Escape velocity for a projectile at earth's surface is $v_e$. A body is projected form earth's surface with velocity $2_{v_e}$. The velocity of the body when it is at infinite distance from the centre of the earth is

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

Let velocity of body when it is at infinity be v and also P.E is zero at infinity.

Apply conservation of energy at surface and infinity-

$\frac{1}{2}m(2v_e{)}^2-\frac{GMm}{R}=\frac{1}{2}m{v}^2$

$\frac{1}{2}m[2\sqrt{\frac{2GM}{R}}{]}^2-\frac{GMm}{R}=\frac{1}{2}m{v}^2$

$\frac{3GMm}{R}=\frac{1}{2}m{v}^2$

$v^2=3\frac{2GM}{R}=3v_e^2$

$⟹v=\sqrt{3}{v}_e$