. 0.5 m/s
3. 15 m/s
4. 30 m/s