What would be the maximum height reached by the body if the body is projected vertically from the surface of the earth of radius R with a velocity equal to half of the escape velocity?
Answer & explanation
Correct answer: option 4
Conservation of Energy : \(E_i= E_f\)
... where E : total energy ; Ei = total energy initial ; Ef = total energy final
\(\Rightarrow T_i + U_i = T_f + U_f\)
... where T : kinetic energy ; Ti = kinetic energy initial ; Tf = kinetic energy final, and
... where U : potential energy ; Ui = potential energy initial ; Uf = potential energy final
\(\frac{1}{2}m \frac{v^2_e}{4} + [-\frac{GMm}{R}] = -\frac{GMm}{R+H}\)
\(\frac{2GMm}{8R} - \frac{GMm}{R} = -\frac{GMm}{R+H}\) ... (\(because v^2_e = \frac{2GM}{R}\))
\(-\frac{3GMm}{4R} = -\frac{GMm}{R+H}\)
\(R+H = \frac{4}{3}R\)
\(\Rightarrow H = \frac{R}{3}\)