A tunnel is dug along the diameter of the earth and a body is released from rest into the tunnel. Find the speed of the body at the center of the tunnel.

7.3 km/sec

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7.9 km/sec

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8.4 km/sec

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9.2 km/sec

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Solution

The correct option is B 7.9 km/sec
Initial gravitation potential,
$V_{i} = - \frac{G M}{R}$
Final gravitation potential,
$V_{f} = - \frac{3 G M}{2 R}$
Change in gravitational potential energy,
$Δ E = m Δ V$
$= m (V_{f} - V_{i}) = \frac{- 3 G M m}{2 R} + \frac{G M m}{R}$
$= - \frac{G M m}{2 R}$
Loss in potential energy = Gain in Kinetic energy
$\frac{G M m}{2 R} = \frac{1}{2} m v^{2}$
$v = \sqrt{\frac{G M}{R}} = \sqrt{\frac{2 G M}{2 R}} = \frac{11.2}{1.414}$
$v = 7.9 k m / s e c$

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