Potential Gravity

Question

The escape velocity applies to any object regardless of its mass. Even very small particles, such as atmospheric gases, can reach escape velocities. Gas molecules at 20°C move with a velocity of approximately 5000 m/s, which is well below the escape velocity for Earth – Phew – we breathe again! The value of a planet’s escape velocity is one of the primary reasons why small planetoids, such as the moon, have no atmosphere. Determine the escape velocity for the moon. Will it be able to trap atmospheric gasses at 20°C? Show all work. (ans. v=2380 m/s)

Answer 1

v_e = escape velocity from moon = ?

 

M = mass of moon = 7.348 x 10²² kg

 

G = universal constant of gravitation = 6.67 x 10^-11 Nm²/kg²

 

I assume you want the escape velocity from the surface of the moon, so we need the mean radius of the moon, R.

 

R = mean radius of moon = 1.773 x 10⁶ m

 

Total mechanical energy per unit mass (i.e. specific mechanical energy) of an object orbiting the moon right at its surface:

 

E = v²/2 - GM/R

 

At v = v_e, the total mechanical energy is zero.

 

0 = v_e²/2 - GM/R

 

v_e = √(2GM/R) = √[2(6.67 x 10^-11)(7.348 x 10²²) / (1.773 x 10⁶)] ≅ 2351 m/s

 

The small difference between my solution and your given answer probably lies in rounding off M and R.  In any case, you can see the logic to the problem.  Since v_e < 5000 m/s, gases at 20°C will escape the surface of the moon.