The ratio of escape velocity of a planet to the escape velocity of...

The formula to calculate the escape velocity is given by

${v}_{e} = \sqrt{\dfrac{2GM}{R}} . . . \left(1\right)$

For the Earth, it is given by

${{v}_{e}}^{E} = \sqrt{\dfrac{2G{M}_{E}}{{R}_{E}}} . . . \left(2\right)$

and, for the planet, it is given by

${{v}_{e}}^{P} = \sqrt{\dfrac{2G{M}_{P}}{{R}_{P}}} . . . \left(3\right)$

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

$\dfrac{{{v}_{e}}^{E}}{{{v}_{e}}^{P}} = \sqrt{\dfrac{{M}_{E}}{{M}_{P}}} \times \sqrt{\dfrac{{R}_{P}}{{R}_{E}}} . . . \left(4\right)$

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

$\begin{cases}\dfrac{{{v}_{e}}^{E}}{{{v}_{e}}^{P}} & = & \sqrt{\dfrac{{M}_{E}}{16{M}_{E}}} \times \sqrt{\dfrac{4{R}_{E}}{{R}_{E}}} \\ = & \dfrac{1}{2} \\ ⇒ & {{v}_{e}}^{P}: {{v}_{e}}^{E} = 2: 1\end{cases}$