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  4. 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

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Q. 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

Gravitation

Solution:

$(K E+P E)_{1}=(K E+P E)_{2}$
$\frac{1}{2} m v^{2}-\frac{3}{2} \frac{G M m}{R}$
$=-\frac{G M m}{R}$
$v=\sqrt{\frac{G M}{R}}=\sqrt{\frac{g R^{2}}{R}}$
$=\sqrt{g R}$

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