Suppose two planets (spherical in shape) of radii R and 2 R , but… - Vidyadip

MCQ

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 planets is $\sqrt{\frac{a}{7} \frac{G M}{R}}$ then the value of $'a'$ is $....$

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

✓
$4$
B
$8$
C
$16$
D
$64$

✓

Answer

Correct option: A.

$4$

a
Assume that at a distance x from the planet of mass M, the net gravitational field becomes zero.

$\frac{G M}{x^{2}}=\frac{G 9 M}{(8 R-x)^{2}}$

$8 R-x=3 x$

$x=2 R$

Apply conservation of energy and consider velocity at $P$ is zero.

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

$\frac{1}{2} v^{2}=\frac{2 G M}{7 R} \Rightarrow v=\sqrt{\frac{4}{7} \frac{G M}{R}}$

$a=4$

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