Question

There is a crater of depth $\frac{R}{100}$ on the surface of the moon (radius R). A Projectile is fired vertically upward from the crater with velocity, which is equal to the escape velocity V from the surface of the moon. Find the maximum height attained by the projectile.

• A. 90R
• B. 95R
• C. 99.5 R
• D. 50R

Answer 1

Answer: (C)

Speed of particle at $A, v_{A}=$ escape velocity on the surface of earth $\sqrt{\frac{2 G M}{R}}$ at highest point $B, v_{B}=0$
Applying conservation of mechanical energy decrease in kinetic energy $=$ increases in gravitational potential energy
$=\frac{1}{2} m v_{A}^{2}=U_{B}-U_{A}=m\left(V_{B}-V_{A}\right)$ $\frac{V_{A}^{2}}{2}=V_{B}-V_{A}$ $\frac{G M}{R}=-\frac{G M}{R+h}-\left[\frac{-G M}{R^{3}}\left[\left(1.5 R^{2}\right)-0.5\left(R-\frac{R}{100}\right)^{2}\right]\right]$ $\frac{1}{R}=-1(R+h)+\frac{3}{2 R}-\left(\frac{1}{2}\right)\left(\frac{99}{100}\right)^{2} \frac{.1}{R}$
Solving this equations, we get $h=99.5 R$