Question

Distance between the centers of two stars is $10a$. The mass of these stars are $M$ and $16M$ and their radii $a$ and $2a$ respectively. A body of mass $m$ is fired straight from the surface of the larger star towards the smaller star. What should be the minimum initial speed to reach the surface of the smaller star? Given $GM/a=4/5)$.

Answer 1

The distance $X$, where gravitational pull of each stars becomes equal and opposite can be obtained
$G\frac{Mm}{X^2}=G\frac{\left(16M\right)m}{(10a-x{)}^2}\Rightarrow x=2a$
So the body will reach the smaller star due to stars gravitational pull is body just crosses point $P$ but using COEm energy at $O=$ Energy at $P$
$-\frac{G.16M,m}{2a}-\frac{GMm}{8a}+\frac{1}{2}m{v}_{min}^2=-\frac{G\left(16M\right)m}{8a}-\frac{GMm}{2a}$
$\Rightarrow V_{min}=\frac{3}{2}\sqrt{\frac{5GM}{a}}$.
$\Rightarrow V_{min}=3$.