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 \[v= \frac{1}{3-t}\]
.