relative velocity

Relative velocity is how fast something moves as seen by a particular observer — and the key insight is that different observers, moving differently, see different velocities for the same object. Walk down the aisle of a moving train and, to a fellow passenger, you stroll at a slow walking pace; to someone standing on the platform, you rush past at the train's speed plus your own. Neither is wrong; velocity is always relative to whoever is measuring.

Precisely, the velocity of an object A relative to an observer B is the object's velocity minus the observer's velocity, all measured in some common frame: v_(A relative to B) = v_A - v_B. Because velocities are vectors, this is a vector subtraction, so directions matter. A handy consequence is that to convert a velocity from one frame to another you just add or subtract the relative velocity between the frames: v_(A relative to ground) = v_(A relative to train) + v_(train relative to ground).

This idea is the whole reason a boat aiming straight across a river drifts downstream, and why a plane must aim into a crosswind to fly a straight course: you add the object's velocity through the medium to the medium's velocity over the ground. At everyday speeds these velocities simply add and subtract as vectors. That simple addition is an excellent approximation but not exactly true near the speed of light, where Einstein's special relativity replaces it with a more careful velocity-addition rule.