A body starts from rest from a point distance R_0 from centre of earth. velocity acquired by body wh

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7.Gravitation

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A body starts from rest from a point distance $R_0$ from the centre of the earth. The velocity acquired by the body when it reaches the surface of the earth will be ($R$ represents radius of the earth).

A

$2\,GM\,\left( {\frac{1}{R} - \frac{1}{{{R_0}}}} \right)$

B

$\sqrt {2\,GM\,\left( {\frac{1}{{{R_0}}} - \frac{1}{R}} \right)} $

C

$GM\,\left( {\frac{1}{R} - \frac{1}{{{R_0}}}} \right)$

D

$2\,GM\,\sqrt {\left( {\frac{1}{R} - \frac{1}{{{R_0}}}} \right)} $

(AIIMS-2014)

Solution

$P.E = \int_{{R_0}}^R {\frac{{GMm}}{{{r^2}}}dr = – GMm\left[ {\frac{1}{R} – \frac{1}{{{R_0}}}} \right]} $

The $K.E.$ acuired by the body at the

$surface = \frac{1}{2}m\,{v^2}$

$\therefore \frac{1}{2}m{v^2} = – GMm\left[ {\frac{1}{R} – \frac{1}{{{R_0}}}} \right]$

$v = \sqrt {2GM\left( {\frac{1}{{{R_0}}} – \frac{1}{R}} \right)} $

Standard 11

Physics

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