The ratio of escape velocity of a planet to the escape velocity of earth will be:-
Given: Mass of the planet is $16$ times mass of earth and radius of the planet is $4$ times the radius of earth.

Question

The ratio of escape velocity of a planet to the escape velocity of earth will be:- Given: Mass of the planet is 16 times mass of earth and radius of the planet is 4 times the radius of earth.

TestHub · Accepted Answer

The formula to calculate the escape velocity is given by

$v_{e} = \sqrt{\frac{2 G M}{R}} . . . (1)$

For the Earth, it is given by

${v_{e}}^{E} = \sqrt{\frac{2 G M_{E}}{R_{E}}} . . . (2)$

and, for the planet, it is given by

${v_{e}}^{P} = \sqrt{\frac{2 G M_{P}}{R_{P}}} . . . (3)$

Divide equation (2) by equation (3) and simplify to obtain the required ratio.

$\frac{{v_{e}}^{E}}{{v_{e}}^{P}} = \sqrt{\frac{M_{E}}{M_{P}}} \times \sqrt{\frac{R_{P}}{R_{E}}} . . . (4)$

Substitute the values of the known parameters into equation (4) to obtain the required ratio.

$\begin{array}{rcl} \frac{{v_{e}}^{E}}{{v_{e}}^{P}} & = & \sqrt{\frac{M_{E}}{16 M_{E}}} \times \sqrt{\frac{4 R_{E}}{R_{E}}} \\ = & \frac{1}{2} \\ \Rightarrow & {v_{e}}^{P} : {v_{e}}^{E} = 2 : 1 \end{array}$

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