### Motion in Space

The vector version of a parametric curve is given by interpreting $\rr=\rr(u)$ as the position vector of an object moving along the curve. The derivatives of position are velocity $\vv$ and acceleration $\aa$: \begin{eqnarray} \vv &=& {d\rr\over du}\\ \aa &=& {d\vv\over du} = {d^2\rr\over du^2} \end{eqnarray} and speed is the magnitude of velocity: $$v = |\vv| = \left| {d\rr\over du} \right|$$

This terminology is most appropriate when the parameter is time, usually denoted by $t$ instead of $u$.