Linear Momentum
Linear Momentum
Momentum p = mv is a vector, and impulse J = ∫F dt = Δp equals the area under a force-time graph. For a chosen system, momentum is conserved whenever the net external impulse is negligible - typically during brief collisions where internal forces dwarf gravity or friction. Collisions are classified by kinetic energy: elastic collisions conserve K as well as p, inelastic ones do not, and perfectly inelastic ones end with the bodies moving together. The center of mass moves as if all the mass and all external forces acted at that point.
Traps: dropping signs in one dimension, since momentum is a vector and a rebound reverses a sign; assuming kinetic energy is conserved in every collision; forgetting that equal and opposite impulses act on two different bodies, so each body's momentum changes; and invoking conservation without checking that external impulses really are negligible.
Worked example: a 2 kg cart at 3 m/s strikes a stationary 4 kg cart and they stick. p_i = 6 kg·m/s, so v_f = 6/6 = 1 m/s. Initial K = 9 J and final K = 3 J, so 6 J is lost to deformation and heat.
Rule of thumb: state the system and the direction, write p_before = p_after as a vector equation, then treat conservation of K as a separate question.