Question

A body is projected vertically upwards from the surface of earth with a velocity equal to half the escape velocity. If $R$ is the radius of the earth, maximum height attained by the body from the surface of earth is

• A. $\frac{R}{6}$
• B. $\frac{R}{3}$
• C. $\frac{2R}{3}$
• D. $\frac{R}{4}$

Answer 1

The correct option is B $\frac{R}{3}$

Let $H$ be the maximum height.
By conservation of energy,
$$\begin{array}{cc}T.E_i=T.E_f& \\ K.E_i+P.E_i=P.E_f& \\ \frac{1}{2}m\frac{V{e}^2}{4}+\left(\frac{-GMm}{R}\right)=\frac{-GMm}{R+H}& \\ \frac{m}{8}\frac{2GM}{R}-\frac{GMm}{R}=\frac{-GMm}{R+H}(\because V_e^2=\frac{2GM}{R})& \\ \frac{-3GMm}{4R}=\frac{-GMm}{R+H}& \\ R+H=\frac{4R}{3}& \\ \therefore H=\frac{R}{3}& \end{array}$$