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A body starts from rest from a point distant $r_{0}$ from the centre of the earth. It reaches the surface of the earth whose radius is $R$ . the velocity acquired by the body is

A

2 G M \sqrt{\frac{1}{R} - \frac{1}{r_{0}}}

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B

\sqrt{2 G M \frac{1}{R} - \frac{1}{r_{0}}}

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C

G M \sqrt{\frac{1}{R} - \frac{1}{r_{0}}}

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D

\sqrt{G M \frac{1}{R} - \frac{1}{r_{0}}}

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Solution

The correct option is B $\sqrt{2 G M \frac{1}{R} - \frac{1}{r_{0}}}$
From conservation of energy:
Gain in kinetic energy $=$ Loss in gravitational potential energy
$⟹ \frac{1}{2} m v^{2} = (- \frac{G M m}{r_{0}}) - (- \frac{G M m}{R})$

$⟹ \frac{1}{2} m v^{2} = G M m (\frac{1}{R} - \frac{1}{r_{0}})$

$⟹ v = \sqrt{2 G M (\frac{1}{R} - \frac{1}{r_{0}})}$

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