Question

A bullet is fired vertically upward with velocity $v$, maximum height attained by the bullet is $9R$ (from surface of earth). Find value of $v$.
Given: Mass of Earth $=M$, Radius of Earth $=R$

• A. $\sqrt{\frac{3}{5}\frac{GM}{R}}$
• B. $\sqrt{\frac{2}{5}\frac{GM}{R}}$
• C. $\sqrt{\frac{9}{5}\frac{GM}{R}}$
• D. $\sqrt{\frac{8}{5}\frac{GM}{R}}$

Answer 1

The correct option is C $\sqrt{\frac{9}{5}\frac{GM}{R}}$

In this question $(h=9R)$

So we have to conserve energy

$(P.E+K.E{)}_{\text{Initial}}=(P.E+K.E{)}_{\text{Final}}$

$\frac{-GMm}{R}+\frac{1}{2}m{v}^2=\frac{-GMm}{R+h}+0$

$\frac{-GM}{R}+\frac{1}{2}{v}^2=\frac{-GM}{R+h}$

$\frac{v^2}{2}=\left(\frac{GM}{R}\right)-\frac{GM}{R+h}$
$\frac{v^2}{2}=\frac{GM}{R}-\frac{GM}{R+9R}$ $h=9R$

$\frac{v^2}{2}=\frac{GM}{R}-\frac{GM}{10R}$

$\frac{v^2}{2}=\frac{GM}{R}\left(\frac{9}{10}\right)$
$\Rightarrow v=\sqrt{\frac{9}{5}\frac{GM}{R}}$