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

To address the given statements about escape velocity, we need to understand the concept of escape velocity in physics. Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a planet without further propulsion.

The formula for escape velocity v_e is given by:

v_e = \sqrt{\frac{2GM}{R}}

Where:

• G is the universal gravitational constant.
• M is the mass of the planet.
• R is the radius of the planet.

Let's analyze each statement:

• 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."

The ratio given is \frac{M}{R}. If this ratio increases, it implies that the escape velocity v_e, which is directly proportional to the square root of this ratio, will also increase. Thus, Statement I is correct.

• Statement II: "Escape velocity is independent of the radius of the planet."

According to the formula, escape velocity v_e is dependent on both the mass M and the radius R of the planet. Specifically, it is inversely proportional to the square root of the radius of the planet. Therefore, Statement II is incorrect.

In conclusion, based on the escape velocity formula:

• Statement I is correct but Statement II is incorrect.

Thus, the correct answer is: Statement I is correct but Statement II is incorrect.