How fast is the point moving? Using calculus, its velocity,
, is defined as the derivative of
with respect to time:
![$\displaystyle v = { dy(t) \over dt } .$](img689.gif) |
(8.1) |
Using numerical computations,
is approximately equal to
, in which
denotes a small change in time and
![$\displaystyle \Delta y = y(t + \Delta t) - y(t).$](img692.gif) |
(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
![$\displaystyle \Delta y \approx v \Delta t .$](img694.gif) |
(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 as8.1
![$\displaystyle y(t) = y(0) + \int_0^t v(s) ds ,$](img697.gif) |
(8.4) |
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
2020-11-11