Linear Momentum
Core ideas
Momentum p = mv is a vector. Impulse J = FΔt equals change in momentum, so a longer collision time reduces average force for the same Δp. In an isolated system, total momentum is conserved in every collision and explosion; kinetic energy is conserved only in elastic collisions.
Candidates must classify collisions (elastic, inelastic, perfectly inelastic) and choose the right conservation statement. Perfectly inelastic collisions stick together and lose the most kinetic energy consistent with momentum conservation. Momentum conservation applies per axis, so 2D problems require x and y equations.
Center-of-mass reasoning helps: the center of mass of an isolated system continues at constant velocity regardless of internal explosions.
Traps
- Assuming kinetic energy is conserved in all collisions.
- Forgetting momentum is a vector - signs matter in 1D.
- Believing a wall "absorbs" momentum in a bounce; the ball reverses, so Δp is larger than for a stick.
- Using impulse = FΔt with average force but substituting peak force.
Rule of thumb
Write p_initial = p_final component by component. Then check whether the collision is elastic; if not, kinetic energy is not conserved and you must not use it.
Worked example
A 2 kg cart at 3 m/s hits a stationary 1 kg cart and they stick. p_i = 6 kg·m/s, so v_f = 6/3 = 2 m/s. Initial K = 9 J; final K = ½(3)(4) = 6 J, so 3 J is lost - perfectly inelastic.