Question

A body is projected vertically upwards from the surface of the earth with a velocity sufficient to carry it to infinity. If the radius of the earth is R, the velocity of the body at a height h is

• A. $\sqrt{gR}$
• B. $\sqrt{\frac{2g{R}^2}{3(R+h)}}$
• C. $\sqrt{\frac{g{R}^2}{R+h}}$
• D. $\sqrt{\frac{2g{R}^2}{R+h}}$

Answer 1

The correct option is D $\sqrt{\frac{2g{R}^2}{R+h}}$

From conservation of energy:
$-\frac{GMm}{R}+\frac{m}{2}(\sqrt{\frac{2GM}{R}}{)}^2=\frac{GMm}{R+h}+\frac{1}{2}m{v}^2\frac{GM}{R+h}+\frac{GM}{R}-\frac{GM}{R}=\frac{1}{2}{v}^2\Rightarrow v=\sqrt{\frac{2GM}{R+h}}=\sqrt{\frac{2g{R}^2}{R+h}}$