1. Escape velocity formula

For a planet of mass $M$ and radius $R$, escape velocity is

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

where $G$ is the gravitational constant.

2. Escape velocity on Earth

Let mass and radius of planet $P$ be $M$ and $R$.

Given:

• Earth mass $= 9M$
• Earth radius $= 2R$

So, escape velocity on Earth is

$v_e = \sqrt{\frac{2G(9M)}{2R}} = \sqrt{\frac{9GM}{R}} = 3\sqrt{\frac{GM}{R}}$

3. Escape velocity on planet $P$

For planet $P$,

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

Now express this in terms of $v_e$.

From above,

$v_e = 3\sqrt{\frac{GM}{R}}$

Thus,

$\sqrt{\frac{GM}{R}} = \frac{v_e}{3}$

Therefore,

$v_P = \sqrt{2}\sqrt{\frac{GM}{R}} = \sqrt{2}\cdot \frac{v_e}{3} = \frac{v_e}{3}\sqrt{2}$

4. Compare with given form

Given required form:

$\frac{v_e}{3}\sqrt{x}$

Comparing with

$v_P = \frac{v_e}{3}\sqrt{2}$

we get

$x = 2$

5. Option check

• A: $1$ ❌
• B: $3$ ❌
• C: $2$ ✅
• D: $18$ ❌

Hence, the correct option is C.