A) \[22.4\text{ }km{{s}^{-1}}\]
B) \[31.7\text{ }km\text{ }{{s}^{-1}}\]
C) \[33.6km{{s}^{-1}}\]
D) none of these
Correct Answer: B
Solution :
Key Idea: 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 \[v=2\sqrt{2}\times 11.2=31.7\,km{{s}^{-1}}\]
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