This is an awesome question.
Brb, doing math.
Fake Edit:
Okay, the plane wouldn't take off.
If you guys want to know how I figured that out in a few seconds, let me know.
I'd dearly love to know.
I could do some maths of my own, but what's the point in making terms for absolute ground speed, relative-to-runway speed, air-speed, and positions relative to air/ground/runway?
Plane whirs engines,of whatever design. These engines, on windless day on a stationary runway would propel the plane forward at V
TO, or the take-off speed necessary. They do this by acting on the air. If the runway is designed so that as the plane moves (say) in one direction by a given speed then the ground moves in the other direction by a given speed... well, it doesn't matter[1], because the engines aren't pushing against the ground. The engines are pushing against the air, which is still assumed to be still. The plane moves in the direction it should do, the rebellious runway moves in the other direction. The
wheels of the plane, free-wheeling, are rotating at twice the speed that they would do in a normal take-off, but unless the bearings catch fire or something else goes wrong this has no (*ahem*) bearing on the speed of the plane... Even in a fully integrated RL example it might mean you'd have some marginal extra friction, but that should be well within the ability of the plane's engines (either through a percentage rise in engine speed, or just taking slightly longer with a smaller acceleration). The aircraft is moving, though, and reaches V
TO, air speed. A figure you'd double to work out the speed relative to the backwards-rolling runway.
Right, in the light of the two most recent replies (ignoring Aquizzar, who has additionally ninjaed while I was suffering delays in getting my preview), I'm going to put this in bigger text...
The plane is not stationary! The plane is moving in one direction, the runway in another.As someone else (several someone else's!) pointed out, if the plane were not moving, the runway would not be moving. In the example of a car on such a runway going at a runway-relative speed of 30mph, the runway would have sped up to 15mph backwards in order to match the 15mph
forwards travel of the car. Or, to put it another way, whatever speed the runway things the car is going, relative to stationary, and thus what speed the runway is going
backwards from stationary, the car is going twice as fast...
So, for a car, to go a certain speed, you'd be driving twice as fast as you normally would need to...
But you're not in car, you're in a plane. The plane doesn't care about the runway[1], it just cares about the air. Which is stationary. So to go forward at a given speed it... uses as much engine power as it
normally needs to go forward at that speed.
(If the air were not stationary, it would be pushing against/with a wind-speed in its attempt to attain a similar relative velocity w.r.t. the air (stationary or not) in order to get lift. The only problems here is if you run out of runway. With a headwind, you'd need less runway anyway. The maths actually comes out that if you had the
wind going one direction as fast as the plane were going in the other direction, then you'd have the "car" example, again, of the plane going half the speed forward and the wind half the speed back in order that the plane's speed relative to the air is the full amount needed to attain lift. Which it could do as easy as (if not easier than) doing so in still air.
Newtons don't really come into it at all[1], for all these examples.
[1] Any friction excepted. Which is going to be
largely irrelevant for any plane without jammed-up wheels in the first place.