Question

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

• A. $v_e$
• B. $2 \, v_e$
• C. $\sqrt{2} \, v_e$
• D. $\sqrt{3} \, v_e$

Answer 1

Answer: (D)

$\sqrt{3} \, v_e$

Answer 2

Let $v$ be the speed of the body when it is at infinite distance from the centre 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,
$\frac{1}{2}mu^{2} - \frac{GMm}{R} = \frac{1}{2}mv^{2} - 0$
or, $v^{2} =u^{2} - \frac{2GM}{R}$
$v^{2} = u^{2} -v^{2}_{e} \left( \because \:\: v_e = \sqrt{\frac{2GM}{R}} \right)$
$v = \sqrt{\left(2v_{e}\right)^{2 } - v_{e}^{2}}$
$= \sqrt{3} v_{e}$