(I'm just gonna' keep on posting it, everytime you say something "smart".)

But just in case you might actually be listening...

A few difinitions for you:

Vp = velocity of plane
Vt = velocity of treadmill

If plane sits on treadmill, and engines turned off, but treadmill starts moving backwards (negative direction):

Vp = Vt, where Vt is some - number

In other words, the plane is moving backwards, at the same velocity as the conveyor.

When plane fires up his engines to prevent backwards movement,

Vp = 0.
Vt = some -#.

Plane isn't moving, now. But it's tires are rotating at an angular velocity necessary to prevent the plane from MOVING backwards with the treadmill. This is not the same as FORWARD movement of the plane. [this is the part you're not grasping]

So when plane starts upping the throttle to actually match the treadmill's velocity, but in the opposite direction,

Vp = some +#
Vt = some -#

And if Vp = some + number, then integral of that velocity (acceleration) will be some + number, too, proving the plane can accelerate, and it will take off.