Is this question answered ok or should I have an equation in t?
The problem is the fact that you write:
$\displaystyle \left(\frac{v}{2}-\frac{v^2}{2}\right)^{-1}=\frac{2}{v}-\frac{2}{v^2}$
which is incorrect.
You can say that:
$\displaystyle \left(\frac{v}{2}-\frac{v^2}{2}\right)^{-1}=\left(\frac{v-v^2}{2}\right)^{-1}=\frac{2}{v-v^2}$