Question

There is a crater of depth $\frac{R}{100}$ on the surface of the moon, where $R$ is the radius of the moon. A ball is projected vertically upwards from the crater with a velocity which is equal to the escape velocity from the surface of the moon. If the maximum height attained by the particle is close to $nR$ , where $n$ is an integer, then find the value of $n$ ?

Answer 1

Speed of particle at $A$,
$v_A=v=\sqrt{\frac{2 G M}{R}}$
At point $B , v _{ B }=0$
By conservation of energy,
decrease in $KE =$ increase in GPE
$\Rightarrow \frac{1}{2} Mv _{\Lambda}^2= U _{ B }- U _{ A }$
$\therefore \frac{1}{2} M \cdot \frac{2 GM }{ R }= MV _{ B }- V _{ A }$
$\therefore \frac{G M}{R}=-\frac{G M}{R+h}-\frac{-G M}{2 R} 3-\frac{R-\frac{R}{100}^2}{R^2}$
$\therefore \frac{1}{ R }=-\frac{1}{ R + h }+\frac{3}{2 R }-\frac{1}{2} \frac{99}{100}^2 \cdot \frac{1}{ R }$
$\therefore \frac{1}{R}=-\frac{1}{R+h}+\frac{3}{2 R}-\frac{1}{2} \frac{0.98}{R} $
$\therefore \frac{1}{R+h}=\frac{1}{2 R}[1-0.98]$
$\therefore 2 R=0.02(R+h)$
$\therefore h =\frac{1.98}{0.02}$
$\therefore h=99 R$