Wednesday, December 05, 2007

The Answer to the Puzzle Question

Yesterday's question was: Imagine a plane is sitting on a massive conveyor belt, as wide and as long as a runway, and intends to take off. 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?

On first encounter this question, which has been showing up all over the Net, seems inane because the answer seems so obvious. However, as with the infamous Monty-Hall-three-doors-and-one-prize-problem (see The Straight Dope: "On Let's Make a Deal" you pick Door#1, 02-Nov-1990), the obvious answer is wrong, and you, Berj, are right--the plane takes off normally, with no need to specify frictionless wheels or anyother such foolishness. You're also right that the question is often worded badly, leading to confusion, arguments, etc. In short, we've got a topic screaming for the Straight Dope.

First the obvious-but-wrong answer. The unwary tend to reason by analogy to a car on a conveyor belt--if the conveyor moves backward at the same rate that the car's wheels rotate forward, the net result is that the car remains stationary. An aircraft in the same situation, they figure, would stay planted on the ground, since there'd be no air rushing over the wings to give it lift.

But of course cars and planes don't work the same way. A car's wheels are its means of propulsion--they push the road backwards (relatively speaking), and the car moves forward. In contrast, a plane's wheels aren't motorized; their purpose is to reduce friction during takeoff (and add it, by braking, when landing). What gets a plane moving are its propellers or jet turbines, which shove the air backward and thereby impel the plane forward. What the wheels, conveyor belt, etc, are up to is largely irrelevant.

Let me repeat: Once the pilot fires up the engines, the plane moves forward at pretty much the usual speed relative to the ground--and more importantly the air--regardless of how fast thec onveyor belt is moving backward. This generates lift on the wings, and the plane takes off. All the conveyor belt does is, as you correctly conclude, make the plane's wheels spin madly. A thought experiment commonly cited in discussions of this question is to imagine you're standing on a health-club treadmill in rollerblades while holding a rope attached to the wall in front of you. The treadmill starts; simultaneously you begin to haul in the rope. Although you'll have to overcome some initial friction tugging you backward, in short order you'll be able to pull yourself forward easily.

As you point out, one problem here is the wording of the question. Your version straightforwardly states that the conveyor moves backward at the same rate that the plane moves forward. If the plane's forward speed is 100 miles per hour, the conveyor rolls 100 MPH backward, and the wheels rotate at 200 MPH. Assuming you've got Indy-car-quality tires and wheel bearings, no problem. However, some versions put matters this way: "The conveyer belt is designed to exactly match the speed of the wheels at any given time, moving inthe opposite direction of rotation." This language leads to a paradox: If the plane moves forward at 5MPH, then its wheels will do likewise, and the treadmill will go 5 MPH backward. But if the treadmill is going 5 MPH backward, then the wheels are really turning 10 MPH forward. But if the wheels are going 10MPH forward . . . Soon the foolish have persuaded themselves that the treadmill must operate at infinite speed. Nonsense. The question thus stated asks the impossible -- simply put, that A = A + 5 -- and so cannot be framed in this way. Everything clear now? Maybe not. But believe this: The plane takes off.

Want an slightly easier answer (surprisingly, from an engineer!):
Bill is a pilot, with multiple engineering degrees,andowns a Cessna T210. "The truth is that, if you could build such a massivec onveyor belt, the plane would want to leave the ground sooner. The deal is that the air next to the conveyor/runway will move backward very near the surface with the effect being reduced as you get further away (higher). As an example, with the airplane moving forward at 40 knots and the conveyor moving backward at 40 knots, the air in the area between the conveyor and the wing will be moving backward at something less than 40 knots but more than 0. If we split the difference and say the air near the wing is being moved backward at 20 knots by the conveyor, the effective airspeed is 60 knots and the (light) plane is already flying.This effect is not imaginary. Remember how much effect there is on the performance of the plane in ground effect which starts half a wingspan above the ground!"

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