1. Use the formula for escape velocity

The escape velocity from the surface of a body is

$v_e = \sqrt{\frac{2GM}{R}}$

So,

$v_e \propto \sqrt{\frac{M}{R}}$

where $M$ is the mass and $R$ is the radius of the body.

2. Relate moon's mass and radius to the planet

Let the planet have mass $M$ and radius $R$.

Given:

• Mass of moon $= \frac{1}{144}M$
• Diameter of moon $= \frac{1}{16}$ times diameter of planet

Since radius is proportional to diameter,

$R_m = \frac{1}{16}R$

3. Write escape velocity ratio

If escape velocity on the planet is $v$, then

$v = \sqrt{\frac{2GM}{R}}$

For the moon,

$v_m = \sqrt{\frac{2G(M/144)}{R/16}}$

Simplify:

$v_m = \sqrt{\frac{2GM}{R} \cdot \frac{16}{144}}$

$v_m = \sqrt{\frac{2GM}{R} \cdot \frac{1}{9}}$

$v_m = \frac{1}{3}\sqrt{\frac{2GM}{R}}$

$v_m = \frac{v}{3}$

4. Check options

• A: $\frac{v}{4}$ ❌
• B: $\frac{v}{6}$ ❌
• C: $\frac{v}{12}$ ❌
• D: $\frac{v}{3}$ ✅

So the correct answer is:

$\boxed{\frac{v}{3}}$