Two planets have masses M and 16M and their radii are a and 2a ,...

Two planets have masses

M

and

16M

and their radii are a and

2a

, respectively. The separation between the centres of the planets is

10a

. A body of mass

m

is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is : Two planets have masses M and 16M and their radii are a and 2a , respectively. The separation between the centres of

Two planets have masses M and 16M and their radii are a and 2a , respectively. The separation between the centres of

Options

A $2\sqrt{\dfrac{GM}{a}}$
B $4\sqrt{\dfrac{GM}{a}}$
C $\sqrt{\dfrac{{GM}^{2}}{ma}}$
D $\dfrac{3}{2}\sqrt{\dfrac{5GM}{a}}$

Correct answer

D. $\dfrac{3}{2}\sqrt{\dfrac{5GM}{a}}$

Step-by-step solution

Two planets have masses M and 16M and their radii are a and 2a , respectively. The separation between the centres of

$\dfrac{1}{2}{mu}^{2} + \left[\dfrac{- G \times 10Mm}{2a} - \dfrac{GMm}{8a}\right] = 0 - \dfrac{G \times 16Mm}{8a} - \dfrac{GM \times m}{2a}$

$⇒ u = \sqrt{\dfrac{45GM}{4a}}$

$u = \dfrac{3}{2}\sqrt{\dfrac{5GM}{a}}$

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