Given below are two statements:
Statement I : For a planet, if the ratio of mass of the planet to its radius increase, the escape velocity from the planet also increase.
Statement II : Escape velocity is independent of the radius of the planet.
In the light of above statements, choose the most appropriate answer from the options given below

We need to evaluate two statements about escape velocity from a planet.

Recall the formula for escape velocity.

The escape velocity from the surface of a planet is derived by equating the kinetic energy of the object to the gravitational potential energy at the surface:

$$ \frac{1}{2}mv_e^2 = \frac{GMm}{R} $$

where $$G$$ is the gravitational constant, $$M$$ is the mass of the planet, $$R$$ is its radius, and $$m$$ is the mass of the object. Solving for $$v_e$$:

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

This can also be written as $$v_e = \sqrt{2gR}$$, where $$g = GM/R^2$$ is the surface gravitational acceleration.

Now evaluate Statement I.

"If the ratio of mass of the planet to its radius ($$M/R$$) increases, the escape velocity increases."

From the formula $$v_e = \sqrt{2GM/R}$$, we can write:

$$ v_e = \sqrt{2G \cdot \frac{M}{R}} $$

Since $$G$$ is a constant, $$v_e$$ is directly proportional to $$\sqrt{M/R}$$. Therefore, if $$M/R$$ increases, $$v_e$$ will also increase. Statement I is correct.

Now evaluate Statement II.

"Escape velocity is independent of the radius of the planet."

From the formula $$v_e = \sqrt{2GM/R}$$, we can clearly see that $$v_e$$ depends on $$R$$ (the radius appears in the denominator under the square root). If we keep $$M$$ constant and change $$R$$, the escape velocity changes. For example, if $$R$$ is halved (planet compressed to smaller size with same mass), $$v_e$$ increases by a factor of $$\sqrt{2}$$.

Note: Escape velocity is independent of the mass of the escaping object ($$m$$ cancels out in the derivation), but it does depend on the radius of the planet. Statement II is incorrect.

The correct answer is Option 2: Statement I is correct but Statement II is incorrect.