# How Much Power Is Required to Hover?

Suppose I have a spaceship that weighs 1,000 kilograms. I take it to the surface of the planet with a gravitational acceleration of 10 meters per second-squared. The planet has no atmosphere and I'm far enough off the surface that there are no abnormal effects from the interactions between exhaust particles and the planet surface. It's going to take 10,000 Newtons of force to keep me aloft.

The spaceship doesn't move from a hover -- does that mean there's no work being done on it? That means the only work being done is on the fuel -- the exhaust gases or plasma we're spewing towards the ground. Is there a fixed power requirement to get the exhaust going fast enough to keep me hovering? Let's assume for the moment that I'm using a negligible amount of fuel, and thus maintaining the same mass. My first thought was to calculate the work done by applying a force over the length of the engine nacelle, for example -- say we're ejecting fuel at a rate of 1 gram per second with an exhaust velocity of 10,000,000 meters per second in order to generate the required 10,000 Newtons to keep us hovering. With an engine nacelle one meter in length, 10,000 Newtons of force over 1 meter will result in 10kJ of work done per second, or 10kW of power. Is that right, and can it change with, say, a shorter engine nacelle?

The force needed to balance weight comes from momentum considerations, not energy:

$$F = dP/dt$$

$$Mg = (dm/dt) V$$

Where $$dm/dt$$ is the mass thrown down in a second, $$V$$ is the velocity of that throw.

Then the power needed is

$$P = 1/2 (dm/dt) V^2$$

There are two more convenient ways to look at this, given that weight is (for now) fixed. You can focus on mass thrown:

$$P = 1/2 (Mg)^2 / (dm/dt)$$

Or throw velocity:

$$P = 1/2 (Mg) V$$

Either way, the lesson is clear: to hover using less power, throw bigger chunks slower.

(Of course, at some point you run out of mass to throw! This has all assumed the lifted mass isn’t changing much)

• So, if I understand this correctly, the answer to the question in title is basically “however much you want”. In limit, zero. – Mormegil Nov 6 at 16:24