Question

A body is projected vertically upwards from the surface of the 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 the earth is

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

Answer 1

Answer: (B)

$\frac{R}{3}$

Answer 2

Here, maximum height attained by a projectile $h = \frac{v^{2}R}{2gR -v^{2}} v= \frac{v_{e}}{2}$ ....(i) Velocity of body = half the escape velocity i.e., $v = \frac{v_e}{2}$ or $v= \frac{\sqrt{2gR}}{2}$ $\Rightarrow v^{2} = \frac{2gR}{4}$ $\Rightarrow v^{2} = \left(\frac{gR}{2}\right)$ Now, putting value of $v^2$ in E (i), we get Height, $h = \frac{\frac{2R}{2} .R}{2gR - \frac{gR}{2}}$ $= \frac{\frac{gR^{2}}{2}}{\frac{3gR}{2}}$ or $h = \frac{R}{3}$