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Offline Centres

Assuming that a tunnel is dug across the earth passing through its centre. Find the time, a particle takes to reach the centre of the earth, if it is projected into the tunnel from the surface of earth with speed needed for it to escape the gravitational field of earth

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$[\sin^{- 1} (\frac{1}{\sqrt{3}}) \sqrt{\frac{R}{g}}]$

$[\sin^{- 1} (\frac{1}{\sqrt{3}}) \sqrt{\frac{3 R}{g}}]$

$[\sin^{- 1} (\frac{1}{\sqrt{3}}) \sqrt{\frac{2 R}{g}}]$

$[\sin^{- 1} (\frac{1}{\sqrt{3}})] \sqrt{\frac{R}{2 g}}$

answer is C.

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Detailed Solution

$V = \sqrt{\frac{g}{R} (3 R^{2} - r^{2})} - \frac{d r}{d t} = \sqrt{3 g R} \sqrt{1 - \frac{r^{2}}{3 R^{2}}} \int_{R}^{o} \frac{d r}{\sqrt{1 - \frac{r^{2}}{3 R^{2}}}} = \int_{0}^{t o} - \sqrt{3 g R} d t$

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