One end of a uniform rigid stick moves with velocity $\vec{v}$ while the other end moves at $\vec{v'}$. It is known that $\vec{v} \cdot \vec{v}' = 0$ and that the velocities and the rod itself are in the same plane. If $|\vec{v}|=|\vec{v'}|= v_0$, the speed of the center of the rod can be written as $\frac{\sqrt{a}}{b}v_0$ where a and b are co-primes. Find the value of $b^a$.