two-dimensional collision
A two-dimensional collision is a glancing hit rather than a head-on one — the objects meet off-center and go spinning off in different directions, like a cue ball clipping another ball at an angle and both rolling away sideways. Now the motion spreads across a plane, so a single line of plus-and-minus signs is no longer enough.
The key idea is that momentum is a vector, and each of its components is conserved separately. So you split every velocity into an x-part and a y-part, and write conservation twice: total x-momentum before = total x-momentum after, and total y-momentum before = total y-momentum after. Two equations, one per direction. If the collision is also elastic, kinetic energy (a single scalar equation) is conserved on top of that. A pretty result: when a moving object strikes an equal-mass one at rest elastically and off-center, the two fly apart at exactly 90 degrees.
Two-dimensional analysis is how real pool shots, car crashes at intersections, and particle-scattering experiments are actually worked out. The method never changes — resolve into components, conserve each — which is why getting comfortable with vector components pays off across all of physics.
A cue ball moving in the +x direction strikes an equal-mass ball at rest, off-center and elastically. Momentum splits between x and y; kinetic energy is also conserved. Working the algebra, the two balls leave along paths that are exactly perpendicular — a right angle you can watch on any pool table.
Equal masses, one at rest, elastic and off-center: the two always separate at 90 degrees.
You cannot conserve 'total speed' or add speeds like plain numbers here — momentum is a vector, so it is the x and y components that each balance. The 90-degree result is special to equal masses with one target at rest and an elastic hit; change any of those and the angle changes.