How can I meaningfully define the energy cost of magical levitation?

Question

In the world I'm building, there is a form of "magic"¹ that effectively amounts to metabolism-powered telekinesis. In particular, there is a maximum power limit of approximately 20 W/kg relative to a person's weight (equivalent to "strenuous exercise", i.e. not something the person can keep up for too long).

For simplicity's sake, I am notionally considering this "magic" to be 100% efficient. This is okay for doing things that involve Work (physics definition), but I run into problems when considering levitation.

Levitation can be thought of as applying a constant "magical" acceleration to an object in order to counteract gravity. Where trouble comes in is that the power required to accelerate an object, in the presence of a 100% efficient "motor", is dependent on velocity. In other words, if I'm just holding an object stationary in mid-air (velocity is zero), the required energy is zero^note! (See also this.) This is "obviously" nonsense.

How can I reframe this in order to be able to make a reasonable statement about how much energy it costs to use my "magic"? Note that I want to be able to do things like pick up and hold small objects with an energy cost comparable to doing the same thing via muscle power. (This is the same magic system as in Would a magic-using society create living batteries?; see that question for more details/examples.)

Ultimately, I want to know if a) people can levitate themselves, and b) if they could slow themselves to a stop in case of a high fall, car accident, or such.

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To put it a little bit differently, if I ignore gravity — actually, if I ignore all static resistances — I can generally express the action of "magic" as doing Work, for which I can reasonably crunch numbers. How, then, would I go about determining a "reasonable" value for how much energy "magic" has to use to counteract static forces (such as gravity)? Ideally, the end result would be a constant whose units are watts per newton.

Note that, while my ultimate goal is coming up with this number (and suggestions on what that number should be are greatly appreciated!), the import part of this question is actually "how did I come up with that number?".

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Doing some spitballing... Earth's gravity is about 10 N/kg (slightly less, but we're approximating). An average person can dead-hang for at least a short time, and people in decent shape can do chin-ups. Let's also say my 20W/kg is a little low, and that someone in really great shape can manage 30W/kg and 20 N/kg. This gives us 1.5 W/N. If I don't get an answer to this question, I'll go with that and (unhappily) call it a day. Can I rationalize this number in some manner other than "I pulled it out my <redacted>"? What I'm looking for is a plausible way of explaining why "magic" takes energy to overcome static forces that, ideally, could be used to calculate a W/N that could give a number in that ballpark. (An explanation that would necessarily give a different number is fine, as long as the number is non-zero and less than, say 5 W/N.)

Answer 1

Simply holding an object above the ground requires no energy; a table can do it all day without requiring any calories. A human holding up an object by is another matter: we cannot simply lock our skeleton in place like a horse, but use our muscles just staying upright.

So, you could have the energy requirement being the same as raising yourself into the air by other means (such as climbing stairs), and the cost of remaining static in space being either the same as standing still, or the same as holding your own weight by your arms, depending on what works best for the plot.

Answer 2

Levitation by sound. Acoustic Levitation

Your magic is nothing else than being able to propagate enough energy on the air to move the item. It has been proven in lab that a ping pong ball can levitate by sound alone.

You need a vector of the same magnitude going the other way.

To cancel gravity you need to apply "Normal Force". That is what a table applies over the object when holding it.

For 1 kilogram you need 9.8 Newtons. or 1 Joule/meter.

Since the air is compressible, your inefficiency is inevitable. In water your magic works better.

In vacuum it does not work!

Have fun :-)

Edited per comment.

Answer 3

Levitation actually doesn't need to consume any energy. A well-constructed levitation spell simply suspends the conversion of gravitational potential energy into kinetic energy. This costs just about nothing. You would only need to spend energy to ascend, and you would recover energy from descending.

The inefficient way to do levitation is to produce a counter-acceleration against gravity. For all practical purposes, it might as well be "Conjure Helicopter Engine." This will drain your energy pool rather quickly.