What kind of propulsion method are known to man that can be used for drone racing on moon land and sky(near surface) ? What role moon atmosphere might play?
Basicly you have 2 Options (and a third hybrid one):
Fastest: Do a sub-orbital hop with a reaction engine. It doesn't matter if this is a cold-gas thruster, a rocket engine, a anti-matter engine... you need a thrust/weight ratio above moon gravity, than the amount of deltaV defines how far you can travel. Keep in mind that you need half of your deltaV to slow down again before impact!
Slowest: move along the ground, using a mechanical device. Wheels, tracks, legs. Your speed won't be too high since you have only the very low moon gravity to stick your movementsystem to the ground. Go to fast and you loose contact and control. But as long as you have electricity(solarcells anyone) you can keep on going.
Hybrid solution: Use a mechanical pusher-plate-system to jump in a sequence of small sub-orbital hops. Because you don't have to stick to the ground you can go faster than a wheeled system, but you don't run out of fuel like a rocket. But you can't slow down before impact so your hops need to stay small. You need some way to control the landing, and you need to make your drone robust enough to survive multiple impacts...
Small automated flying probes have been considered for the exploration of lunar lava tubes.
From Flying, Hopping Pit-Bots for Cave and Lava Tube Exploration on the Moon and Mars. by J. Thangavelautham et al:
Fuel is non-cryogenic - RP1 and Hydrogen Peroxide. This feeds a single rocket used mainly for lift and hovering. For attitude control there are several attitude control jets powered by Hydrogen Peroxide.
The paper also gives an overall mass budget of 3 Kg for this small probe, overall size of perhaps 30cm. Naturally for proposed racing applications, something larger would be preferred for more than a few minutes of endurance.
For the interested there is a second paper which gives less information about the robots, but some information on the overall mission profile for exploring a lava tunnel.
The moon has no atmosphere, so there is not much point in talking about drone racing. There are only 2 ways to race drones on the moon:
Hold the races inside (in a large sealed cave or dome that is filled with air). In that case a normal drone will do the job, but it will be able to fly longer because, due to the fact that lunar gravity is only one sixth of the earth's, the engines will use less power than on earth.
To hold the races outside, you need to use rocket engines, just as you would for anything else that needs to lift off from the lunar surface. There are no rocket-propelled "drones" on earth. The military use some jet-powered ones, but these rely on air to burn the fuel, so they will not work on the moon either.
About the only thing we currently have that could be called a "rocket powered drone" are the re-usable rockets of SpaceX and Orbital. They can take off under rocket power and land again. I'm guessing that's not what you had in mind.
You can improve on a reaction engine by (essentially) throwing a rubber ball at the surface.
The rubber ball is just reaction mass, of course, but over rock, if accurately thrown, it will bounce back, conveying further momentum when you catch it, as well as preserving reaction mass.
Over dust, of course, it dissipates its energy instead of bouncing, and if inaccurately thrown, it will rebound off target. So some secondary propulsion (or a supply of rubber balls) is essential.
If tethered, recovery is possible but inconvenient, but your height is limited to the tether. This becomes equivalent to the "pusher plate" in another answer.
Overall, it's unlikely to be a practical means of transport, despite the reaction mass saving, but it may be capable of providing stable flight at a focus point above an ideally shaped crater, for example.
There is no atmosphere on the moon, unless you use rockets you're looking at ground based racing. There are two interesting variations on ground based racing to consider, taking advantage of the lower gravity.
The low gravity will make magnetic levitation a lot easier than on Earth. Vehicles like the maglev train will be a lot more practical. One of the issues with magnetic levitation is the need for a special track, which on Earth is expensive.
Night time temperature on the moon may be low enough to keep some superconductors below their tranistion temperature (on earth superconducting magnets have to be cooled as they lose their power at temperature well below ambient.)
With alternating magnetic fields it is possible to hover over a simple metal surface such as aluminium using only eddy currents (an alternating magnetic field generates currents in metals that repel the magnet.) This is impractical on earth but would be more practical on the moon. A large aluminium arena could be built.
Could also be referred to as "ground effect rocket."
A hovercraft could float on a cushion of air (or better, hydrogen) expelled directly into the vacuum. Such a device would be less efficient than a combustion system, but would possibly be viable on the moon.
Tracks for wheeled racers on the moon would have to be much more steeply banked than on Earth. A 45 degree angle is needed when centripetal acceleration matches gravitational acceleration. This would occur at a lower speed on the moon than on Earth. This acceleration is given by