Question

Assertion A: The escape velocities of planet are same. But $A$ and $B$ are of unequal masses.

Reason R: The product of their masses and radii must be same.$M_1{R}_1=M_2{R}_2$

In the light of the above statements, choose the most appropriate answer from the options given below:

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

Answer 1

The correct option is B

A is correct but R is not correct

The explanation for the Assertion statement:

Escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is given by $v_e=\sqrt{\frac{GM}{R}}$ , Where$G$ is gravitational constant and $M$ is the mass of body escaping and $R$ is distance from centre of mass.

Given escape velocities are same, we have

$$\begin{gathered} v_e=\sqrt{\frac{G{M}_1}{R_1}}=\sqrt{\frac{G{M}_2}{R_2}} \\ \Rightarrow \frac{M_1}{R_1}=\frac{M_2}{R_2} \\ \Rightarrow M_1{R}_2=M_2{R}_1 \end{gathered}$$

Where $M_1$ is mass and $R_1$ is distance of first planet and $M_2$ is mass and $R_2$ is distance of second planet. Since planets in general have different radius the masses will also be different

Explanation for Reason statement:

Also, we have derived in assertion that $M_1{R}_2=M_2{R}_1$, but reason statement states that the product of their masses and radii must be same$M_1{R}_1=M_2{R}_2$.

Therefore, the reason is not correct for the given assertion.

Hence, option (B) is the correct answer.