Two-Dimensional Collisions

Momentum is a vector, so the conservation law applies to each direction independently. If two objects collide at some angle, you don't need a single complicated equation — you need two simple ones. Add up all the x-components of momentum before the collision, set that equal to the sum of x-components after. Do the same for y. Each equation looks exactly like a one-dimensional momentum problem, because that's exactly what it is: the collision's effect on horizontal motion, and its effect on vertical motion, don't interfere with each other.

This is why pool balls flying off at odd angles aren't actually mysterious. Each ball's momentum splits into a north-south piece and an east-west piece, and each piece obeys its own simple conservation rule. Once you solve for the final x-momentum and final y-momentum of the system, you combine them with the Pythagorean theorem to get the total momentum's magnitude, and inverse tangent to get its direction.

This connects directly to the one-dimensional collision work earlier in the unit: same conservation law, same classification into elastic and inelastic, just broken into components first. It also sets up circular motion and rotation later, where splitting vectors into perpendicular components becomes the standard move for almost every problem. The habit you build here — resolve into components, conserve each one, recombine at the end — is the same habit you'll use for forces on inclines and for projectile motion.