The masses and radii of the earth and moon are $M_{1}, R_{1} a n d M_{2}, R_{2}$ respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is

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

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

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

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

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Solution

The correct option is A
$2 \sqrt{\frac{G}{d} (M_{1} + M_{2})}$

Gravitational potential at mid point
$V = \frac{- G M_{1}}{\frac{d}{2}} + \frac{- G M_{2}}{\frac{d}{2}}$
Now, $PE=m×V=frac−2Gmd(M1+M2)$
[m = mass of particle]
So, for projecting particle from mid point to infinity
KE=|PE|
$\Rightarrow \frac{1}{2} m v^{2} = \frac{2 G M}{d} (M_{1} + M_{2}) \Rightarrow v = 2 \sqrt{\frac{G (M_{1} + M_{2})}{d}}$

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The masses and radii of the earth and moon are $M_{1}, R_{1} a n d M_{2}, R_{2}$ respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is