AP Physics 1: Conservation of Momentum (collisions made simple)

Conservation of momentum is the AP Physics 1 topic where students lose the most points to setup errors, not physics errors. The math is one equation. The discipline is in the diagram.

The one equation

For any collision with no external forces (or where external forces are negligible during the collision):

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

Total momentum before equals total momentum after. That's it. Every AP Physics 1 collision problem starts here.

The 3-step protocol

1. Draw before and after. Two diagrams. Label each object's mass and velocity in each. Use a consistent positive direction.
2. Write the conservation equation using the diagram. Velocities to the left get negative signs.
3. Plug in and solve. If the collision is perfectly inelastic, the two objects stick together and have a common final velocity, so the equation has one unknown.

Worked example — perfectly inelastic

A 2 kg cart moving right at 3 m/s collides with a stationary 1 kg cart. They stick together. Find the final velocity.

Conservation: (2)(3) + (1)(0) = (2 + 1)v_f. 6 = 3v_f. v_f = 2 m/s to the right.

Worked example — elastic

A 2 kg cart moving right at 4 m/s collides elastically with a stationary 4 kg cart. Find final velocities.

Two equations: momentum (2)(4) = 2v_1f + 4v_2f, and kinetic energy (1/2)(2)(4)² = (1/2)(2)v_1f² + (1/2)(4)v_2f².

For elastic collisions, you can also use the relative-velocity rule: v_1i − v_2i = −(v_1f − v_2f). So 4 − 0 = −(v_1f − v_2f), giving v_2f − v_1f = 4.

Combined with the momentum equation: 8 = 2v_1f + 4(v_1f + 4). 8 = 6v_1f + 16. v_1f = −4/3 m/s. v_2f = 4 − 4/3 = 8/3 m/s.

Common AP Physics 1 momentum mistakes

1. Forgetting to make velocity negative for leftward motion. Momentum is a vector. Direction matters.
2. Confusing elastic and inelastic. Elastic conserves KE; inelastic doesn't. Perfectly inelastic = stuck together.
3. Treating an explosion problem as a collision. Same conservation, but the initial momentum might be zero (object at rest exploding into two pieces).
4. Missing impulse-momentum theorem connection. Average force times time equals change in momentum. Many AP problems use this instead of conservation.

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