Would an object with negative mass levitate on the moon?

[PirateMonk]

Curious as to how that falls out of the equations? As far as I recall, the gravitational attraction between two bodies is proportional to both masses (and this is equal and opposite per Newton), so if one mass were negative and one positive then they ought to repel, while if both were negative I would expect them to attract. This is assuming that anything like a negative mass can exist in the first place, of course, but I'm pretty sure we're supposed to just go with that assumption given the question being asked.

You are correct about the directions of the gravitational forces, but an object with negative inertial mass (which, as NichG points out, leads to huge problems) accelerates in the opposite direction of any force exerted on it. So a negative and positive mass experience gravitational repulsion, but the negative mass responds to that repulsion by moving towards the positive mass, so if they start at rest and their masses are equal except for sign, the negative mass will just chase after the positive mass forever, with both of their speeds approaching infinity. Similarly, if you have two negative masses, they will be attracted, but respond to that by accelerating away from each other.

This also means that a negative mass won't actually float in an atmosphere. The air pressure on the bottom will be slightly higher than on the top, so the net force from the air will be upwards, and thus it will accelerate downwards faster than in a vacuum. On the moon, there is no air to push on it, so the only force with be gravity, pushing it up and causing it to fall down. Once it hits the ground, the ground pushing it up will cause it to go down even faster, with effects roughly like Battleship789 described.

An object with positive inertial mass but negative gravitational mass, in contrast, would be repelled by positive masses (both in terms of forces and accelerations) and so would float with or without an atmosphere (slightly more with one).