Question

Given below are two statements :
Statement I: For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. 
Statement II: Escape velocity is independent of the radius of the planet. 
In the light of above statements, choose the most appropriate answer from the option given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but statement II is incorrect
• D. Statement I is incorrect but statement II is correct

Answer 1

The Correct Option is C

The formula for escape velocity is: \[ v_e = \sqrt{\frac{2GM}{R}}, \] where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet. From the formula, it is clear that \( v_e \propto \sqrt{\frac{M}{R}} \). - As the ratio \( \frac{M}{R} \) increases, the escape velocity \( v_e \) increases. Hence, Statement I is correct. 

- However, \( v_e \) depends on \( R \) as seen from the formula, so escape velocity is not independent of the radius of the planet. 

Hence, Statement II is incorrect. Thus, the correct answer is \( \boxed{(3)} \).