In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion.

The corresponding displacement vector can be defined as the difference between the final and initial positions:

Study of these higher order derivatives can improve approximations of the original displacement function.

Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics.

The instantaneous speed, then, is distinct from velocity, or the time rate of change of the distance travelled along a specific path.

The velocity may be equivalently defined as the time rate of change of the position vector.

If one considers a moving initial position, or equivalently a moving origin (e.g. an initial position or origin which is fixed to a train wagon, which in turn moves on its rail track), the velocity of P (e.g. a point representing the position of a passenger walking on the train) may be referred to as a relative velocity; this is opposed to an absolute velocity, which is computed with respect to a point and coordinate axes which are considered to be at rest (a inertial frame of reference such as, for instance, a point fixed on the floor of the train station and the usual vertical and horizontal directions).