It seems to me that it would be pretty easy to take a run, start levitating and skid along rather than walking.

The benefit over roller skates is twofold:

1. It requires no equipment, so anyone can do it whenever they like.




2. You don't accelerate down hill. so presumably you don't slow down going up hill?!

Even if we disregard the conservation of energy loophole and take into account that it's still tiring to do. I think you would see a huge change in the way people got around.

This might seem a small thing at first, but after a while we would see huge changes in urban planning and environments to accommodate the change from walking to 'skidding':

• pedestrianise city streets as people can travel further to get to car parks, bus stops etc




• 'skidways' introduced to allow fast travel without crashing. Vertical poles to grab and swing around when exiting or cornering




• crashpads on walls at corners for people who go too fast!




• electromagnetic launch strips to start you off super fast!




• various skid-assisting devices, can you push yourself along or steer with a walking stick? Bags with wheels so you don't have to levitate them too.

High speed transportation

Think mag-lev for humans, but magnets doing the pushing and the person doing the repulsion. Even with 15 mins of levitation time, this is plenty sufficient for moving people around the city.

Alternatively, small portable propulsion means could be provided for off-roading experiences, obviously with less oversight and control so therefore more risk of dying.

Free energy generation

If you don't accelerate going downhill, then there is missing potential energy somewhere in the system. Harness it.

It seems that ground level is all in your mind, hence the failure to accelerate downhill. This should allow you to perform some interesting rescue feats.

• Little Johnny too close to the edge of the dangerous cliff? Levitate out there to get him back without putting pressure on the ground.


• A sheep stuck on the side of a mountain? Again you won't accelerate down the hill so you're safe holding position without risk of falling.


• Little Johnny is a pain and he's fallen through thin ice this time. Again you can get out there without taking any great risk yourself.


• Horse trapped in a swamp? Again you're good to go.

Many of the difficult or inaccessible areas that once required specialist equipment or significant risk are now accessible.

We're not reinventing the wheel in the short term, as you've said it's exhausting in a way that riding a bicycle isn't, but we are reinventing the dangerous area rescue.

Traverse the minefields

You could also play "floor is lava" game but getting across area that may contain landmines is probably quite useful application. Use small electrical propeller to provide thrust.

Military and Police applications

Quick and quiet movement

Ever tried to move quickly and quietly as you approached a cults/terrorist/suspects bunker in the middle of the woods to rescue some hostages? Probably not, but as you can guess it takes a lot of time and a lot of work to make sure you are not heard as you get into position. Floating would would prevent a lot of noise when sneaking up and probably take near the same amount of effort. Even if it was only as far as someone could push you from the perimeter - time is a huge factor in this.

Less worry about landmines and IEDs

The actual procedure from exiting a vehicle in Iraq, involved checking the ground to make sure you would not get blown up as soon as you put your foot down. Getting blown up by IEDs or landmines would be a lot less of a worry (and worth the effort) if you could hover. Even for just a short 5 minute area check, esp. if you were transporting a VIP and were only the first part of the team or in a police situation when approaching that same bunker from the first part.

I wonder how this would work for those with physical disabilities. Say someone who cannot use their legs is able to levitate. I assume levitation of a centimeter still applies to your feet even if you normally sit in a wheelchair. So someone who normally has to sit at half of their normal height and wheel around to move would now be able to levitate at their full height and move around even without using their legs.

Levitating instead of walking for even only a few minutes would be revolutionary for those who normally require a wheelchair/crutches/etc. to get around.

• Cross bodies of loose consistency/poor grip (calm water, marshy ground, snow, ice, etc.)

When you now need a bridge to cross a river or a lake, or some kind of support to go on ice or snow, you can simply step on the surface and, while being above it, simply paddle or propel yourself on it. It saves the money needed for building pedestrian bridges, in the case of lakes/rivers.

• Perform actions on surfaces which cannot be walked on

Think things like maintaining a zen garden where you don't want to leave footprints or inspecting the 10 meter diameter mirror of a telescope without putting your boots on it.

It feels like, because of the curious no-downslope-acceleration rule, like we should be able to make a perpetual motion machine, but I'm not sure how that would work.

Essentially the hovering is just fancy rollerskates/maglev, apart from this one thing.

Normally, if you push something on rollerskates up a hill, it takes more force than pushing them on a flat plane, because you are storing potential energy in that thing.

And if you push it back down, it's easier, and you're recovering energy.

In this case, however, you gain no energy pushing someone downhill. That implies there's no gravitational force affecting you while you hover.

Consider a tramline, or cable car. One car goes up, one down. In the "up" car, people are requested to levitate. In the down car, they are requested not to. The down car will be "heavier", so the system can work indefinitely from gravity, so long as there are people wiling to help out.

However, since this takes work on the part of the passengers, and since infeasible amounts cannot be lifted this is no different in practice to a cable car that has pedals and asks people to propel it by pedalling.

[Edit: with clarifications, the universe doesn't act as if the mass is cancelled out, just that the reaction force is separated from the sole of the foot, as if on a small pedestal or stiletto heel. So the absence of acceleration downhill isn't due to no force; just that the force is vertical, with no tangential component. So, this cable-car thing doesn't work: the force is applied to the cable car just as if you were on rollerskates in the car.]

However, to continue the hypothesis that this essentially makes the person immune to gravitational effects within a short distance from local-ground... this could make space flight vastly cheaper, depending how it works.

Immediately obvious is that the mass of the astronaut does not need to be taken into account when accelerating, if they can levitate.

Far less obvious is that if the user essentially has no mass, and changing their local ground does not require force and hence a momentum change (which the no-slope-acceleration thing kind of requires in order to be possible)... then you can now launch astronauts at insane accelerations. You can subject them to explosive forces and they'd be invulnerable to them. The limit of acceleration would no longer be the squishies, but what the equipment could withstand. That means concepts like https://en.wikipedia.org/wiki/Nuclear_pulse_propulsion become reasonable.

This would of course introduce risk to the launch - what if the astronaut's levitation lapsed, even for a moment? They'd be a thin paste on the floor before they could blink.

[Edit: Again, the clarified mechanism means there's no protection from acceleration granted. Rats!]

More interesting yet, though, is what this implies about physics. The in-universe H=WP/m means that the units of willpower are the meter/kilogram, and every physicist on the planet will be looking at a way to make that unlock the energy of that mass.

Rearranging, you get m=WP/H. Combining with e=mc^2, you get e=WP * C^2 / H. At very low heights, you potentially have very high energy indeed.

[Edit: this bit remains. For a start, on a smooth and hard enough surface, you can lift near-infinite masses.

You can lift twice the mass, at half the height. In theory, if you can lift your weight by an inch, you can levitate a thousand times your own weight to a thousandth of an inch.

The downside is, the force is probably applied at your feet. In reality, it will work no better than a pair of rollerskates with a lift capacity of a thousand bodyweights: you can't pick up and carry that weight, it would crush all your bones, so having mental rollerskates that strong makes no difference to your real lifting capacity.

But the units of willpower being meters/kilogram (from distance = WP / mass) does mean that in any expression that involves mass, we can see if replacing it with (WP/Height) to try to exploit that exponential power of the distance. Again, though, I can't see a sane way to do it. It doesn't break physics in any way, and it doesn't achieve anything that a magnet or a wheel can't do for longer and cheaper.]

This would be fantastic for the elderly. Especially for elderly women. Now people wouldn't have to worry about falling down and breaking bones. You say that this power won't prevent damage from falling from height but what most of these people need is just a little help with balance. This effect is boosted by the clarification that your head rises first when laying down. That means people who are off balance will be set up on their feet.

Skimming the answers, but it appears so far no one has thought of the obvious:

• Hold on to a car, levitate just as it drives off - there you go at high speed. This could be transport, or it could be sports
  


• Speaking of sports, kids and adults would invent a lot of new games and sports that involve levitation. Rotating sticks that you hold on to, then let go. For distance, direction, whatever.


• Take a long stick with you, levitate, push yourself off the ground or a nearby object with the stick - free movement.
  
  
  
  
  • Instead of personal sticks, you can add handholds to walkways to people who are levitating can pull themselves along to add speed.
  
  
  • and finally you can put the handholds on rotating poles so people don't even have to pull themselves anymore, just hold on and be pulled, then let go. Can be used for direction changes as well.
  
  
  
  


• Avoid making footprints, avoid slipping on a wet floor, avoid dirt on your new shoes - there are many instances where you would levitate for a short time to cross a terrain that you don't want to touch with your feet (or shoes).


• 1-2 cm is sometimes all you need to reach something on the top shelf.

Since it is an ability anyone has at any time anywhere, the main uses would be situational. It's like having a thumb - whenever it helps, you just use it without thinking much about it, the rest of the time you forget how special it makes homo sapiens.

A rather more domestic option - laying carpet or painting or washing floors. These tasks are often a hassle when you have to ensure that you work towards the exit or box yourself in. Potentially much easier when you just levitate out.

I think it might have some limited possibilities in creating a sterile work environment if you don't have to touch the floor while moving around.

Also potentially theft and murder, being able to enter and leave somewhere without touching the floor would leave very little evidence and leave someone unsure which directions you moved in.

OK the way I understand it is that levitating is like someone has put a thick invisible mattress beneath you. The downside of the mattress is sticky and sticks to whatever surface it's attached to. The top side is slippery like ice but it's where you're standing. The top side is also perfectly horizontal, hence no movement without propulsion. So far, not much violating of any physical laws (except the existence of the mattress itself).

So the question is - what applications would benefit from you not touching the ground, although force is still applied to the ground?

Well... I can immediately see applications in sports and entertainment (like ballet). That's cute but not exactly world changing.

Hmm... how about firefighters? Would it be a great boon to be able to float over ground that is too hot to walk on otherwise? Come to think of it they already have thick boots, and if the floor is too hot even for that... But maybe there is some application there.

One more idea - this could find use also in common everyday scenarios, like getting over big puddles or spills. Kids would use this while playing. Builders could use it to get over areas that should not be stepped on directly.

Hmm... what about rescues on ice? How big of an area does the weight get distributed on? Maybe this can help rescuers to get closer to someone who has broken ice and fallen in water.

Levitation without upwards motion.

H = WP/M implies that energy is spent only once a mass is off the ground. A much more common application would be to reduce the weight of walking individuals while leaving them on the ground. A 250 pound person who effectively weighed 60 pounds would be very fast. Walking would become possible for people with weak legs who were effectively paraplegic at their normal weight. Morbidly obese persons or those with bad arthritis could get around much easier.

Even easier, one could levitate select parts of the body for improved appearance and cosmetic effect. This would otherwise require either plastic surgery or special garments.

Fantastic intercourse.

That would be the overwhelming advantage.

Also, if it's possible to do this form of levitation in your sleep, you could finally avoid the discomfort of beds - wrinkles pressing in to you, never being comfortable, etc.

New forms of dance.

The society in question would pioneer fantastical forms of low-altitude dance.

Could you levitate over water? possible uses for coastal indigenous culture who was adept at would be for advanced fishing and hunting.

In a more technological culture, you could use it to avoid detection by pressure pad traps.