Question 183 of 223

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 $\sqrt{\dfrac{a}{7}\dfrac{GM}{R}}$, then the value of $a$ is
[Given : The two planets are fixed in their position]

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Answer: 4

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