Question

The magnitude of their relative velocity just before they collide is $\sqrt{\frac{xGM}{r}}$. Find $x$.

Answer 1

Just before they collide, their velocities are perpendicular to each other as shown in figure.
Let their velocities be $V_1,V_2$. Given that their orbital radius(say r) is equal.
So, $V_1=V_2=\sqrt{\frac{GM}{r}}$.
The vectorial addition of the equal magnitude perpendicular vectors $=2^{0.5}{V}_1=2^{0.5}{V}_2=\sqrt{\frac{2GM}{r}}$.
Therefore$,x=2$