Question

A ball is thrown vertically upwards with a velocity equal to half the escape velocity from the surface of the earth. The ball rises to a height $h$ above the surface of the earth. If the radius of the earth is $R$, then the ratio $\frac{h}{R}$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. 2
• D. 3

Answer 1

Answer: (B)

Let the body be thrown vertically upwards with velocity v. Then by the law of conservation of mechanical energy,
$\frac{1}{2} m v^{2}-\frac{G M m}{R}=-\frac{G M m}{(R+h)}$
or $v^{2}=\frac{2 G M}{R}\left(\frac{h}{R+h}\right)$
Given, $v=\frac{v_{e}}{2}=\frac{1}{2} \sqrt{\frac{2 G M}{R}}$
$\left(v_{e}=\left(\frac{2 G M}{R}\right)^{1 / 2}\right)$
$\therefore \frac{1}{4} \frac{2 G M}{R}=\frac{2 G M}{R}\left(\frac{h}{R+h}\right)$
or $R+h=4 h$ or $\frac{h}{R}=\frac{1}{3}$