Finally, An Answer

Several years ago, I posted a problem describing a plane on a massive conveyor belt:
From www.boingboing.net

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

Similar, but not exact, question was busted on Mythbusters:
From mythbustersresults.com:

An airplane cannot take off from a runway which is moving backwards (like a treadmill) at a speed equal to its normal ground speed during takeoff.

Every since that episode aired, I had serious doubts about the validity of the test; although I could never fully articulate those doubts, even to myself. The biggest issues I had was the speed at which the conveyor belt (or treadmill) was moving.

And that’s the rub. The first question posits that the “belt is designed to exactly match the speed of the wheels, moving in the opposite direction,” while the second questions says “a speed equal to its normal ground speed“. These, my friends, are two entirely different questions.

I thought all hope had been lost, until Randall “xkcd” Munroe became my hero. He asks the version of question I originally posted a couple years ago. It’s actually almost the exact same wording, only he adds in a bit about it being a 747 and then provides an answer:

From blag.xkcd.com:

The practical answer is “yes”. A 747’s engines produce a quarter of a million pounds of thrust. That is, each engine is powerful enough to launch a brachiosaurus straight up (see diagram). With that kind of force, no matter what’s happening to the treadmill and wheels, the plane is going to move forward and take off.

But there’s a problem. Let’s take a look at the statement “The conveyor belt is designed to exactly match the speed of the wheels”. What does that mean?

You think you have it all wrapped up in a nice little package, and then you get to the “But there’s a problem.” And you utter to yourself, “Crap.”

I’m not going to spoil Randal’s rather excellent explanation of the problem. However, he does do a pretty good job explaining it, and you really should read it.

I will, however, add a couple of footnotes:

  • A PID controller is a Proportional, Integral, Derivative controller, which is a type of feedback controller. For example, let’s say you’re running on a treadmill and you start running faster, a PID controller can measure the speed you’re running at and automatically increase the speed of the treadmill so that you don’t run off the end.
  • I’m not sure, but I believe the “‘JetBlue’ scenario” that Randall mentions may refer to the JetBlue Flight 292 incident of a few years ago.

If you also remember the discussion from last time, I think Chris Barnhart is the winner.

P.S. Mr. Munroe claims that xkcd doesn’t stand for anything. However, if you assign each letter a number (A=1, B=2, C=3, etc), X+K+C+D => 24+11+3+4 = 42. Check that out, you can’t make up that shit. And no, I didn’t figure that out all by myself.

Image Credit: Randall Munroe

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