Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Earth has atmosphere whereas moon doesn't have any atmosphere.
Reason R : The escape velocity on moon is very small as compared to that on earth.
In the light of the above statements, choose the correct answer from the options given below:

Solution

Physics sir · 5 steps

Compare escape speed with how fast gas molecules move, and you see why the Moon loses its air.

1. Escape velocity. A body escapes a planet if its speed is at least vₑ = √(2GM/R). The Moon has about 1/81 of Earth's mass and about 0.27 of its radius.

   v_(e,moon)/v_(e,earth) = √((1/81)/0.27) ≈ 0.21

2. Check R. So vₑ on the Moon is about 2.4 km s⁻¹, against 11.2 km s⁻¹ on Earth. R is true.
3. Gas molecules. Gas molecules move fast from thermal energy. For example, the rms speed of H₂ at 300 K is about 1.9 km s⁻¹. Some molecules always move faster than the average, so many of them go above 2.4 km s⁻¹ but almost none reach 11.2 km s⁻¹.
4. Check A. Over a long time the Moon's gas molecules leaked away into space, while Earth's gravity held its gas. So A is true, and the low escape velocity is exactly the reason.
5. So. Both statements are correct and R correctly explains A, which is option (c).

The idea to keep. A body keeps an atmosphere only if the speeds of its gas molecules stay well below its escape velocity. A small escape velocity means the gas slowly leaks away.

Answer from the NTA final answer key. Solution by PrepWiser.