Question:

Can we make a drone fly on the Moon by using cylinders with compressed air (or cold helium)?

Wyatt: 3 days ago

Suppose we want to make a drone fly on the Moon (the gravity on the Moon is 1/6 of that on Earth), only by making use of its rotors and air.
The drone is as light as possible ($m_{min}(kg)$, has dimensions $l(m)$ long, $w(m)$ wide $h(m)$ high. The rotor blades have an as big as possible length of $r(m)$ (which is of course smaller than the length $l$, ca va sans dire). The cylinders are as small and light as possible but big enough to give a constant air output for some time. Say the output of air is $a(kg/sec)$ at a speed of $v(m/s)$.
The cylinders inside the drone release their air through tubes that lead the air to an (as light as possible) construction of tubes with holes around the drone, out of which the air is sprayed towards the drone so there is a cloud of air around the drone (the mass of construction is included in the drone's mass).

To ignore ground effects we'll hang the drone high somehow and let it go as soon the rotors start and the air is released (zo the initial upward velocity is zero). Things are constructed in a way that no net momentum is given to the drone.

Can the drone get an upward lift while the rotor blades are rotating in the surrounding air? Although the air spreads out very fast (see a comment made this loss of air can be compensated by the air that's provided by the cylinders.

The air is in a steady-state (going towards the drone to be sucked out by the vacuum on the Moon). So the four rotor blades have an amount of air beneath them.

Now the most probable velocity of an atom (or molecule) is $\sqrt{\frac{2kT}{m}}$, and for air (at about 300 Kelvin, the velocity at which it escapes into the vacuum is about 1500 (m/sec). So what if we used cold helium at a temperature of, say 4,5 Kelvin, and sprayed the helium to constantly fill the gap left by the escaping helium. The speed of expanion would be reduced to about 200 (m/sec), taking into account the reduced $m$ in the formula just mentioned. The weght of the cylinders would be reduced (as would the lifing power though).

For sure, it has to work in outer space where no gravity is present.

It's not my intention to propose a new kind of propulsion technique. Obviously, letting a device fly in a vacuum is much more efficient by just putting thrusters on it. It's just the physics I'm interested in.

Answer:
Isabella: 3 days ago

The question reveals a misunderstanding of how a drone stays aloft. In fact, a drone, helicopter, or airplane stays aloft by shoving air downward. That is, the upward force due to an airfoil moving through air is equal in magnitude to the downward momentum change per unit time in the air the airfoil is moving through. That momentum change is $v dm/dt + m dv/dt$ and amounts to a deflection of the air downward.

In the scheme you suggest, the rotors of the drone would "shove" air downward very inefficiently; and the drone would quickly run out of air. Better just to direct the air downward at high velocity right from the compressed air container through a suitable nozzle.

However, an even more practical approach would be to let the drone use lunar dust as reaction mass. The dust could be carried by the drone and accelerated downward electrostatically a little bit at a time to provide upward thrust. Whenever the drone runs out of dust, it lands and scoops up some more.