Last month, New York Times technology columnist David Pogue set the Web abuzz by rekindling an old brainteaser about whether a theoretical airplane would be able to take off from a theoretical treadmill. The puzzle was “ripping around the Internet” (Pogue’s words), and appeals for clarification quickly reached Ask the Pilot’s in box. Will it or won’t it fly, people wanted to know, imploring me to weigh in.
Belatedly, and grudgingly, I will now do so. Such topics tend to induce the rapid closure of my eyelids, and while I’d like to tell you this is the kind of shop talk that keeps aviators engaged and alert in those quiet midnight hours high above the ocean, nothing could be further from the truth. (Mostly they’re just bemoaning the loss of their pensions and talking about movies.) Nevertheless, here goes …
“Imagine a plane is sitting on a massive conveyor belt,” poses Pogue’s Dec. 11 Times blog, “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?”
When I last checked, more than 860 people had posted their opinions, split about 50/50 between those who say the airplane will fly and those who insist it can’t. If you look carefully you can locate my own contribution, flatly declaring that no, absolutely not, the aircraft will not get off the ground. “If this is truly ‘ripping around the Internet,’” I snarked, “then heaven help us. The plane will not fly. Of course it won’t fly.”
And why should it? How can it fly if it’s not moving? For an aircraft to get and stay aloft, it needs lift; it needs air passing above and below its wings. And for that it needs to move. For a cursory lesson on how this works, simply shove your arm out the window of a speeding car. Shape your hand into an approximation of a wing, angle it slightly into the oncoming wind, and voilà, it’s flying.
Now, imagine you are in that same car, on a treadmill. The car’s wheels are spinning — be it at 60 mph or 600 mph — but when you put your hand out the window, does it rise up? Of course not. Your hand won’t fly, and the plane won’t fly either for exactly the same reason: because for all its efforts, the vehicle isn’t moving. You have zero relative speed and zero lift.
This seemed so obvious that it needed a caveat: “On the other hand, if you were able to generate a tremendous enough amount of thrust,” I noted in a follow-up post, “and redirect the vector of that thrust downward, you could, conceivably, lift the plane off like a rocket. Heck, you can make anything ‘fly’ if you stick enough power under it. But that isn’t fair to the spirit of the premise.”
Except, wait a minute, what is the premise?
Go back and read it. “Imagine a plane is sitting on a massive conveyor belt,” it says, “as wide and as long as a runway.” I’d glanced right over those key words, “as long as a runway.” I was so caught up in the image of a motionless plane on a regular old treadmill — like the kind you might see at the gym — that I missed the whole question. Looking back, that does seem a dull and senseless riddle: Can a plane fly if it can’t move? Obviously not. There has to be more to it.
And there is. At heart, this has nothing to do with the principles of lift but, rather, with those of friction and acceleration. The gist of the question is better understood as follows: Will an airplane, under its own power, remain motionless on a 10,000-foot-long treadmill, or will it roll forward? Will it accelerate and fly?
Turns out the answer is yes. A distinctly theoretical yes, for reasons we’ll get to shortly, but for all intents and purposes of the puzzle, that’s yes enough.
A car won’t accelerate on a treadmill; the belt will always match the rotation of the tires. You will not accelerate on a treadmill; the belt always runs in sync with your footfalls. But an airplane is different. An airplane’s wheels are not powered by gears or a drive train. They hang inertly below, and are free-spinning. The thrust force that moves the plane along couldn’t care less about the ground. The engines are not fighting against the surface, they are fighting against the air.
Confusing, I know, and in the interest of full disclosure, physics was the one class in high school that I outright failed and had to take twice (you try ciphering out equations while listening to Minor Threat on a pair of clandestinely strung ear buds). And some of you might remember what happened the last time I combined things aeronautical and mathematical. So let’s get somebody else to explain.
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.
Whose idea was this crazy problem?
Meanwhile I can’t decide if this is good or bad news for the conveyor belt industry and treadmill enthusiasts worldwide. Though, as they say, any publicity is good publicity.
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