1. Assertion A: Earth has atmosphere whereas moon doesn't have any atmosphere.

   This is essentially true at the JEE level.

   • Earth has a significant atmosphere because its gravity is strong enough to retain gas molecules.
   • The Moon has extremely weak gravity and hence cannot retain an atmosphere in any substantial amount.

   So, Assertion A is correct.

2. Reason R: The escape velocity on moon is very small as compared to that on earth.

   Escape velocity is given by $v_e = \sqrt{\frac{2GM}{R}}$

   • For Earth: $v_{e,\text{Earth}} \approx 11.2\ \text{km/s}$
   • For Moon: $v_{e,\text{Moon}} \approx 2.4\ \text{km/s}$

   Thus, the Moon’s escape velocity is much smaller than Earth’s.

   So, Reason R is also correct.

3. Does R explain A?

   Yes.

   • Gas molecules in an atmosphere have random thermal speeds.
   • If the escape velocity of a celestial body is too small, many gas molecules can escape into space over time.
   • Since the Moon has a much smaller escape velocity, it cannot retain atmospheric gases, whereas Earth can.

   Hence, Reason R is the correct explanation of Assertion A.

4. Evaluation of options

   • A: $\mathbf{A}$ is false but $\mathbf{R}$ is true $\to$ Incorrect
   • B: Both are correct but $\mathbf{R}$ is not the correct explanation $\to$ Incorrect
   • C: Both are correct and $\mathbf{R}$ is the correct explanation $\to$ Correct
   • D: $\mathbf{A}$ is true but $\mathbf{R}$ is false $\to$ Incorrect

Therefore, the correct option is: $\boxed{\text{C}}$