Question

Escape velocity for a projectile at earth's surface is $V_e$. A body is projected from earth's surface with a velocity $2V_e$. The velocity of the body when it is at infinite distance from the center of the earth is?

• A. $V_e$
  
• B. $2V_e$
  
• C. $\sqrt{2}{V}_e$
  
• D. $\sqrt{3}{V}_e$
  

Answer 1

The correct option is D $\sqrt{3}{V}_e$


Let $v$ be the speed of the body when it is at infinite distance from the center of the earth and $u$ be speed of projection of the body from the earth's surface.

According to law of conservation of mechanical energy,

$\frac{1}{2}m{u}^2-\frac{GMm}{R}=\frac{1}{2}m{v}^2-0$

$v^2=u^2-\frac{2GM}{R}$

$v^2=u^2-V{{}_e}^2(\because V_e=\sqrt{\frac{2GM}{R}})$

Given that $u=2V_e$
So, $v=\sqrt{(2V_e{)}^2-V_e^2}=\sqrt{3}{V}_e$