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Masses and radii of earth and moon are $M_{1}, M_{2}$ and $R_{1}, R_{2}$ respectively. The distance between theircentre is 'd'. The minimum velocity given to mass 'M' from the mid point of line joining their centre so that it will escape.

A

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

B

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

C

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

D

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

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Solution

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Correct option is A)

From conservation of energy, since the velocity will be 0 at infinity,

$\frac{1}{2} m v^{2} - \frac{G m M _{1}}{d / 2} - \frac{G m M _{2}}{d / 2} = 0$

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

Answer is option A.

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