Suppose two planets (spherical in shape) of radii R and 2 R , 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

Question

Suppose two planets (spherical in shape) of radii $R$ and $2 R,$ 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 $\sqrt{\frac{a}{7} \frac{G M}{R}}$ , then the value of $a$ is
[Given : The two planets are fixed in their position]

TestHub · Accepted Answer

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.

$\frac{1}{2} m v^{2} - \frac{G M m}{R} - \frac{G 9 M m}{7 R} = 0 - \frac{G M m}{2 R} - \frac{G 9 M m}{6 R}$
$∴ V = \sqrt{\frac{4}{7} \frac{G M}{R}}$

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