Question

The radius of the earth is R and acceleration due to gravity at its surface id g. The minimum speed with which a body must be thrown from the surface of the earth so as to reach a height of R/a is

• A. $\sqrt{gR}$
• B. $\sqrt{\frac{gR}{2}}$
• C. $\sqrt{\frac{gR}{5}}$
• D. $\sqrt{\frac{2gR}{5}}$

Answer 1

The correct option is D $\sqrt{\frac{2gR}{5}}$

$F=\frac{GMm}{x^2}$

$dW=Fdx,$

$dW=\frac{GMm}{x^2}dx$

$W=\underset{R}{\overset{5R/4}{\int }}\frac{GMm}{x^2}dx$

$=GMm\underset{R}{\overset{\frac{5R}{4}}{\int }}\frac{1}{x^2}dx$

$=GMm\left[\frac{4-5}{5R}\right]=\frac{GMm}{5R}$

$\frac{g{R}^{2}m}{5R}=\frac{1}{5}gRm$

$W=\frac{1}{2}m{v}^2$

$\frac{1}{2}m{v}^2=\frac{1}{5}gRm,v=\sqrt{\frac{2}{5}gR}$