First slide

Gravitaional potential energy

Question

A particle is dropped from height 3R from surface of earth. Then, the velocity of particle at height R is (given R is radius of earth, g is acceleration due to gravity on the earth’s surface) :

Moderate

A

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

$\frac{2}{3} \sqrt{gR}$
C

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

Path of particle is parabolic

Solution

Principle of conservation of energy is to be applied. Since acceleration is variable
$\begin{array}{l} - \frac{GmM}{4 R} = - \frac{GmM}{2 R} + \frac{1}{2} {mv}^{2} \\ v = \sqrt{\frac{GM}{2 R}} = \sqrt{\frac{gR}{2}} \end{array}$

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