## Mathematical Recreations

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There is a longstanding thought-experiment riddle about an airplane on a treadmill. There are some good responses, but I really haven't seen a satisfactory treatment yet. There are two competing phrasings. I'll give the one that has a clean answer first:

A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane's speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?

The answer to this version is an unambiguous yes. The treadmill and the plane both move, and they move in opposite directions. The wheels turn at twice the speed of the airplane. Any airplane should be able to have its wheels turn at twice the takeoff speed without damage.

The other phrasings are more problematic, and many definitive responses provide the same knee-jerk yes. The problem with the knee-jerk response is that forces need to balance, and the phrasing seems to imply (to the knee-jerk responders) that the forces can't balance, so we shouldn't even try. That is wrong. The forces will balance, even if something has to break down. In fact, nothing has to break down, and the forces will balance even in the ideal case.

A plane is standing on a runway that can move (some sort of band conveyer). The conveyer belt exactly matches the speed of the wheels at any given time, moving in the opposite direction of rotation. Can the plane take off?

The best answer to this version no—the airplane cannot take off normally. This phrasing is quite clear. The wheels do not move. (That is, they rotate, but do not have any translational motion.) If the wheels do not move, then the airplane does not move. You can pick any mechanism you want for this; for example, the wheels spin fast enough to increase the friction in the bearings so that the thrust of the plane is balanced by the friction in the bearings. Now, some "authorities" will claim that the bearings can't provide enough friction to prevent the plane from moving forward, but this plainly contradicts the statement of the paradox, which says that the rotation rate of the wheels is exactly balanced by the speed of the conveyer belt. So what breaks down?

Let's look closely at how the forces balance in the ideal case where the wheels are frictionless, the conveyer belt is perfectly able to match the rotation of the wheels, and the wheels do not slip on the conveyer belt. The plane's propeller provides forward thrust. This is balanced by a rearward force from the wheels onto the axles. This rearward force must be provided by the rearward force of the conveyer belt on the wheels at the point of contact. These forces are applied at an offset equal to the wheels' radius. That produces a torque on each wheel, causing the wheel to spin. That is, the torque on the wheels is balanced by the angular acceleration of the wheels. Later, the friction of a non-ideal wheel may increase enough to help balance the torque, but it is negligible at the outset. Note (and this is important) that the forces will balance, and the plane will not take off even if the wheels are frictionless, because the inertia and angular acceleration of the wheels will balance the thrust from the propellers and the force of the conveyer on the wheels. If you assume that the conveyer belt is perfect and the airplane does not disintegrate at the bearings, then the standoff can continue indefinitely (although the conveyer belt might reach relativistic speeds).

Although this is a thought experiment, it's still instructive to ask what the treadmill needs to do to successfully prevent a normal airplane from taking off. These are some realistic values for the key characteristics of the airplane: Forward thrust = 1000 Newtons = 1000 kg m/s2. Wheel radius = 20 centimeters = 0.20 m. Wheel inertia = 0.5 kg m2, which is equivalent to a mass of 13 kilograms located at the radius of the wheels. If there is no source of friction, then all of the thrust goes into accelerating the two large wheels at a rate of (0.5*1000 N)(0.20 m)/(0.5 kg m2) = 200 s−2. The outside of a wheel (at a radius of 20 cm) accelerates at 40 m/s2 ≈ 4g, where g is the acceleration due to gravity. This is the acceleration rate of the treadmill. Now, 4g is pretty strong acceleration, but it certainly doesn't violate any laws of physics. In fact, you might actually be able to build a treadmill with available technology that accelerates that fast for a few seconds. It would take just over eight seconds for the treadmill to reach a supersonic speed. Although an actual treadmill might be able to accelerate that quickly, I'm reasonably certain that regulating the acceleration to exactly match the speed of the wheels is beyond existing technology.

If someone tells you that the airplane will take off because the coefficient of sliding friction between the melting rubber of the wheels and the conveyer-belt runway will drop to nothing, and the plane will take off with the wheels sliding on the runway instead of turning, then give them 100% credit. That is a good answer. (The paradox says that the conveyer belt exactly matches the (rotational) speed of the wheels, but it is perfectly consistent to say that the wheels will slide.

Similarly, if someone says that the bearings will explode and the conveyer belt will stop because there are no wheels to match speeds with, then the plane takes off sliding on its belly, give them 100% credit.

But if they say the plane takes off normally, then they are ignoring the constraints of the problem, because they do not understand how the physics of the system can be consistent with the statement that the conveyer belt (successfully) matches the speed of the wheels.

The poorest phrasing of the paradox is as follows:

A plane is standing on a runway that can move (some sort of band conveyer). The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving in the opposite direction of rotation. Can the plane take off?

This phrasing is probably supposed to be the same as the second phrasing, but the "is designed to" modifier really confuses the issue. It invites a discussion of real conveyer belts and their capabilities. If that's what you mean by the question, then the answer is probably It is impossible to build a conveyer belt that can accelerate quickly enough to stop an airplane from taking off. Although four g isn't that ridiculous…

This is certainly not the only correct answer to this puzzle out there. I've found a few others that are worthy of mention. The source that seems to have gotten the correct answer most concisely is rracecarr-ga at Google Answers, although you have to sift through a lot of other poster's nonsense to find rracecarr's clear responses. Cecil Adams at The Straight Dope has it correct in his second column on the subject, although he still seems uncomfortable with the answer. In other discusion at The Straight Dope, zut and Paradoxic also have it correct (and zut actually helped Cecil get it right).

In addition, zut adds some valid points about air entrained by the conveyer belt that will produce lift for the stationary plane. Give full credit to anyone who agrees that the plane will go nowhere until the windspeed produced by the moving conveyer belt is great enough to lift a stationary plane. At that point, the idealized conveyer will eventually travel fast enough to create sufficient airflow across the wings to lift the stationary plane. But what happens then? Well, the plane lifts off the runway with zero groundspeed. At that point, the wheels are no longer being accelerated by the conveyer belt, but they will keep spinning if they are frictionless. A skilled pilot should be able to gradually leave the entrained air stream near the conveyer belt and successfullly fly away—at least after a few tries. Even if the wheels have normal friction, but no degradation, it will take some time for the wheels to spin down, and our skilled pilot should be able to fly away. But the plane does not take off normally.

So, that's my final answer. The plane will not move forward while it is in contact with the conveyer belt. The belt accelerates rapidly in order to satisfy the constraints of the paradox. This produces a force on the plane which is transferred through the wheels' axles, and produces a torque which accelerates the wheels rapidly in concert with the conveyer belt. Eventually, the plane will be able to take off when the windspeed produced by the moving conveyer belt is great enough to lift the stationary plane. It's the pilot's job to keep the plane under control and gradually move out the layer of entrained air above the conveyer belt and fly away. That's probably impossible due to the turbulent nature of the flow, but we have an idealized conveyer belt with idealized wheels, so we'll give that capability to the pilot.

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Thanks,
Steve