Question

An object is projected vertically upward from the surface of the earth of mass M with a velocity such that the maximum height reached is eight times the radius R of the earth. The initial speed of projection is

• A. $\frac{4}{3}\sqrt{\frac{Gm}{R}}$
• B. $\frac{5}{4}\sqrt{\frac{Gm}{R}}$
• C. $\sqrt{\frac{3Gm}{R}}$
• D. None of these

Answer 1

The correct option is D None of these

Let, the initial velocity be $V$ and final velocity at maximum height is zero.
Gravitational force is conservative force and so total energy of earth and body remains constant.
Initial gravitational potential energy $P_i=\frac{-GMm}{R}$
Final gravitational potential energy $P_f=\frac{-GMm}{8R}$
Initial kinetic energy $K_i=\frac{m{V}^2}{2}$
Final kinetic energy $K_f=0$
Therefore $P_i+K_i=P_f+K_f$
$\frac{-GMm}{R}+\frac{m{V}^2}{2}=\frac{-GMm}{8R}$
Solving for $V,$ we get $V=\sqrt{\frac{7GM}{4R}}$