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

  $\frac{R}{3}$
• Correct answer
  C

  $\frac{2 R}{3}$
• D

  $R$

Solution

Maximum height attained by a projectile
Velocity of body $=$ half the escape velocity $v=\frac{v_e}{2}$ $\begin{aligned} \text{or} \quad v=\frac{\sqrt{2 g R}}{2} \Rightarrow v^2=\frac{2 g R}{4} \\ \text{or} \quad v^2=\left(\frac{g R}{2}\right) \end{aligned}$ Now, putting value of $v^2$ in Eq. (i), we get $\begin{aligned} h & =\frac{\frac{g R}{2} \cdot R}{2 g R-\frac{g R}{2}}=\frac{g R^2 / 2}{3 g R / 2} \\ \text { or } \quad h & =\frac{R}{3} \end{aligned}$

Hence, the correct answer is option: 'C'.