The mass of the moon is $$\frac{1}{144}$$ times the mass of a planet and its diameter $$\frac{1}{16}$$ times the diameter of a planet. If the escape velocity on the planet is $$v$$, the escape velocity on the moon will be:

We need to find the escape velocity on the moon relative to the planet.

Since $$M_m = \frac{M_p}{144}$$, $$d_m = \frac{d_p}{16}$$, we have $$R_m = \frac{R_p}{16}$$ and the escape velocity on the planet is $$v$$.

We start with the escape velocity formula: $$v_e = \sqrt{\frac{2GM}{R}}$$

Next, the ratio of the escape velocity on the moon to that on the planet is given by $$\frac{v_m}{v_p} = \sqrt{\frac{M_m/R_m}{M_p/R_p}} = \sqrt{\frac{M_m \cdot R_p}{M_p \cdot R_m}}$$

Substituting the expressions for mass and radius yields $$= \sqrt{\frac{1}{144} \times 16} = \sqrt{\frac{16}{144}} = \frac{4}{12} = \frac{1}{3}$$

This gives $$v_m = \frac{v}{3}$$.

The correct answer is Option 1: $$\frac{v}{3}$$.