Conservation of Momentum (Two Bodies)

With no external force acting, the total momentum of two colliding bodies before the impact equals the total after: always, whether they bounce, stick, or shatter. Kinetic energy is not so obliging; it survives only in a perfectly elastic collision. The principle was nailed down in 1668 when the Royal Society set the collision problem as a challenge and John Wallis, Christopher Wren and Christiaan Huygens independently sent in the answer.

Signs are everything in one dimension: pick a positive direction and stick with it, so a body moving the other way carries a negative velocity. A 2 kg cart at 5 m/s striking a stationary 3 kg cart and slowing to 1 m/s leaves the second cart at (10 + 0 − 2) ⁄ 3 ≈ 2.67 m/s. Because momentum is conserved in every collision, crash investigators use exactly this equation with skid-mark evidence to back out pre-impact speeds, and it works equally well for recoil: the rifle and the bullet start with zero total momentum and must end with zero.