Energy, power and staying still.

You can hover in a rocket, or a balloon or a helicopter. For the first two cases the calculation is straightforward.

For a rocket, the lift is given by the momentum of the expelled matter per second, i.e. rate of change of momentum. If this is equal to mg you'll hover.

 

There is no minimum power required. The balloon uses zero. Even the 'helicopter' can use arbitrarily small amounts of power if it is allowed to have arbitrarily large 'rotors'.

 

You have to consider the method of elevation. For instance, hovering in a balloon requires no power.

If you assume your rocket is perfect then the calculation above can be taken as a minimum requirement.

 

Even for the rocket you can't pin down a minimum power without other considerations. You can write the thrust force as v*dm/dt where v is the exhaust velocity and dm/dt is the rate of exhaust mass ejection. Power is (1/2)*v^2*dm/dt. So by making v very small and dm/dt very large you can hover at as low a power as you wish. Of course, you can't do it for very long. This is basically the flight strategy of taking an anvil with you as fuel and then when it's time to hover, dropping it and trying to stand on it.

 

Dick, thanks for clarifying that.

I don't see the logic here because there's a constraint that the lift force must be above a certain minimum.

 

Set v*dm/dt=M*g (M is the mass of the rocket - which will start changing but we don't worry about that because we aren't concerned with hovering for any significant period of time). Then v=(M*g)/(dm/dt). So power P is proportional to 1/(dm/dt). Sending v to zero and dm/dt to infinity gives a fixed force for arbitrarily small power (albeit also for arbitrarily small times). PS. I've given this a lot of thought - a biologist buddy used to try to convince me there MUST be a minimum power requirement.