If I fire a bullet at the moon, is there a speed at which that bullet stays in orbit around the moon forever?

Answer

If we neglect the relief on the moon :

The gravitational acceleration at the lunar surface is GM / R² where M and R are the mass and radius of the moon.

The centrifugal acceleration in a circular orbit is v² / R, with v being the speed.

If we want to make a bullet fly in a circular orbit just above the moon’s surface, those two accelerations must neutralize each other. Gravitation is downward, centrifugal acceleration upward. By equating the two sizes, we find the required speed.

With G = 6.6727 10-11 Nm²/ kg²R = 1.738 10⁶ m and M = 7.35 10²² kg

from which : v = 1680 m/s