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

Answer: 99

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