How fast is the point moving? Using calculus, its *velocity*, , is defined as the derivative of with respect to time:

Using numerical computations, is approximately equal to , in which denotes a small change in time and

(8.2) |

In other words, is the change in from the start to the end of the time change. The velocity can be used to estimate the change in over as

(8.3) |

The approximation quality improves as becomes smaller and itself varies less during the time from to .

We can write to indicate that velocity may change over time. The position can be calculated for any time from the velocity using integration as^{8.1}

which assumes that was known at the starting time . If is constant for all time, represented as , then . The integral in (8.4) accounts for being allowed to vary.

Steven M LaValle 2019-03-14