Mdthbs

In my world, some individuals have the ability to absorb and distribute momentum from sources around them. I am considering what would happen physically if someone was taking all of the momentum from a falling object (like a rock for example). My question is, would the falling object just stop midair or would it keep falling slowly?

I ask this question as if the object is stopped completely couldn't this magic user just have an unlimited power source from continuously absorbing the gravitational force on the object?

A falling object has momentum proportional to its mass and velocity ($p = mv$). If a person could magically absorb that momentum, they could decrease it to 0 by reducing the velocity to 0. The object would stop, but at that point it would have no more momentum to absorb. They could then let the object fall again, accumulating more momentum, before harvesting the momentum and bringing it back to 0. Alternatively, you could partially harvest the momentum at any point, causing the object to decelerate but not stop. You can't continuously harvest momentum from a stationary object, since it has zero momentum to begin with.

In short, you can harvest momentum though a series of drops with full stops, or do it continuously by causing the object to fall slower than it would under gravity alone. But you can't get momentum out of a stationary object - as you point out, this would be an unlimited source of energy (i.e. a perpetual motion machine). As an example, a hydroelectric dam generates no power by simply holding the reservoir upstream - it generates power by allowing some of the water to fall and harvesting its kinetic energy.

Momentum is p=mv. V is velocity. I have to ask you:Velocity relative to what? For example, Earth is around you, and has quite a lot of momentum. Assuming that your character can absorb Earth's momentum and redistribute it to much smaller things around him you can use a very small fraction of Earth's momentum to accelerate, say, a stone, to velocities, where very intersting things happen.

"But won't Earth fall into the sun if it loses all its velocity" - Once again, frame of reference matters: Is the Sun the frame of reference? If it is Earth will become a static object at zero m/s relative to the sun, neither falling nor orbiting. Maybe that will slowly disturb the other planets orbits?. If the frame of reference is, say, the galaxy, then earth will eventually leave the solar system as it stays behind. Not good.

You already have a great set of answers here: I'd just like to give you some insight into why this is the way physics is.

Objects falling under gravity have a constant force and a varying momentum. As pointed out by @Nuclear Wang and @Geronimo,

$ Momentum = mass * velocity $

But force:

$ F = mass * acceleration
= mass * frac{v-u}{time}$

where v is your final velocity and u is your initial velocity.

When you rob an object of momentum, you do not rob it of acceleration. You simply set its velocity to 0. So if you did this once your object would stop for a moment in mid air before beginning to accelerate again due to gravity.

If you prefer a mathematical proof:

$ Distance~~Travelled = (u*time) + frac{1}{2}*acceleration*time^2 $

The equation above (which was devised by Newton) shows that taking away momentum (which is just speed) doesn't stop an object. Newton demands you deal with acceleration too.

To answer your question, you need to travel towards the source of gravity every so slightly to generate momentum, so you can't harness this forever.
One final piece of math:

$ Distance~~Travelled = frac{(final~~velocity)^2-(starting~~velocity)^2}{2 * grav~acceleration} $

If the distance travelled is zero, you haven't changed your velocity (which is zero when you're holding the object in the air). If your velocity is zero, your momentum is zero. Gotta give if you wanna get.

By the way, your comment on splitting up 'magical energies' actually has some bases in science. It's widely hypothesised, if not believed, that every force in our universe is some combination of these four. Electromagnetic covers heat, magnets, electricity and light. Graviton for gravity, and the strong nuclear holds protons and neutrons together in the atomic shell. The weak nuclear is responsible for some flukey stuff, and honestly you won't deepen your world knowledge by reading into it. Freel free to though.