"According to Paul J. Camp, a professor in the department of physics at Spelman College, it's all pretty simple. "At first, the conveyor will hold the plane still. But only to a certain point, after which, driven by thrust from its engines, the craft will accelerate."
But the problem clearly states: The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction.
"The key is in the behavior of friction," Camp says. "Friction is a peculiar force in that it has an upper limit. For instance, push an object on your desk, but not hard enough to move it. Why doesn't it move? Because the friction force exactly balances the force of your push. At some point you push hard enough to set the object in motion. This is the point where friction has topped out and is not capable of growing any larger."
With the airplane and treadmill, there is, at the outset, friction force capable of rotating the tires at the proper speed to keep the plane stationary. However, as the thrust is increased, that force eventually maxes out. (Two separate frictions are at play here, actually, one between the tires and belt, the other between the plane's axles/bearings and its wheels. The first will max out before the second.)
"And at that point the wheels no longer roll, they slide," says Camp. "Or rather, they roll and slide at the same time. Tire motion is now decoupled from the belt motion. No matter how much you whiz up the treadmill, you won't add any more rotational velocity to the wheels because friction is already doing everything it is capable of. The plane skids toward takeoff -- likely accompanied by much smoke and a powerful rubbery stink."
And there you have it, at least on paper. Bear in mind that for a plane to reach that point of decoupling would require two things above and beyond the pale of normal engineering. First, a remarkable amount of power -- far more than any jetliner, and probably any military plane, is capable of developing. The illustration on Pogue's blog is of an Airbus A320; some sort of rocket plane would be more appropriate. Second, no existing aircraft tires could take such abuse. The rotational velocity required before reaching the friction limit would have them bursting within seconds, causing the plane to be flung backward. Believe it or not, landing gear isn't engineered with giant treadmills in mind, and pilots need to adhere to maximum groundspeed limits, lest their tires wind up like this. These limits occasionally present problems during tailwind operations or in the case of flap and slat malfunctions -- scenarios dictating the need for unusually high takeoff or landing speeds.
For good measure, the treadmill itself, as described, could never be built. It can't "exactly match the speed of the wheels," because the wheels will turn at the speed of the treadmill plus the speed of the plane relative to the ground. When the speed of the plane is greater than zero (which it is the moment its wheels start to spin; otherwise they would never move), then the problem becomes impossible. By definition, the wheels have to be turning faster than the treadmill. "
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