Question

Distance between centres of two stars is $10a$. The masses of these stars are M and $16$M and their radii are a and $2$a, respectively. A body is fired straight from the surface of the larger star towards the smaller star. What should be its minimum initial speed to reach the surface of the smaller star?

Answer 1

Let P be the point on the line joining the centres of the two planets such that the net field at that point is zero.
Then, $\frac{GM}{r^2}-\frac{G16M}{(10a-r{)}^2}=0$
$\Rightarrow (10a-r{)}^2=16r^2$
$\Rightarrow 10a-r=4r$
$\Rightarrow r=2a$
Potential at point P,
$v_p=\frac{-GM}{r}-\frac{G16M}{(10a-r)}=\frac{-GM}{2a}-\frac{2GM}{a}$
$=\frac{-5GM}{2a}$
Now if the particle projected from the larger planet has enough energy to cross this point, it will reach the smaller planet.
For this, the KE imparted to the body must be just enough to raise its total mechanical energy to a value which is equal to PE at point P, i.e.,
$\frac{1}{2}m{v}^2-\frac{G\left(16M\right)m}{2a}-\frac{GMm}{8a}=m{v}_p$
or $\frac{v^2}{2}-\frac{8GM}{a}-\frac{GM}{8a}=-\frac{5GMm}{2a}$
or $v^2=\frac{45GM}{4a}$
or $v_{min}=\frac{3}{2}\sqrt{\frac{5GM}{a}}$.