The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!' but 'That's funny...' --Isaac Asimov
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Airplanes and Treadmills and bloggers Oh My!

September 26th, 2008 by eric

Apparently I am the last person on earth to have become acquainted with the … Airplane on the … Treadmill problem, which is often stated thus:

"Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway. The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the plane take off?"

Think about it…

a little longer…


David Pogue at the NYT says “no, because the plane will not move relative to the ground and air, and thus, very little air will flow over the wings. However, other people are convinced that since the wheels of a plane are free spinning, and not powered by the engines, and the engines provide thrust against the air, that somehow that makes a difference and air will flow over the wing.”

Admittedly, these were my first thought as well, and, admittedly, it seemed to me that anyone who didn’t get that was a moron. Briefly. I had to read several explanations before I finally hit on the misconception that I had been holding in my head, which was that it was somehow possible to build a magical conveyor belt that would stop the aircraft from moving forward. That’s how I read it, and in one interpretation this is what the problem states, but no such conveyor belt can exist, for various reasons. If such a conveyor belt could exist then it is obvious (if you know what creates the lift) that with no forward velocity relative to the air then the plane cannot take off, no matter how hard its engines are pushing. I was focused on the “taking off” part and not on the “forward motion” part. In any semi-realistic system (e.g. the conveyor belt does not move at the speed of light, the rubber and tarmac have non-infinite coefficients of friction) the thrust from the engines could overcome the extra friction from the conveyor belt and therefore accelerate the aircraft, meaning that the plane would take off anyway, barring mechanical failure.

After you recognize the physical impossibility of the situation, the right ‘answer’ becomes a question about what you want to get out of the ‘thought puzzle’. If the point is to demonstrate that on an airplane the lift is provided by relative motion of the wings through the air (i.e. an airplane is not a rocket [1]), then you don’t need a conveyor belt, you can just chain the imaginary plane to an imaginary wall. If the point is to demonstrate that on an airplane thrust is provided by the jet engines acting on the air and not by driving the wheels (i.e. an airplane is not a car), then you don’t need a conveyor belt either, you just need to give the imaginary plane and pair of imaginary skis. If you just want to confuse people, then go ahead and continue to ‘analyze’ the impossible situation in ever-increasingly-minute unrealistic detail. Otherwise, you can restate the poorly devised problem. I found that restating it in the following way made it easier for me to understand where my misconception was:

Imagine an airplane on a runway.  Is it possible for the ground to exert enough backwards force to overcome the forward force of the jet engines?

Then you can draw the simple free body diagram and show that static friction is the only resisting force if the plane is stopped. (If the plane starts moving, other forces like rolling friction and air resistance also come into play). When the wheels do not rotate the question might become:

Imagine the pilot of an airplane cranks the engines to 100% on takeoff but forgets to turn off the parking brake.  Can the plane take off?

Which is a question that can be answered by engineering or experiment, as opposed to the imaginary conveyor belt “designed to exactly match the speed of the wheels”.

What would happen in this case? I don’t know! I imagine the thrust would be great enough to 1) overcome the static friction between the rubber wheels and the ground, so that the plane would start sliding, or 2) break whatever mechanism the parking brake relies upon, releasing the wheels to roll. In the first case, the wheels would probably melt from the frictional heat generated by dragging them across the tarmac, and the plane would crash. In the second case, the wheels would roll normally and the plane would take off as normal.

Nevertheless, the result is that this last question is answerable in reality, whereas the original ‘puzzle’ is not. My main point in raising the whole debate (debacle?) is because I think it is useful to realize that even the scientifically trained, the science educators, the alleged ‘smart people’, can and do hold misconceptions about the way things work, and that those misconceptions are sometimes hard to spot in isolation, without an external viewpoint. This is especially true when the problem is inexactly or confusingly stated, and double-plus especially true when we talk to people who are not scientifically trained, or who may not have the background to solve the problem in the first place. In those cases we need to be particularly exact, exceedingly clear, and strive to help identify their misconceptions without condescension or appeals to authority. In my case, I understood that the plane will just take off in my brain before I believed it in my gut, but it’s the gut that needs to be reached if the lesson is to be retained for any length of time. Brains are notoriously easy to manipulate. Instincts are harder to change.

[1] Actually, commercial jets are similar to horizontal rockets…

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