First slide

Gravitaional potential energy

Question

A body starts from rest from a point distance R_o from the centre of the earth. The velocity acquired by the body when it reaches the surface oft he earth will be (fire presents radius of the earth)

Moderate

A

$2 GM (\frac{1}{R} - \frac{1}{R_{0}})$
B

$\sqrt{2 GM (\frac{1}{R} - \frac{1}{R_{0}})}$
C

$GM (\frac{1}{R} - \frac{1}{R_{0}})$
D

$2 GM \sqrt{(\frac{1}{R} - \frac{1}{R_{0}})}$

Solution

$P . E . = \int_{R}^{R_{0}} \frac{GMm}{r^{2}} dr = GMm [\frac{1}{R} - \frac{1}{R_{0}}]$
The K.E. acquired by the body at the surface $= \frac{1}{2} {mv}^{2}$
$\begin{array}{l} ∴ \frac{1}{2} {mv}^{2} = GMm [\frac{1}{R} - \frac{1}{R_{0}}] \\ v = \sqrt{2 GM [\frac{1}{R} - \frac{1}{R_{0}}]} \end{array}$

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