Question

Assertion The escape velocity for a planet is $v_e=\sqrt{2gR}$. If the radius of the planet is doubled, the escape velocity becomes twice (i.e. $v_e{}^{\prime }=2v_e$ )
Reason In the relation for escape velocity, $v_e=\sqrt{2gR}$, 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, $v_e=\sqrt{2gR}$,
where $g=\frac{GM}{R^2}$.
$\Rightarrow v_e=\sqrt{\frac{2GM}{R}}$
i.e. $v_e\propto \frac{1}{\sqrt{R}}$
So, if radius is doubled, i.e. $R^{\prime }=2R$
$v_e^{\prime }=\sqrt{\frac{2GM}{(2R)}}=\frac{1}{\sqrt{2}}\sqrt{\frac{2GM}{R}}=\frac{v_e}{\sqrt{2}}$
Therefore, Assertion is incorrect but Reason is correct.