Ok i have the best example I can think of:

If a plane is sitting on the equator on a runway can it take off against the counter-rotation of the earth?

Hopefully that should help some people.

Thing is that this is not about a theoretical plane over a theoretical threadmill - it's about a real plane over a real threadmill. Othewise, I could also say that, in a perfect system, rolling friction doesn't increase with speed. And this perfect system where the plane does take off (i.e., your "correctly performed experiment") simply put doesn't exist.

Edit: The only thing that matters is the speed of atmosphere relative to the plane.

If a plane is sitting on a treadmill with its wheels moving at a speed corresponding to the speed of the treadmill the atmosphere is stationary relative to the plane.

In this situation if the plane attempts to accelerate it can and will take off. However it will not take off just by sitting there.

Edit2: Unless you are implying that the "negative" windspeed relative to the plane prevents it from accelerating.  That doesn't sound right to me, but if you have a good argument for it I am interested. 

I meant that the atmosphere is stationary relative to the plane.  I guess I didn't make that clear.  (have cold...can't articulate clearly)

"If a plane is traveling at takeoff speed on a conveyor belt, and the belt is matching that speed in the opposite direction, can the plane take off? "

If the plane needs to move 60mph for takeoff speed, and the conveyor moves 60mph in the opposite direction, as prescribed in the above statement, then the friction from the wheels might make the plane move only 55~57mph. The plane isn't going to just sit still like some people want to think.

The wheels are just attached with bearings, no gears, drivetrain, or anything that will actually truely effect the speed of the plane. They might as well be water ski's or ice ski's which a plane can take off wearing also.

I've posted this twice now and people still argue around me, but noone seems to refute this: A plane with wheels on a conveyor is the same as a plane with water ski's on water or ice ski's on ice. It will not just "sit" there. It will move forward at "almost" the same speed as if the conveyor was not there at all.

Okay nick, imagine if you have some skis on and wings attached to your arms and somehow you are going foward and your on a treadmill. If their is no airforce to pick you up their is no chance of you flying up.

Simple as that

Right, but the point is the treadmill isn't going to stop you from going forward and getting the airforce needed.  The wheels just have bearings so the treadmill shouldn't effect anything.  Am I missing what you're trying to explain to me somehow?  Relative to the ground and earth outside of some "landingstrip lengthed treadmill" you're still definitely moving just like you would on solid pavement.  That's all that is need for takeoff yeah?

A plane sitting on the equator has no relative airspeed.

A plane on a treadmill with its wheels spinning at the speed of the treadmill has no relative airspeed.

I avoid saying the plane is "traveling" on the treadmill because it really isn't travelling relative to anything except the treadmill.

Let me know if I am missing something, but I think the analogy works.

Once  _again_, a stationary plane WILL take off if its engines create airflow along its wings, provided the following conditions are fullfilled:

1. The plane's engines are strong enough that they can accelerate the plane to the equal and opposite speed of the conveyor belt underneath the plane (keeping the plane stationary to someone standing at the belt).  It is completely irrelevant what is holding the plane on the belt, be it wheels, sleds, skis, whatever. As long as the plane can match the speed of the belt by overcoming the sum of the resistive forces, we are ok.

2. The engine(s) must be mounted horizontally, otherwise we can cheat and actually build a rocket...

3. The engine(s) must move air along its wings to create momentum directed upwards.  

4. The speed of the conveyor belt can be increased to a critical speed. The critical speed is defined as the speed at which the momentum generated by the engine(s) on its wings overcome the weight of the plane. At this point, the plane will lift off the belt and jolt.

What most people here seem to misunderstand is that even though the plane is stationary to someone standing at the belt (and the air surrounding her/him), the air is not stationary in the engine(s)/wing system. In order to match the conveyor belt movement backwards, the plane has to move/roll/scratch/ski/whatever forward at the same speed with momentum mass(plane) x velocity(plane), the engines have to create the backward momentum m(air) x velocity(air). Now velocity(air) is _a lot_ higher than velocity(plane/belt), therefore there is considerable airflow around the engine(s).

Now a plane takes off if the airflow on its wings is high enough so that the upward force created is higher than the weight of the plane. If we attach the engine(s) to the plane at a place so that the airflow does not flow along its wings, there will be no upward force created and the plane does nothing. If we mount propellers to the wings, the airflow by the propellers _will_ create uplift, and given the critical speed can be achieved, uplift will cancel weight and the thing takes off.

Whether this theoretical situation can actually be performed is a completely different matter and depends on fullfilling all the conditions above. I'm certainly not going to stand next to the plane if somebody tries it, I still value my life somewhat.

So for the final time: Yes, at least in theory the plane _will_ take off the conveyor belt.