Question

Distance between the centre of two stars is $10a$. The 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 larger star towards the smaller stars. What should be its minimum initial speed to reach the surface of the smaller star? Obtain the expression in terms of $G,M$ and $a$.

Answer 1

The object needs enough energy to make it to the position where the gravitational forces from the two stars are equal. Beyond that, it will accelerate to the smaller stars.

Let the distance of the equilibrium point from the large star be $x$

$\frac{GMm}{{(10a-x)}^2}=\frac{G\left(16M\right)m}{x^2}\Rightarrow x=8a$

Least $K.E$ with which the body starts to accelerate towards the small star $=\frac{45GMm}{8a}=\frac{1}{2}{mv}^2$

$\therefore v=\frac{3}{2}\sqrt{\frac{5GM}{a}}m{s}^{-1}$