Question

A spaceship is launched into a circular orbit of radius R close to the surface of earth. The additional velocity to be imparted to the spaceship in the orbit to overcome the earths gravitational pull is : (g = acceleration due to gravity)

• A. 1.4147 Rg
• B. 1.414 $ \sqrt{Rg} $
• C. 0.414 Rg
• D. 0.414 $ \sqrt{Rg} $

Answer 1

The Correct Option is D

Let a spaceship is launched in a circular orbit of orbital velocity $ {{v}_{o}}. $ That spaceship should have escape velocity $ {{v}_{es}} $ to overcome the earths gravitational pull. Now suppose v is the additional velocity to be imparted to the spaceship. Then according to above statement $ {{v}_{0}}+v={{v}_{es}} $ $ \left[ \begin{align} & \because \,{{v}_{0}}=\sqrt{Rg} \\ & {{v}_{es}}=\sqrt{2}Rg \\ \end{align} \right] $ or $ v={{v}_{es}}-{{v}_{o}} $ $ v=\sqrt{2}{{v}_{o}}-{{v}_{o}}={{v}_{o}}(\sqrt{2}-1)={{v}_{o}}(1.414-1) $ $ =0.414\sqrt{Rg} $