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If 'd' is the distance between the centers of the earth of mass $M_{1}$ and moon of mass $M_{2}$ , then the velocity with which a body should be projected from the mid point of the line joining the earth and the moon, so that it just escapes is :

A

$\frac{G ( M _{1} + M _{2} )}{d}$

B

$\frac{G ( M _{1} + M _{2} )}{2 d}$

C

$\frac{2 G ( M _{1} + M _{2} )}{d}$

D

$\frac{4 G ( M _{1} + M _{2} )}{d}$

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Solution

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Correct option is D)

Using law of conservation of energy

Kinetic energy minimum $= -$ potential

PE at mid point

$= \frac{- 2 G M m}{d} (M_{1} + M_{2})$

$= \frac{1}{2} m V_{e}^{2} = \frac{2 G m}{d} (M_{1} + M_{2})$

$\Rightarrow V_{e} = \frac{4 G ( M _{1} + M _{2} )}{d}$

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