First slide

Gravitational potential and potential energy

Question

Assuming earth as a uniform sphere of radius R, if we project a body along the smooth diametrical chute from the centre of earth with a speed v such that it will just reach the earth's surface then v is equal to

Moderate

A

$\sqrt{gR}$
B

$\sqrt{2 gR}$
C

$\sqrt{\frac{gR}{2}}$
D

none of these

Solution

$\begin{array}{l} (KE + PE)_{1} = (KE + PE)_{2} \\ \frac{1}{2} {mv}^{2} - \frac{3}{2} \frac{GMm}{R} = - \frac{GMm}{R} \\ v = \sqrt{\frac{GM}{R}} = \sqrt{\frac{{gR}^{2}}{R}} = \sqrt{gR} \end{array}$

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