The masses and radii of the earth and the Moon are $M_{1}$ , $R_{1}$ and $M_{2}$ , $R_{2}$ , respectively. Their centres are at distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway the two centres so as to escape to infinity is?

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

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\sqrt{\frac{4 G (M_{1} + M_{2})}{d}}

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\sqrt{\frac{4 G M_{1} M_{2}}{d}}

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

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Solution

The correct option is B $\sqrt{\frac{4 G (M_{1} + M_{2})}{d}}$
Potential energy of mass $m$ when it is midway between masses $M_{1}$ and $M_{2}$ is
$U = - \frac{G M_{1} m}{d / 2} - \frac{G M_{2} m}{d / 2}$
$= \frac{2 G m}{d} (M_{1} + M_{2})$
According to law of conservation of energy,
$\frac{1}{2} m v_{e}^{2} = \frac{2 G m}{d} (M_{1} + M_{2})$
Therefore escape velocity
$v_{e} = \sqrt{\frac{4 G (M_{1} + M_{2})}{d}}$

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The masses and radii of the earth and 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 the two centres so as to escape to infinity is given by