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 ' $m$ ' is projected from the surface of the planet of mass '$M$' directly towards the centre of the second planet. The minimum speed '$v$' required for the satellite to reach the surface of the second planet is $\sqrt{\frac{a}{7}\frac{GM}{R}}$ then the value of '$a$' is_____
[Given: The two planets are fixed in their position]

Answer 1

Answer: 4

Acceleration due to gravity will be zero at $P$ therefore,
$\frac{GM}{x^2}=\frac{G9M}{(8R-x{)}^2}$
$8R-x=3x$
$x=2R$
Apply conservation of energy and consider velocity at $P$ is zero.
$\frac{1}{2}m{v}^2-\frac{GMm}{R}-\frac{G9Mm}{7R}=0-\frac{GMm}{2R}-\frac{G9Mm}{6R}$
$\therefore V=\sqrt{\frac{4GM}{7}\frac{GR}{R}}$