The Airplane and the Treadmill

Anybody who is familiar with the xkcd fora is probably aware that bringing up the legendary Airplane-on-a-Treadmill scenario comes with the risk of facing the Banhammer. The reason for it is that the subject has already created two threads and 450 posts of uncut stupid. So at this moment, I'll be discussing the ever-confusing Airplane-on-a-Treadmill problem here, in an effort to elucidate the problem. There are a few typical descriptions of it, but the key points are this:

1. You have a typical airplane on an infinitely long treadmill.

2. The treadmill can accelerate to any speed.

3. Structural integrity will not be considered as a restriction (i.e. No structural failures of the airplane, tires, or treadmill.)

4. The axles on the airplane's wheels are, for this case, frictionless.

5. The airplane will start its engine and attempt a takeoff roll as the treadmill begins moving in the opposite direction, matching speeds with the wheels of the plane.

6. The airplane will either take off as normal, or will be stopped by the treadmill.

Now, the major source of confusion with this problem seems to be in item number 5; many people have differing interpretations of just what this statement about 'the speed of the wheels' means, as follows:

1. The speed of the wheels as measured by a speedometer connected to the wheel. This is dependent on the rotation rate of the wheel.

2. The speed of the wheels as measured by a stationary observer off the treadmill as they move through space, i.e. The translation speed of the wheels.

3. The ground speed of the airplane, which is identical in principle to case b.

So we'll deal with the misconceptions that abound in case a. Many people think that since the treadmill speeds up, it's going to draw the airplane back as it tries to accelerate; this is not the case. The issue here is that a false analogy to a car is being drawn. A car engine delivers its power to the wheels, which rotates them; that rotation then translates into a static friction force between the tires and the ground, which is what propels the car forward. A car on a treadmill would be stopped by the treadmill because it operates with respect to the ground. An airplane, and the wheels thereof, however, behave differently.

What IS the difference, I hear you cry? Well, the difference is an airplane isn't driven by rotating the wheels; it's driven by pushing the entire aircraft through the air via direct, and the wheels simply rotate freely on the axle to allow the plane to move more easily along the ground. A car's driving force goes through the ground it rests on, while an airplane's driving force goes through the air. The airplane operates in a different reference frame than a car, speficially, with respect to the air, not the ground.

Now, time to analyze the forces involved in the plane powering up. Brakes are off, throttle wide open, which means at our initial situation of a stationary plane and a stationary treadmill, we have two forces acting – thrust, and static friction. Thrust acts through the structure of the plane, accelerating it forward, and the friction acts on the tires. That friction, however, translates very little into the structure of the plane, and mostly only serves to rotate the tires along the ground. Additionally, that friction exists whether the treadmill is moving or not; it exists just the same on a paved, static runway, and airplanes take off from those all the time.

Now, what's the biggest problem with case a? This situation fundamentally defines the speed of the airplane as zero for all time; that's the only case where this situation would be true, and the only case where it can be true is if the engine is not running, or if you're picky, at low idle. Since it makes that definition, it's not physically significant, and once the airplane begins moving through space in reality, then you get a situation where the treadmill is constantly accelerating to match the speedometer connected to the wheel, which accelerates the wheel (but fails to decelerate the plane), which accelerates the treadmill. So with this feedback loop happening, you'll eventually get a situation that is eventually going to become physically untenable, but still failing to stop the acceleration of the plane through space.

So, that being said, case a is physically trivial. There's no important physics happening there because we've demanded in the question that the airplane remains stationary, and it has essentially reduced to “Demand that the airplane remains stationary. Does the airplane take off?” It creates a tautology by framing the question poorly or incorrectly.

So now that we've shown why that definition not only runs afoul of sense, but makes no difference to the physics of the problem as well, let's talk about case b, where the treadmill runs at the speed of the wheels through space. So, for a plane traveling at any given speed, v, along the ground, the treadmill will be running at the same speed, v, in the opposite direction. Now, taking into account what I mentioned well above about the friction, the wheels, will be rolling – and freely spinning on the axle, remember – at a relative speed of 2v, so for a plane traveling at 50 knots, the treadmill rolls back at 50 knots, and the wheels rotate at the equivalent of 100 knots. All the treadmill does is speed up the wheels, slow down the plane.

So, after all that, no matter how you rig it up, as long as your axle is frictionless and your wheels don't explode, then you can get moving forward, get lift, keep accelerating, and take off, no matter how you rig it up.

Still don't believe me? Well, here is some experimental verification, with a real plane on a moving tarp acting as a conveyor belt. You don't even need the ideal scenario!

So hopefully after reading this, you'll be able to find the signal for all the noise on this subject. Fare thee well, and hopefully we've learned something interesting!