Plane on a Treadmill, redux

Cecil's Columns/Staff Reports
121

That’s completely reasonable. However, note that Cecil does give an alternate phrasing of the question at the end of his first column. In addition, some other people have argued “plane speed” could very well mean “speed with respect to the treadmill”–that’s sort of reasonable if you think of looking out of the bomb bay doors, for example, at the treadmill/runway underneath. But that’s other people; your argument that “plane speed” should refer to “fuselage speed with respect to the ground” is reasonable.

Exactly!

IF the wheel assembly can minimally withstand the forces required, then the plane will remain stationary, right? As long as the plane is stationary, the “wheel speed” and the “belt speed” are the same. In order to keep the plane stationary, the belt has to accelerate constantly, but it doesn’t have to instantly go to infinite speeds as Cecil said in his first column.

122

I wanted to believe. I really did. I sat here for ten minutes thinking, my god, he’s right. Then reality set in.

Think about it this way. The wheel is attached to the axle of the plane. The plane moves forward, th axle moves forward and as a result the wheel rotates around the axle. If the ground is stationary, and the axle moves forward 1 foot in one second, then the outer diameter of the tire, in contact with the non moving ground, will also move forward 1 foot in one second. If you had put a mark under the tire on the runway when you started and another under the tire when you stopped, the distance between the marks would be 1 foot.

Enter the treadmill.

If the treadmill is to move the same speed as the tire, then it must move backward 1 foot in 1 second. Place your marks. The axle moves forward 1 foot in 1 second, the treadmill moves backward 1 foot in 1 second, and the distance between the marks is 2 feet. The outer diameter of the tire, in contact with the treadmill, must have moved 2 feet in 1 second. Oops. We didn’t move the treadmill fast enought. Kick the treadmill up to 2 feet in a second and the marks are … um… three feet apart. Gotta go faster. See the problem? Infinite acceleration of the treadmill.

123

In the wheel speed paradox, the conveyor is trying to match the speed of the wheels as determined by the acceleration of the plane.

In BR#2 the conveyor does not attempt to match the wheel’s speed as determined by the plane’s acceleration. The treadmill arbitrarily accelerates to create enough drag on the assembly to counteract the engine’s thrust. We also accept and ignore the forces on the tire and wheel, which should tear them apart, and the fact that a treadmill that acellerates at a constant speed indefinitely will reach infinite (and theoretically impossible) speeds. It is simply a fantasy construct, and I am willing to accept it as such.

Very different.

124

Make that accelerates at a constant rate.

125

Error can not compute. Tires are no where mentioned in the question. Error can not compute.

IT says the speed of the plane. the only relevant SPEED of the plane is airspeed since THAT is what determines if it will take off or not.

Answer barring the other issues I raised the airplane will move and will take off.

Chris Taylor
http://www.nerys.com/

126

I hate nitpicking myself, and this is the last time I’ll ever do it. Just accept that I work hard and reply fast.

I meant to say the outer circumference of the tire is in contact with the ground/treadmill. Not the diameter.

127

You said:

I used tire because that’s the part of the wheel that touches the ground. Wheel consiting of a tire and a hub. I see several mentions of the wheel in your post.

If the treadmill is to match and oppose the speed of the planes wheels and not the speed of the plane then we have a situation requiring infinite acceleration. The heavens will rend asunder and suck up all of creation before the plane gets a chance to take off.

If the wheels are going 10 and the treadmill is going 10 then the plane is stationary, which I reject.

128

If the conveyor tries to match the speed of the wheels, it will increase acceleration until it transmits enough force (through the acceleration of the wheels) to counteract the thrust of the plane’s engines. By adjusting the belt acceleration (ignoring the forces on the tire and wheel, which should tear them apart, as you say), you can keep the plane stationary, which keeps the conveyor speed and the wheel circumference speed matched.

BR#2 isn’t different from the “wheel speed paradox” (which isn’t a paradox at all, really)–it’s the solution to the wheel speed paradox.

That’s not true, unless you assume completely massless wheels and/or no frictional coupling between the plane and the treadmill.

You do realize that you’re arguing semantics, don’t you?

129

The original question states “This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction).” Not almost the same, not converging to the same, but exactly the same. As I tried to illustrate in post #122, if one substitutes plane speed with wheel speed, the treadmill will never go fast enough, because of the treadmill! PARADOX! But, if we change the question, then we can get to BR#2.

Totally different. The wheels are turning faster than the treadmill, the treadmill does not care, and the plane does not move.

130

First problem the result might be infinite acceleration but you forget something - I dont care how fast the conveyor is going since as soon as the plane hits take off speed it wont be TOUCHING the conveyor anymore and it is free to accelerate to whatever speed it wants. but the DELTA of this acceleration is not infinite. it is finite based on the airplanes acceleration and it will not reach a sufficient force to stop the airplane from taking of before the airplane reaches take off speed (about 20-30 seconds)

So either way the plane takes off.

Semantics ? your kidding right. Semantics is arguing over whether something is Melon Red or Brick Red.

issues that FUNDIMENTALL alter the discussion at hand (speed of the plane or speed of the wheels) are not semantics :slight_smile:

and its a mute discussion since the question clearly states speed of the plane which means airspeed.

Chris Taylor
http://www.nerys.com/

131

Yes it does. the tread mill can never produce enough friction in the time that it has. if the plane does not move then the wheels never move and therfore the treadmill never moves

an average plane can reach take off speed in a matter of seconds… in theory if the treadmill speed were altered to be that of the WHEELS “percieved” rotational speed yes the theoretical top speed would be infinite but its not 0-infinite its on an acceleartion curve and the plane will attain flight speed before this curve gets to the point of producing enough drag to keep the airplane on the ground.

and this is easily solved. I just put skis on my plane at which point your treadmill once again becomes irrelevant.

(no where in the questions does it say this plane has wheels so…)

Chris Taylor
http://www.nerys.com/

132

You lose me at “either way”. Matching wheel speed sets up an impossible scenario.

Well, I’ll buy into matching ground or air speed. The airplane will not depart until the necessary indicated air speed is reached. We are basically in agreement there.

133

That was in response to BR#2 which I previously stated as false for the “stationary” aspect in post #111, but I was willing to move on from there because BR#2 is a fantastical solution. I agree, the plane must move for the treadmill to start.

Takeoff speed in a matter of seconds? Certainly more than an inconsequential number of seconds. Feels like days. You say the treadmill can’t. I say it most certainly can. This treadmill can move faster than the speed of light in a nano-second. This is the perfect treadmill. This treadmill is God (wrong thread).

OK, we keep saying treadmill. The question states “(some kind of band conveyor)”. Ok, so it doesn’t say some kind of band conveyor covered in grease / made of ice, but then again it doesn’t exclude that either. Just to be on the safe side, I’ll stick with wheels.

134

Every Plane I have been in from Cessana’s to Airliners to C130’s was able to go from Standstill to airborne in under 30 seconds.

No matter how fast the conveyor can go its acceleration is LINKED TO and THERFORE LIMITED TO that of the airplane and its wheels.

If you lock its speed to the wheels one of 2 things will happen.

1 the airplane will move forward and leave the ground before the wheels fail from friction.

2 the airplane will move forward the wheels will fail leaving the rims on the conveyor and still almost certaintly be able to leave the ground (since now the frction will be lower and the wheels might even stop spinning if they jam up and now the conveyor will also STOP since the wheels have stopped) either way the plane has a VERY good chance of getting of the ground (a JET might not since they tend to be heavier but a fighter or a propeller driven airplane would almost certaintly still rise from the ground.

Either way the only thing limiting the planes ability to Take Off and fly is the “unknown” variables of the Conveyor. where does the boundry layer the conveyor create stop (and it will create a nice breeze mind you)

in fact even in your scenario of where the plane can someone produce enough friction against this ever faster conveyor the plane WILL STILL take from even if stationary since the conveyor itself will generate the airflow needed to fly the airplane. (no reality excemption here the conveyor WILL move a layer of air above it. if its frictioneless enough not to move the air its frictionless enough to be NO hinderance to the airplane taking off no matter how fast it goes.

No matter how you look at it the plane takes off because it is NOT affected by the conveyor for is thrust.

Chris Taylor
http://www.nerys.com/

135

Um… BR#2 does not lock the speed of the conveyor to the wheels.

BR#2 simply states that IF the conveyor can produce enough force against the axle of the wheel assembly to counteract the thrust of the powerplant, the plane will not move. Cecil then admits that this treadmill has some serious problems, summarily rejects that reality and substitutes his own.

So you see, boys and girls, in Cecil Land (or is it Cecil World; I aways get those two mixed up) there is a treadmill that can stop a plane from taking off.

I’ll buy that for a dollar.

136

What if the plane is a jump jet?

137

Define “wheel speed.” I was defining “wheel speed” as “the speed of the circumference of the wheel with respect to the axle.” In that case, the treadmill speed and the “wheel speed” are identical if and only if the plane doesn’t move with respect to the ground. That’s where this whole BR#2 comes from; it’s not some completely sperate problem that Cecil threw in to confuse people. It’s a condition that arises naturally from extending the original thought experiment.

Again, for the fourth time, it depends on your assumptions about the problem. If you assume the conveyor matches the fuselage speed of the plane with respect to the ground, as you’re doing, the plane will take off. Absolutely. In fact, so far as I can tell, no one disagrees with that at all.

For the third time, there are different versions of the question which are worded different ways. Any particular wording can also be interpreted different ways. Of course the interpretation is fundamental to the problem, but your argument is basically, “my interpretation is correct and no other interpretations are allowed because planes are equipped with an airspeed sensor.” That’s semantics.

138

I suppose I should repost something again from the previous thread:

There are plenty of answers to this question, because the key to the question is what you assume from the beginning. The assumptions are the key. There are no assumptions that are particularly wrong, but you need to be clear up front what assumptions you’re making, and how you’re interpreting the problem. Let’s start off at the top:

A. Suppose we actually built a treadmill and put a 747 on it, and had the treadmill match the speed of the plane. Would the 747 take off? If “exactly matching the speed of the plane” means that the treadmill matches the speed of the fuselage with respect to the ground, then yes. The treadmill simply accelerates in the opposite direction that the plane does. The wheels wind up rotating twice as fast as they normally would (and it’s possible the tires might blow from overspeeding), but the plane will take off, leaving a treadmill behind that’s rotating in the opposite direction.

B. But that problem is trivial. Let’s assume that the speed matching means the treadmill matches the speed of the tire circumference with respect to the hub, like at the end of Cecil’s first column. Would the 747 take off? Almost certainly it would, but only because we can’t build a treadmill capable of keeping up with the thrust transmitted to the plane by the engines–in other words, we violate the spirit of the question, because the treadmill isn’t matching the wheel velocity.

C. OK, that’s stupid. It’s a thought experiment. Posit a magic treadmill that can accelerate as fast as desired. And it doesn’t break. Hum. Well, it’s possible the wheels will skid on the treadmill, because the friction wouldn’t be able to transmit the necessary force. Depends on the total thrust of the plane, I guess. If that’s the that case, we again violate the spirit of the question, and–

D. It’s a thought experiment, smart guy. Assume there’s enough friction to rotate the tires as much as we want. All right. When the engine lights off, the treadmill will accelerate until the force transmitted through the wheel hub to the plane (from inertial wheel acceleration, friction, and perhaps tire hysteresis) exactly balances the thrust. The plane will stay stationary (with respect to the ground) as the treadmill (and wheels) are accelerated to ever-increasing speeds. Eventually, either the bearings would overheat, the tires would blow, or the wheel would rip itself apart due to inertial forces. After that, the plane crashes and burns. Then you’ve destroyed a rather expensive magic treadmill.

E. Thought experiment, I said! Let’s posit ultra-strong and heat resistant tires. All right. It turns out the real world is rather complicated. If the treadmill is a long, runway-sized treadmill, it will eventually, running thousands of miles an hour, pull in air at high enough velocity that the plane will lift off at zero ground speed (but substantial air speed). However, now you’re running into trans-sonic compressibility effects…

F. No speed of sound effects! And assume magic air that doesn’t become entrained with the treadmill motion. And don’t throw in any other crazy stuff, either. In that case, the treadmill speeds up (still balancing the plane’s thrust force) and the plane stays in place until the engines run out of fuel. I imagine the treadmill goes pretty fast at that point. The plane stays put until the fuel’s gone, at which point the magic treadmill whips it backwards.

G. Backwards, shmackwards. Now we’re getting somewhere. What if we had infinite fuel? Then the wheels keep going until they’re running near light speed, and relativistic effects take over. The wheels get smaller, I suppose…

H. None of that! No relativity-- Hey, wait a minute. Back up. Suppose we have zero friction bearings and tires. That doesn’t seem so unreasonable for a thought experiment. Well, zero friction tires would mean they just slip on the runway like skis, since nothing turns them. So the plane will take off, tires motionless, and the treadmill won’t move.

I. Hey! Quit it! I already said the tires don’t skid! Sorry. Just friction on the tire/treadmill interface, then, but none in the bearing or sidewall. With zero friction in the bearing, you lose the friction coupling between the treadmill and the jet. But you still have inertial coupling. The jet power goes into accelerating the wheels, and you have the same case as you do with friction. The jet stays stationary as the wheel accelerates; the wheel just accelerates faster.

J. Well, how about the other way around? Massless wheels, but you still have friction? Here it starts to get complex. As you accelerate the wheels, the bearings will change shape and heat up and so forth, so it’s reasonable to guess that the “friction coefficient” goes up with increasing speed. If that’s the case, then when the engines start, the treadmill accelerates up to whatever speed will give enough friction to balance the thrust. The plane stays stationary, wheels rotating at some reasonably constant (but large) velocity, dissapating the engine power through friction.

K. But I want massless wheels and a constant coefficient of friction. Indestructable wheels, remember? None of this hand-waving “it’s gonna get bigger” crap. OK. It is a thought experiment. With a limited “friction coefficient,” only a limited amount of energy can be absorbed by the friction. When the engine lights off, the treadmill instantly accelerates to infinite speed. It’s never able to counteract the thrust force, and thus plane takes off, leaving the infinite-speed treadmill behind.

L. Ah. OK, one last step. What if we had no bearing friction and massless tires? What happens then? Pretty much the same thing. There’s now no energy losses in the wheels and tires, no coupling between the treadmill and the plane–no bearing friction, no inertial effects, no air resistance, and no way for the treadmill to affect the plane’s motion. The same thing would happen as above, with the plane taking off, leaving the infinite-speed treadmill behind. However, there’s one added interesting thing: This is now an unstable runaway system. There’s no resistance to treadmill motion, and a positive feedback circuit. Imagine the poor mechanic who bumps a wheel, setting it in motion. A very slight roll by the tire is sensed, and the treadmill luches forward. The tire goes faster, the treadmill goes faster, the tire goes faster… Since we’ve posited an instantly-accelerating treadmill and no relativity and no air resistance and no wheel inertia, the treadmill goes from zero to infinity in no time flat. Try to keep your balance on that.

Pick your scenario–they’re all correct.

139

Part of the the problem is that the question is not only interpreted two ways. The wording has changed from the original question from speed of the wheels to speed of the plane.

The question Cecil answered

A subtle difference. But enough to certainly change how you look at the problem.

140

Even given this wording, it is possible to imagine a simpler solution. Picture, instead, a treadmill that is driven by the wheels—a well-greased treadmill poised in equilibrium, which reacts to anything moving on top of it by sliding in the opposite direction. A car on this treadmill won’t move, since the car’s wheels will just push the belt out from under the car. The same will happen with anything that pushes against the ground to move, like a runner. But a plane’s wheels won’t turn on this treadmill; they’ll just drag the treadmill along with them. The “speed of the wheels” is always zero, until the plane takes off.

This meets the constraints of the original problem, requiring only a different assumption about how the treadmill will “match the speed.” By removing motive power from from the treadmill it eliminates a lot of complexity.