Question

A body falls freely towards the earth from a height $2 R$, above the surface of the earth, where initially it was at rest. If $R$ is the radius of the earth then its velocity on reaching the surface of the earth is:

• A. $\sqrt{\frac{4}{3} g R}$
• B. $\sqrt{\frac{2}{3} g R}$
• C. $\frac{4}{3} g R$
• D. $2 g R$

Answer 1

Answer: (A)

Initial energy of the body
$=-\frac{G M m}{(R+2 R)}$
Final energy of the body $=-\frac{G M m}{R}+\frac{1}{2} m v^{2}$
where $m$ is the mass of the body and $M$ is the mass of the earth. By conservation of energy
$-\frac{G M m}{3 R}=-\frac{G M m}{R}+\frac{1}{2} m v^{2}$
$\Rightarrow \frac{1}{2} m v^{2}=\frac{2 G M m}{3 R}$
$\Rightarrow v^{2}=\frac{4 G M}{3 R}$
$\Rightarrow v=\sqrt{\frac{4 G M}{3 R}}$
$\Rightarrow v=\sqrt{\frac{4 G M R}{3 R^{2}}}=\sqrt{\frac{4}{3} g R}$