Question

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The escape velocities of planet A and B are same. But A and B are of unequal mass.
Reason R : The product of their mass and radius must be same. $M_1R_1 = M_2R_2$ In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both A and R are correct and R is the correct explanation of A
• B. Both A and R are correct but R is NOT the correct explanation of A
• C. A is correct but R is not correct
• D. A is not correct but R is correct

Hint:

Escape velocity depends only on the ratio $M/R$. Remember: $v_e \propto \sqrt{M/R}$. If mass increases by a factor, the radius must also increase by that same factor for $v_e$ to remain constant.

Answer 1

The Correct Option is C

Step 1: The formula for escape velocity is given by $v_e = \sqrt{\frac{2GM}{R}}$.
Step 2: For the escape velocities of two planets to be equal ($v_{e1} = v_{e2}$), we must have $\frac{M_1}{R_1} = \frac{M_2}{R_2}$.
Step 3: This implies the ratio of mass to radius must be the same, not the product. Thus, Assertion A is correct because unequal masses can have equal escape velocities if their radii are also different in the same proportion.
Step 4: Reason R states $M_1R_1 = M_2R_2$, which is mathematically inconsistent with the escape velocity formula.