The escape velocity from a planet is v_ e . A tunnel is dug along a...

The escape velocity from a planet is

v_{e}

. A tunnel is dug along a diameter of the planet and a small body is dropped into it at the surface. When the body reaches the centre of the planet, its speed will be

Options

A $\mathrm{V}_{\mathrm{e}}$
B $\mathrm{v}_{\mathrm{e}} / \sqrt{2}$
C $\mathrm{v}_{\mathrm{e}} / 2$
Dzero

Correct answer

B. $\mathrm{v}_{\mathrm{e}} / \sqrt{2}$

Step-by-step solution

\frac{1}{2} \mathrm{mv}_{\mathrme^{2}=\frac{\mathrm{GMm}}{\mathrm{R}} ; \mathrm{v}_{\mathrm{e}}=\sqrt{\frac{\mathrm{GMm}}{\mathrm{R}}}=\sqrt{2 \mathrm{gR}}

In tunnel body will perform SHM at centre

\mathrm{V}_{\mathrm\max}=\mathrm{A\omega}

(see chapter on SHM)

=\frac{\mathrm{R} 2 \pi}{2 \pi \sqrt{\mathrm{R} / \mathrm{g}}}=\sqrt{\mathrm{gR}}=\frac{\mathrm{v}_{\mathrm{e}}}{\sqrt{2}}

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