Gravitation Question 350

Question: The radius of the earth is reduced by 4%. The mass of the earth remains unchanged. What will be the change in escape velocity?

Options:

A) Increased by 2%

B) Decreased by 4%

C) Increased by 6%

D) Decreased by 8%

Show Answer

Answer:

Correct Answer: A

Solution:

  • 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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