Question

The escape velocity of a projectile on the earth's surface in $ 11.2\text{ }km{{s}^{-1}} $ . A body is projected out with thrice this speed. The speed of the body far away form the earth will be:

• A. $ 22.4\text{ }km{{s}^{-1}} $
• B. $ 31.7\text{ }km\text{ }{{s}^{-1}} $
• C. $ 33.6 \,km{{s}^{-1}} $
• D. none of these

Answer 1

Answer: (B)

Conservation of energy holds in the universe.
By law of conservation of energy,
energy at surface of earth = energy at infinity i.e.,
$ {{(U+K)}_{surface}}={{(U+K)}_{\inf inity}} $ or $ -\frac{GMm}{R}+\frac{1}{2}m{{(3{{v}_{e}})}^{2}}=0+\frac{1}{2}m{{v}^{2}} $
or $ -\frac{GM}{R}+\frac{9v_{e}^{2}}{2}=\frac{1}{2}{{v}^{2}} $
But escape velocity, $ {{v}_{e}}=\sqrt{\frac{2GM}{R}} $
$ \therefore $ $ -\frac{v_{e}^{2}}{2}+\frac{9v_{e}^{2}}{2}=\frac{1}{2}{{v}^{2}} $
Or $ {{v}^{2}}=8v_{e}^{2} $
Or $ v=2\sqrt{2}\times 11.2$
$=31.7\,km{{s}^{-1}} $