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The masses and radii of the earth and the Moon are $M_{1}$ , $R_{1}$ and $M_{2}, R_{2}$ respectively. Their centres are a distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway between teh two centres so as to escape to infinity is $n \frac{G ( M _{1} + M _{2} )}{d}$ . The value of $n$ is:

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Correct option is A)

From the law of energy conservation,

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

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

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

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

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