Question

Assertion The escape velocity for a planet is $v_{e}=\sqrt{2 g R}$. If the radius of the planet is doubled, the escape velocity becomes twice (i.e. $v_{e}{ }^{\prime}=2 v_{e}$ )
Reason In the relation for escape velocity, $v_{e}=\sqrt{2 g R}$, the acceleration due to gravity $g$ is inversely proportional to radius of the planet.
Thus, $v_{e} \propto \frac{1}{\sqrt{R}}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
• B. Both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
• C. Assertion is correct but Reason is incorrect.
• D. Assertion is incorrect but Reason is correct.

Answer 1

Answer: (D)

Escape velocity, $v_{e}=\sqrt{2 g R}$,
where $g=\frac{G M}{R^{2}}$.
$\Rightarrow v_{e}=\sqrt{\frac{2 G M}{R}}$
i.e. $ v_{e} \propto \frac{1}{\sqrt{R}}$
So, if radius is doubled, i.e. $R^{\prime}=2 R$
$v_{e}^{\prime}=\sqrt{\frac{2 G M}{(2 R)}}=\frac{1}{\sqrt{2}} \sqrt{\frac{2 G M}{R}}=\frac{v_{e}}{\sqrt{2}}$
Therefore, Assertion is incorrect but Reason is correct.