[aardvark291]

The power required to fly at constant altitude and speed is velocity multiplied the force of drag.

Energy is consumed to exert a force over a distance, but we're not moving against gravity ("constant altitude"), so no energy is directly expended to fight gravity.

Now, that argument cheats a little, because there is a relationship between lift and drag: compare induced drag (drag created as a result of producing lift) to parasitic drag.

[LeifCarrotson]

Consider a helicopter nearly hovering, but moving forward at a walking pace.

To maintain the hover and prevent the helicopter from falling out of the sky, the engines are consuming large amounts of power. To move it forward at a sedate pace only requires a small expenditure of energy to overcome drag.

If it was on wheels, a human could push it across a hangar with little effort. A human definitely could not hold a conventional helicopter in the air by lifting or by pedaling to turn the rotors.

The reactive force lifting the helicopter should not be thought of as drag.

[tgb]

Are you actually claiming that a helicopter doesn't use any energy to hover in place? Your argument describes the work done on the helicopter (or plane), but not the work the helicopter does on the air, which it pushes downward considerably. So, too, does an airplane's wings 'push' air downwards. Contrast to the vehicle sitting on the ground, where the ground is incompressible and no work is done on it.