## Finding an Expression For Displacement When Velocity Given In Terms of Displacement

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

Now separate variables.
$\frac{dx}{x^2}=x^{-2}dx=dt$

Now integrate in the usual way.
$\int^x_1 x^{-2}dx = \int^t_2 dt$

$[- \frac{1}{x} ]^x_1 = t-2$

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

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