Conservation of Linear Momentum

A system keeps its momentum when its net external impulse is zero (or negligible over a short interval): outside forces may be absent, may cancel, or may be too brief to matter, and in every such case p_before = p_after. In a collision the forces the objects put on each other are internal and cancel in pairs, so the total runs straight through. Kinetic energy is a separate ledger: it survives a perfectly elastic collision but drops in an inelastic one, where the objects stick and the lost energy turns to heat. A system at rest that bursts apart keeps its total at zero, so the pieces fly off with equal and opposite momenta.

The errors here cluster around the word isolated. Students treat conservation as automatic, setting momentum-before equal to momentum-after for a system that isn't isolated — a falling ball, a sliding block, a ball off a wall — or expecting a single object to keep its own momentum, when conservation needs zero net external impulse. They botch recoil and explosions, scoring the parts by equal speeds or equal kinetic energies (or letting the total grow) instead of equal and opposite momenta that sum to the start, so the lighter part comes out faster. And they confuse the two ledgers, assuming kinetic energy is conserved in every collision or reading an inelastic energy drop as a broken law, when momentum is conserved either way and kinetic energy only in an elastic collision.