I have a magic system with very specific physical rules (hence why the question is science-based), but I need help working out a specific scenario. Also feel free to suggest better terminology where it can use improvement; explaining this system has always given me difficulties.
It works like this: a witch can make the normal force apply without physical contact. For example, imagine that you are an evildoer walking up towards a witch you want to murder. If she is inside a building, then she can take the wall behind her, and manifest its physical presence in front of your nose. You'll bump into what feels like solid rock to you, and probably break your face.
And a witch can go on the offense too. She needs to make a slight hop, and then as she is about to land, projects her own body right on top of your head. If you didn't anticipate it, then the sudden impact of the mass of a human body on your head can break your neck. There's more sophisticated uses of this force; a skilled witch in active combat will seem to be doing parkour in the air whilst people around her are kicked without ever coming into physical contact - at least not in a way you can perceive with your eyes.
Other aspects and limitations:
- This kind of sorcery only carries over the normal force on the super-atomic scale, not photonic interaction, chemistry, etcetera; so the projections are tangible but invisible.
- A projected surface has all the friction of its source object (at least the amount of friction resultant from physical ridges in the object, not that from any chemical bonding or whatnot).
- It does not create a vacuum where you project the item; a projection is an infinitely thin shell.
- You can make interactions with the projections as if they were the real objects; project a small cup and you can fill the projection with water, and see water hovering in the air, whilst the source cup feels heavier.
- Force on the projection can move and/or destroy the source object as if it were interacting with the source object. This also works vice versa.
- The direction of any forces involved is never changed by the magic.
- The projection ceases to be once a witch stops actively maintaining it.
Let me know if this makes sense so far.
Now, the specific use case I want to figure out is flight. One of the things a witch can do is seemingly walk on air, by actually walking on the floor below, which she projects higher up, right below her feet. That way she can create staircases anywhere.
But she can also change the scale of the projections. Surgery is made much easier by projecting an upscaled version of the patient's body, covering the projection with some sand so she can see it, and then manipulating the enlarged body with her bare hands, allowing high precision.
And that allows for a much cooler form of flight. A witch can walk on the air itself; if she projects an enormous surface of air below her feet. It should be a large enough surface that she can step on the air, push herself off it, and take another step with her other foot before the first foot sinks too far.
And because this projection is perfectly smooth (it is an artificial gathering of air along a perfectly flat, slightly inclined plane), it is frictionless. Once reaching cruising altitude (basically just high enough not to bump into buildings), a witch can lie down and glide on the projected air, and traverse the sky at hawk speeds. She only occasionally needs to get up and gain more altitude; even more skilled witches can use the (projected) wind for that instead and basically take off without expending any energy.
How large is the stairway to heaven?
The stepping process is the specific use case I need to sort out. The question is: in terms of horizontal surface area, how much air is needed to walk upon, particularly to lift oneself up? Think of it like the witch wearing enormous sandals, perfectly stiff and weightless, of such an area that you can step on the air the way you can walk on snow with snowshoes.
I want to sort this figure out because it will allow me to put an upper limit on the size of both projections and source objects. A formula with as input an arbitrary weight would be most useful, as I am also considering the witches building "boats" to fly on the frictionless projected air, and then I would need to know the required surface area to support the boat too.