4.4 Conservation of Linear Momentum

Momentum is conserved in isolated systems

For a closed, isolated system (no net external force), total linear momentum is conserved: total momentum before equals total momentum after. Internal forces (like collisions or explosions) redistribute momentum between objects but cannot change the system's total. This principle follows directly from Newton's third law — internal forces come in equal/opposite pairs that cancel.

Applying conservation to collisions

Use p_total,before = p_total,after to solve collision and explosion problems. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not — objects may stick together (perfectly inelastic), losing the most KE. Always define a consistent positive direction since momentum is a vector.

External forces change system momentum

If an external force (like friction or gravity from outside the system) acts, momentum is not conserved unless you include the source of that force in the system. The impulse-momentum theorem shows any net external impulse equals the system's change in total momentum, Δp_total.

Key terms

Conservation of linear momentum

In an isolated system (no net external force), total momentum before an interaction equals total momentum after.

Elastic collision

A collision in which both momentum and kinetic energy are conserved.

Inelastic collision

A collision in which momentum is conserved but kinetic energy is not; objects may or may not stick together.

Perfectly inelastic collision

Objects stick together after colliding, moving with a common final velocity; maximum kinetic energy is lost.

Why internal forces don't change total momentum

By Newton's third law, internal forces between objects in a system are equal and opposite, canceling out when summed over the whole system.

Try an AP-style question

A 2 kg cart at 3 m/s hits a 1 kg cart at rest, and they stick. What is their final speed?

1. 1 m/s
2. 3 m/s
3. 6 m/s
4. 2 m/s