3.3 Linear momentum and its conservation

For a closed system with negligible external impulse, total momentum before an interaction equals total momentum after it: Σp_before=Σp_after.

Define the system and a positive direction first. Momentum is a vector, so opposite directions must carry opposite signs.

Two carts initially at rest apart can exchange equal and opposite momentum; for a collision, the signed total before and after must match.

Momentum is not automatically conserved for every chosen object: an external force can change that object’s momentum.

In one dimension, write m₁u₁+m₂u₂=m₁v₁+m₂v₂, with velocities signed along one chosen axis.

List known masses and velocities, keep units consistent, then solve for the unknown. A negative result means motion opposite to the chosen positive direction.

A 2 kg cart at 3 m s⁻¹ sticks to a 1 kg cart at rest: 6=(3)v, so the joined speed is 2 m s⁻¹ in the original direction.

Do not replace momentum with speed, and do not drop the sign of a cart moving in the opposite direction.

An elastic collision satisfies momentum conservation and also conserves total kinetic energy: KE_before=KE_after.

Use momentum and kinetic-energy equations together only when the collision is identified as elastic; otherwise kinetic energy may become thermal, sound or deformation energy.

For two identical smooth balls in a head-on elastic collision, exchanging their velocities satisfies both conservation laws.

Momentum conservation alone does not prove a collision is elastic; most real collisions are not perfectly elastic.

If the system is isolated, momentum remains conserved in an inelastic interaction, but total kinetic energy after is smaller because energy is transferred to deformation, heat or sound.

Check whether objects separate or stick, then use the momentum equation. Track the lost kinetic energy as a change of energy store, not as lost total energy.

A lump of clay hitting a stationary block and sticking has one common final velocity; the difference in kinetic energy becomes internal energy.

“Kinetic energy is not conserved” does not mean energy disappeared, and inelastic does not mean momentum is unbalanced.