The mass of the moon is \(\frac{1}{144}\) times the mass of a planet and its diameter is \(\frac{1}{16}\) times the diam…

Explanation

The formula for escape velocity is:

\[v = \sqrt{\frac{2GM}{R}}\]

From the given problem, the ratios for the moon compared to the planet are:

Mass: \({M}_{m}=\frac{{M}_{p}}{144}\)

Radius: \({R}_{m}=\frac{{R}_{p}}{16}\) (the ratio of radii is the same as the ratio of diameters)

Now, we can find the escape velocity on the moon (\({v}_{m}\)) by setting up a ratio:

\[\frac{v_m}{v_p} = \sqrt{\frac{M_m}{M_p} \times \frac{R_p}{R_m}}\]

Substitute the given values:

\[\frac{v_m}{v} = \sqrt{\frac{1}{144} \times 16}\] \[\frac{v_m}{v} = \sqrt{\frac{16}{144}}\] \[\frac{v_m}{v} = \sqrt{\frac{1}{9}}\] \[\frac{v_m}{v} = \frac{1}{3}\] \[v_m = \frac{v}{3}\]

Correct Option: D