Distance between the centres of two starts is $10a$ . The masses of these stars are $M$  and $16M$  and their radius $a$ and $2a$ respectively. A body of mass$m$  is fired straight from the surface of the larger star towards the surface of the smaller star. Now if its minimum initial speed to reach the surface of the smaller star in terms of $G, M$ and $a$ is $\frac{3}{2}\sqrt{\frac{\beta GM}{a}}$, then find value of $\beta$.

A body moving radially away from a planet of mass$\mathrm{M}$, when at distance $\mathrm{r}$ from planet, explodes in such a way that two of its many fragments move in mutually perpendicular circular orbits around the planet. What will be

(i) Then velocity in circular orbits?
(ii) Maximum distance between the two fragments before collision and
(iii) Magnitude of their relative velocity just before they collide?

A satellite$P$ is revolving around the Earth at a height $h=$radius of Earth $\left(R\right)$ above equator. Another satellite $Q$, is at a height $2 \mathrm{h}$ revolving in opposite direction. At an instant the two are at same vertical line passing through centre of sphere. If the least time after which again they are in this situation is given by$\frac{2\pi R^{{\displaystyle \frac{3}{2}}}6\sqrt{\alpha }}{\sqrt{GM}(2\sqrt{2}+3\sqrt{\beta })}$, then find $\alpha +\beta$.