## An Expression For the Velocity When Acceleration is Given in Terms of Velocity

Suppose the acceleration of a body is given in terms of the velocity. How can we find an expression for the velocity?
If
$a=2v^2$
and
$v=1$
when
$t=2$
(in the appropriate units), then we can write
$a= \frac{dv}{dt}=2v^2$

Now separate variables.
$\frac{dv}{v^2}=v^{-2}dv=dt$

Now integrate in the usual way.
$\int^v_1 v^{-2}dv = \int^t_2 dt$

$[- \frac{1}{v} ]^v_1 = t-2$

$- \frac{1}{v} - ( \frac{1}{1} )=t-2$

Making
$v$
the subject gives
$v= \frac{1}{3-t}$
.