POSITION, VELOCITY AND ACCELERATION

The connections between position, velocity, and acceleration are one of the most important themes in differential and integral calculus, especially when these quantities vary with time.

Position is the mathematical identification (a vector) of an absolute or relative location of a point or body in space, P(x, y, z, t). It has the units of length.

Velocity is the rate of change (in time) of the position of a point or body. It is defined as the derivative of position: V(x, y, z, t) = (∂P)/(∂t) = Ṗ. It has both a magnitude and direction, so it is a vector. It has the units of length/time.

Acceleration is the rate of change (in time) of the velocity of a point or body. It is defined as the derivative of velocity. a(x, y, z, t) = (∂v)/(∂t) = V̇ or the second derivative of the position a(x, y, z, t) = (∂²P)/(∂t²) = P̈. It has both a magnitude and direction, so it is a vector. It has the units of length/time/time.

These are partial derivatives as they can be functions of more than one independent variable. The converse of these definitions is also true: Acceleration can be integrated in time to find velocity, which can be integrated to find the position.

Because these are vectors, the magnitude of each can be found by dotting the vector with itself and taking the square root. For example, the speed of a point can be computed by dotting the velocity V = xi + yj + zk with itself: Speed = |V⋅V| = √(x² + y² + z²). While velocity is a vector, the speed is a scalar.