Acceleration due to gravity on a planet is $12$ times the value on the earth. Escape velocity for the planet and the earth are $V_{p}$ and $V_{e}$ respectively. The ratio of the radii of the planet and the earth is $1 : 2$ , then

V_{p} = \frac{V_{e}}{\sqrt{6}}

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V_{p} = \sqrt{5} V_{e}

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V_{p} = \sqrt{6} V_{e}

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V_{P} = \frac{V_{e}}{6}

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Solution

The correct option is C $V_{p} = \sqrt{6} V_{e}$
Given,
$g_{p} = 12 g_{e}$ and $\frac{r_{p}}{r_{e}} = \frac{1}{2}$
Escape Velocity on earth is given as
$V_{e} = \sqrt{2 g_{e} r_{e}}$
Similarly, for an planet, escape velocity is given as,
$V_{p} = \sqrt{2 g_{p} r_{p}}$
or, $\frac{V_{p}}{V_{e}} = \frac{\sqrt{2 g_{p} r_{p}}}{\sqrt{2 g_{e} r_{e}}}$
or, $\frac{V_{p}}{V_{e}} = \sqrt{\frac{g_{p}}{g_{e}} \times \frac{r_{p}}{r_{e}}}$
or, $\frac{V_{p}}{V_{e}} = \sqrt{12 \times \frac{1}{2}}$
or, $V_{p} = \sqrt{6} V_{e}$

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