Two-Dimensional Collisions

Momentum is a vector, so when a collision sends objects off at angles instead of along one line, you don't write one momentum equation — you write two. Pick x and y axes, usually with x along the incoming velocity. Total momentum in, in the x-direction, equals total momentum out, in the x-direction. Same for y, where the incoming momentum is usually zero if one object starts at rest. That gives two equations and lets you solve for two unknowns, like a final speed and an angle.

The famous right-angle result — two identical pool balls flying apart at 90 degrees after a glancing elastic hit — isn't a coincidence, it's what happens when you combine both conservation laws for equal masses with one ball initially still. Momentum conservation fixes the vector sum of the two final velocities; kinetic energy conservation fixes the sum of their squared speeds. Together, for equal masses, the only way to satisfy both is for the two paths to be perpendicular. Change the masses or make the collision inelastic, and that clean 90-degree geometry disappears.

This topic sits right after impulse and 1D collisions: same bookkeeping, just doubled. It also sets up the kinetic energy question that follows naturally — momentum conservation comes from Newton's third law and never fails in an isolated system, but kinetic energy conservation is a bonus feature of elastic collisions only, not a law of nature.