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,
$\frac{1}{2}m{v}^2-\frac{GMm}{R}=-\frac{GMm}{(R+h)}$
or $v^2=\frac{2GM}{R}\left(\frac{h}{R+h}\right)$
Given, $v=\frac{v_e}{2}=\frac{1}{2}\sqrt{\frac{2GM}{R}}$
$\left(v_e={\left(\frac{2GM}{R}\right)}^{1/2}\right)$
$\therefore \frac{1}{4}\frac{2GM}{R}=\frac{2GM}{R}\left(\frac{h}{R+h}\right)$
or $R+h=4h$ or $\frac{h}{R}=\frac{1}{3}$