Question

Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass $M$ and $9M$ respectively have a centre to centre separation $8R$ as shown in the figure. A satellite of mass ${}^{\prime }{m}^{\prime }$ is projected from the surface of the planet of mass ${}^{\prime }{M}^{\prime }$ directly towards the centre of the second planet. The minimum speed ${}^{\prime }{v}^{\prime }$ required for the satellite to reach the surface of the second planet is $\sqrt{\frac{a}{7}\frac{GM}{R}}$ then the value of ${}^{\prime }{a}^{\prime }$ is _____.

[Given : The two planets are fixed in their position]

Answer 1

Let net force experienced by the mass $m$ at point $\text{P}$ be zero.

$\therefore \frac{GMm}{x^2}=\frac{G\left(9M\right)m}{(8R-x{)}^2}$

$8R-x=3x$

$\therefore x=2R$

So, we have to give at least enough velocity to reach point $\text{P}$, because after that bigger planet will attract mass $m$ towards itself automatically.

Now, applying conservation of energy between projection and point $\text{P}$ and consider velocity at $\text{P}$ is zero.

$\frac{1}{2}m{v}^2-\frac{GMm}{R}-\frac{G\left(9M\right)m}{7R}=0-\frac{GMm}{2R}-\frac{G\left(9M\right)m}{6R}$

$\therefore v=\sqrt{\frac{4}{7}\frac{GM}{R}}$

Hence, correct answer $:4$