A) Increased by 2%
B) Decreased by 4%
C) Increased by 6%
D) Decreased by 8%
Correct Answer: A
Solution :
[a] Escape velocity \[=v=\sqrt{\frac{2GM}{R}}\] \[\Rightarrow {{v}^{2}}=\frac{2GM}{R}\] ....(i) \[{{v}^{2}}=\left( 2GM \right){{R}^{-1}}\] Differentiating both sides, we get, \[2v\frac{dv}{dR}=-\frac{2GM}{{{R}^{2}}}\Rightarrow v\frac{dv}{dR}=\frac{GM}{{{R}^{2}}}\] ....(ii) Dividing (ii) by (i), \[\frac{1}{v}\frac{dv}{dR}=-\frac{1}{2R}\] \[\Rightarrow \left| \frac{dv}{v} \right|\times 100=\frac{1}{2}\times 4%=2%\] \[\therefore \]If the radius decreases by 4%, escape velocity will increase by 2%.
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