Distance between the centres of two stars is 10a. Masses of these stars are M and 16M and their radii are 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. The minimum initial speed of the body to reach the surface of the smaller star is $\frac{3\sqrt{\mathrm{x}}}{2}\left(\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}}\right)$ . Find ‘$\mathrm{x}$’.

Let there be two stars 1 and 2 as shown below.

Let P be a point between C₁ and C₂, where gravitational field strength is zero.
Hence
$\frac{\mathrm{GM}}{{\mathrm{r}}_1^2}=\frac{\mathrm{G}\left(16\mathrm{M}\right)}{{\mathrm{r}}_2^2};\frac{{\mathrm{r}}_2}{{\mathrm{r}}_1}=4,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\mathrm{r}}_1+{\mathrm{r}}_2=10\mathrm{a}$
$\therefore {\mathrm{r}}_2=\left(\frac{4}{4+1}\right)\left(10\mathrm{a}\right)=8\mathrm{a}$
${\mathrm{r}}_1=2\mathrm{a}$
Now, the body of mass m is projected from the surface of large star towards the smaller one. Between C₂ and P it is attracted towards 2 and between C₁ and P it will be attracted towards 1.
Therefore, the body should be projected to just cross point P because beyond that the particle is attracted towards the smaller star itself. 

From conservation of mechanical energy $\frac{1}{2}{\mathrm{mv}}^2$
= potential energy of the body at P - potential energy at the surface of larger star.
$\therefore \frac{1}{2}{\mathrm{mv}}_{\min }^2=\left[\frac{\mathrm{GMm}}{{\mathrm{r}}_1}-\frac{16\mathrm{GMm}}{{\mathrm{r}}_2}\right]\mathrm{-}\left[-\frac{\mathrm{GMm}}{10\mathrm{a}-2\mathrm{a}}-\frac{16\mathrm{GM}}{2\mathrm{a}}\right]$

$\frac{1}{2}{{\mathrm{mv}}^2}_{\min }=\left[-\frac{\mathrm{GMm}}{2\mathrm{a}}-\frac{16\mathrm{GMm}}{8\mathrm{a}}\right]-\left[-\frac{\mathrm{GMm}}{8\mathrm{a}}-\frac{8\mathrm{GMm}}{\mathrm{a}}\right]$
$\Rightarrow \frac{1}{2}{\mathrm{mv}}_{\min }^2=\left(\frac{45}{8}\right)\frac{\mathrm{GMm}}{\mathrm{a}}$
$\Rightarrow {\mathrm{v}}_{\min }=\frac{3\sqrt{5}}{2}\left(\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}}\right)$.

So x=5.