[Coda]

Originally Posted by Potironette

So.. something like this can't happen? :

That could happen, but that's not what the equations you set up represent. That would be more like:
Ball: y = 10 - v*t - (1/2)g*t^2
Platform: y = 0 + (1/2)a*t^2

I used " d, " with the comma for the platform, since I was worried I'd forget that I was trying to think about positions.

Oh, I thought the comma was a typo because you didn't use it anywhere else. ^^() Usually when you want to distinguish two related variables, you use a prime (') or a subscript (but Trisphee doesn't have bbcode for that; sometimes it's typed with _, for example using "v_0" to mean "initial velocity").

Ball: y = d - (1/2)g*t^2
Platform: y = e + (1/2)a*t^2
y = d so the ball ends up where it started
Ball: d = d - g*t^2
0 = -g*t^2
meaning that for the ball to hit the ground where it began, no time has passed, since knowing only the initial and final position, it's that the ball has not moved at all.

The main problem here is that you're lacking an initial velocity term. Without an initial velocity, the ball will never go up, which means the only time it can be at its starting position is at the start.

If you DID have a velocity term in there, then the equation would have two solutions (as I described with the quadratic formula), representing the two different times the ball was at position d.

But..
Platform:
d = e + r where r is the distance between d and e
d = e + (1/2)a*t^2
e + r = e + (1/2)a*t^2
r = (1/2)a*t^2
In this case the time is not zero because this equation has more information on what the ball is doing?

This isn't quite representing the same system -- the previous system is a special case of this one where r = 0, but this system of equations can represent more possibilities.

That means that if somehow, there was an accelerating box on the ground moving all around earth somehow, and there was a person in it, the person would feel that gravity was whatever direction opposite where the person was accelerating in? Would they think the ground was the side of the box accelerating towards them and they'd feel like the box was moving to side pushing them against the side of the box on the ground :o? Actually, this reminds me sort of the NASA centrifuge I keep hearing mentions of in class, though I'm not sure how people feel inside there.

Not quite. The person would feel gravity pointing towards the surface of the Earth, and they would ALSO feel ANOTHER force coming from the side of the box pushing them around, in the same direction as the box's acceleration. (If the box were maintaining a constant speed, they wouldn't feel that.) You can add the force vectors together (this is a two-dimensional operation) to determine the net force that the person would feel. You could theoretically put a diagonal platform inside that box perpendicular to that net force vector, and the person could stand on it.

If you were standing on the floor of a rocket steadily accelerating through outer space, then the back wall of the cockpit would feel like the "floor" from the perspective of the occupant because of the direction of the acceleration.

The centrifuge is a substantially more complicated bit of math because it's a rotating system, and one of the forces comes from the fact that the end of the centrifuge ISN'T moving relative to the passenger so it's pushing back on the passenger to keep him from flying out the end from sliding down the side. At that point you're getting into calculus if you want to derive the forces from first principles; in practice, that work has already been done for you and you work with a different set of force equations for rotating systems.