Question

Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass $M$ and $9\, M$ respectively have a centre to centre separation $8\, R$ 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

Acceleration due to gravity will be zero at $P$ therefore,
$\frac{G M}{x^{2}}=\frac{G 9 M}{(8 R-x)^{2}}$
$8 R-x=3 x$
$x=2 R$
Apply conservation of energy and consider velocity at $P$ is zero.
$\frac{1}{2} m v^{2}-\frac{G M m}{R}-\frac{G 9 M m}{7 R}=0-\frac{G M m}{2 R}-\frac{G 9 M m}{6 R}$
$\therefore V=\sqrt{\frac{4 G M}{7} \frac{G R}{R}}$