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 $\sqrt{\frac{2GM}{R}}$ at highest point $B,v_B=0$
Applying conservation of mechanical energy decrease in kinetic energy $=$ increases in gravitational potential energy
$=\frac{1}{2}m{v}_A^2=U_B-U_A=m\left(V_B-V_A\right)$ $\frac{V_A^2}{2}=V_B-V_A$ $\frac{GM}{R}=-\frac{GM}{R+h}-\left[\frac{-GM}{R^3}\left[\left(1.5R^2\right)-0.5{\left(R-\frac{R}{100}\right)}^2\right]\right]$ $\frac{1}{R}=-1(R+h)+\frac{3}{2R}-\left(\frac{1}{2}\right){\left(\frac{99}{100}\right)}^2\frac{.1}{R}$
Solving this equations, we get $h=99.5R$