Question

A particle is projected vertically upwards from the surface of the earth (radius $R)$ with a speed equal to one-fourth of escape velocity. The maximum height attained by it from the surface of the earth is

• A. $\frac{R}{5}$
• B. $\frac{R}{10}$
• C. $\frac{R}{15}$
• D. $\frac{R}{20}$

Answer 1

The correct option is C $\frac{R}{15}$

Escape velocity, $v_e=\sqrt{2gR}$
$v=\frac{\sqrt{2gR}}{4}=\frac{\sqrt{gR}}{2\sqrt{2}}$
From conservation of energy
$\frac{1}{2}m{v}^2-\frac{GMm}{R}=0-\frac{GMm}{R+h}$

Using $g=\frac{GM}{R^2}$
$\frac{1}{2}m\frac{gR}{8}-\text{mg R}=-\frac{g{R}^{2}m}{R+h}$
$\frac{1}{16}=1-\frac{R}{R+h}$
$\frac{1}{16}=\frac{h}{R+h}$
$R+h=16h$
$R=15h$
$h=\frac{R}{15}$