Plane on a Treadmill, redux

Cecil's Columns/Staff Reports
101

You have your scenario wrong man. Let me try to explain.

an bug as 1 speed. Ground speed.
and airplane has TWO speeds Ground and AIR speed

Place both on a Conveyor as described (note this is impossible since all speeds MUST be 0mph in this quesion IF we are talking about GROUND force dependant items such as cars and bugs I will explain this if you desire)

Now back to the bug and the airplane. on the conveyor its not the same

the bug now has TWO speeds Ground speed and Conveyor speed

the airplane now has THREE speeds Ground Air and Conveyor speed

for an airplane to fly ground speed is utterly irrelevant. the only reason ground speed is an issue for airplanes is that you need groundspeed to translate from point a to point b in the X and Y axis.

so if the bug was moving its FEET at 10mph force (note the careful wording) the conveyor moved backwards at 10mph the bugs speed would be as follows

Feet Speed 10mph Conveyor speed -10mph ground speed 0mph

the airplane has no “feed speed” as it has no feet (locomotive feet)

it thrust forward at 10mph the ground moves AWAY at 10mph the conveyor picks up to 10mph backwards. this with the excemtion of bearing drag at this speed does nothing to inhibit the AIRSPEED of the airplane and since the air is calm it also does not inhibit the ground speed (since in calm air air and ground speed for an airplane are EQUAL) what does change is something that is normally irrelevant to an airplane the speed of its WHEELS they would be going 20mph since the airplane is moving FORWARD at 10mph (not thrust its actually MOVING because of that thrust)

NOW the way to make your answer correct would be to make the CONVEYOR an AIR TUNNEL

if you put an airplane in an airtunnel if it was flying forward at 10mph and the windtunnel blew air back at 10mph it would have a ground speed of ZERO and an airspeed of of 10mph

but this can not work as worded just like this puzzle can not be used for cars.

you would have to chance Same speed as the airplane to same THRUST as the airplane.

speed and thrust for an airplane are INDEPENDANT

speed and thrust for a car are always EQUAL (except when on a treadmill)

you could never do this for a car or bug for the following reason

if the car is at 0 the treadmill is at 0

if the car tried to move forward the puzzle says that the treadmill will react in perfect sync in the opposite direction

but if it reacts perfectly in the opposite direction the car NEVER attain a forward speed but if its forward speed is 0 the treadmill speed must also be zero

this is called a paradox. it CAN NOT exists its simply impossible.

the only way to rememdy the paradox is to say the treadmill will always match the speed of the cars WHEELS.

NOW you have a problem that can be solved. the ground speed of the car will always be zero while the speed speed will always be whatever speed the speedometer says. the conveyor speed will always be the same as the spedometer speed and the ground speed will always be zero since the conveyor speed will always cancel out the speedometer speed.

This puzzle works because it is comparing the speed of INDEPENDANT variables.

PLANES speed (not air speed not wheels speed not THRUST force but the actual forward speed of the airplane) with wheel speed (conveyor speed) which is irrelevant to an airplanes airspeed.

Think of an airplane on the ground as HOVERING. the ground is supplying the hovering energy. as the airspeed increases LIFT takes over the job of lifing the airplane up instead of the ground. Once the lift exceeds in upward force the mass of the airplane it moves on the Z access away from the ground. simple.

the airplanes propeller is thrusting against the AIR not against the conveyor so the conveyor is irrelevant if we ignore the other reality variables.

Here is how you can test this yourself. the only way the conveyor can have ANY effect on the airplane is if the airplane must PUSH AGAINST the conveyor.

so if I can prove that an airplane does not need to push against a flat plane (ground or conveyor) at all in order to move forward I can prove the conveyor will not stop the plane from moving forward.

Easy.

Get yourself a balsa $2 Windup airplane. WInd it up and point it straight up. now let it go. DOnt throw it just point it up and release it getting your hands out of the way once you let it go.

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

102

Let me try this analogy one more time. This is actually an easy system to envision wherein the speed of the conveyor will match the speed of the plane (relative to the ground). Not the wheels. [sub]whateverintheheckthatmeans[/sub]

This is for BR#1 (basic realization #1) To understand this, you must understand that the wheels of an airplane ‘freewheel’ and have nothing to do with accelerating the plane.

One thing that seems to be hard for some folks to wrap their brains around is how a conveyor automatically and exactly matches the speed of the plane.

Here are two simple ways to look at it-

#1

Place our plane on the tread mill, a runway length treadmill.

From the rear of the plane, attach a cable from the rear of the plane, and attach it to the roller at the beginning of the runway/conveyor behind the plane. Run the cable UNDER the roller and have enough excess so that you can wrap a few thousand feet around the roller.

Fire up the plane. As the plane moves forward (relative to the ground) the cable will turn the roller on the treadmill and pull it at exactly the same speed the plane moves forward in the oposite direction the plane is moving. The plane WILL move forward relative to the ground.

We have an exact match of conveyor and plane speed. And the plane is moving relative to the ground and the air around it (not just the speed realive to the conveyor).

We have a plane moving down a ‘runway’ with a lot of resistance because it’s pulling the runway beneath it, and the wheels are spinning twice as fast, but we have a plane moving. And the plane is moving relative to the ground and air. Which means air speed and flight. (no tail winds OK?)

Simple envision #2

We can do the same thing with a system on the ground that accelerates the conveyor at the same speed that the plane is moving.

To visualize it, we will use a powerful truck on the ground. The plane sits on the conveyor. Now, attach a cable from the back of the truck to the underside of the rear roller on the treadmill.

Fire up the plane and the truck. As the plane begins to move on the conveyor, have the truck simply keep up with the plane. The Truck pulls the roller in the opposite direction that the plane is moving. Since the truck (on the ground) and the plane on the conveyor/runway are moving at the same speed, and the truck is pulling the conveyor backwards (because the cable is wrapped underneath the roller) we know that the conveyor is moving backwards at the same speed that the plane is moving forward.

And we know that the plane is moving forward relative to the ground, because the truck and the plane are matching speeds. If the truck is moving on the ground, and the plane is going at the same speed as the truck, the plane is moving relative to the ground and air, just like the truck is.

If the plane is moving forward, it will achieve lift.

103

Well, a couple things: First of all, there are different versions of this question, both in Cecil’s columns and in other places. The different versions are phrased differently, and imply different things as far as what velocity is being matched to the treadmill–the speed of the plane, the speed of the wheels, and so forth. Second, one could interpret the “speed of the airplane” as being the speed with respect to the treadmill as opposed to the speed with respect to the ground.

Granted, this is a semantic argument, but the point is that you need to be clear about just what it is that you’re assuming up front. Nothing wrong with assuming that the treadmill matches the speed of the plane with respect to the ground, as you do, However, alternate interpretations are hardly “not possible” or “woefully mistaken.”

Yes assumptions. There are two fundamentally different ways of interpreting this problem–you’re interpreting it in one way, and nerys (for example) is interpreting it in another way. Neither way is clearly “correct” or “incorrect,” they’re just different assumptions about the structure of the problem.

Mmmm… so far as I know, no one is making this particular argument. Certainly no one is conceiving of an “imaginary infinite resistance.” If you’d like a more detailed explanation of the physics, pick a particlar point and ask.

You’re mixing up you interpretations here. Again, nothing wrong with that particularly, but I should point out that when you’re analyzing the case of the car on the treadmill, you’re requiring the treadmill to keep the car stationary with respect to the ground–the interpretation you said was “not possible” when analyzing the airplane.

104

“Well, a couple things: First of all, there are different versions of this question, both in Cecil’s columns and in other places. The different versions are phrased differently, and imply different things as far as what velocity is being matched to the treadmill–the speed of the plane, the speed of the wheels, and so forth. Second, one could interpret the “speed of the airplane” as being the speed with respect to the treadmill as opposed to the speed with respect to the ground.”
Actually its pretty irrelevant. it does not matter what the conveyor matches speed with or which direction it goes. Aircraft thrust is independent of the surface the aircraft sits atop so unless is physically keeps the airplane from moving the airplane will move forward and it will take off (barring other issues)

“Granted, this is a semantic argument, but the point is that you need to be clear about just what it is that you’re assuming up front. Nothing wrong with assuming that the treadmill matches the speed of the plane with respect to the ground, as you do, However, alternate interpretations are hardly “not possible” or “woefully mistaken.””
Sure they are The question was actually pretty clear on the points we are discussing right here. what the conveyor or treadmill matches speed with it irrelevant (barring the excemptions I pointed out regarding materials failure and turbulent airflow and length of conveyor) - It also quite clearly said matched speed of the airplane. and airplanes speed is measured based on the air it travels through not the surface it resides on so the question was more than clear enough on this specific point.

“Yes assumptions. There are two fundamentally different ways of interpreting this problem–you’re interpreting it in one way, and nerys (for example) is interpreting it in another way. Neither way is clearly “correct” or “incorrect,” they’re just different assumptions about the structure of the problem.”

Your gonna have to clarify this. My interpritation is correct in every aspect that I could forsee based on the wording used ? the parts that were not clear I spelled out pretty clearly.

If the issue is WILL the plane “move” then there is no interpritation the answer is YES it will move.

“You’re mixing up you interpretations here. Again, nothing wrong with that particularly, but I should point out that when you’re analyzing the case of the car on the treadmill, you’re requiring the treadmill to keep the car stationary with respect to the ground–the interpretation you said was “not possible” when analyzing the airplane”

NO I am not requiring the car to remain stationary the conditions of the question are.

If the treadmill always matched the speed of the car and since a car is GROUND PLANE dependant for its locomotion by definition it will result in a paradox unless you differenciate between ground speed and wheel speed.

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

105

Let me try one more example:
Put a plane and a car on a runway. Tell them both to go 10mph. I hope everyone will agree that they both move down the runway side by side. Both of their speedodometers show them going 10mph. They both see the white dashes going by on the runway showing visually that they are moving forward.

Let’s say that this runway is so huge that there no other visual clues that you are moving, just the ones listed above. (maybe you’re on the salt flats on a hazy day)

Now what if I told you that this whole runway was on a conveyor belt, and this belt was going backwards at 10mph. To help visualize things, shrink this huge conveyor belt down to the size of the plane and car. Does this affect the speed of the plane or car at all? If you say yes, I can’t reason with you any further.

To an outside observer (standing on the ground beside the now shrunken down conveyor belt) you see a plane and a car beside each other on the conveyor belt. Since the plane is not going faster than the car (and the rotation of the belt) it’s not going through the air and I think everyone agrees that if a plane doesn’t move through the aitr, it can’t attain lift and fly.

106

Not irrelevant at all. Surely you see that the speed of the plane with respect to the ground and the speed of the plane with respect to the belt itself are different things? That’s pretty much the heart of the difference between the two interpretations: If the conveyor matches the plane fuselage’s speed with respect to the ground, the plane simply rolls down the runway with the wheels spinning faster than they normally would. If the conveyor matches the plane fuselage’s speed with respect to the belt itself, the plane stays stationary.

There you go–you just made my point for me. If the treadmill transmits enough force to the airplane to counteract the engine thrust, the plane won’t move. One interpretation of the question is that the plane stays stationary, held by the force transmitted from the belt.

So you’re differentiating between the case of an automobile and the case of an airplane based on the fact that an auto has a speedometer cable connected to the wheels and the plane has an airspeed sensor installed? Again, there’s nothing particularly wrong with that, but characterizing another interpretation of the question as “woefully mistaken” because it doesn’t conform to a specific methodology of measurement that changes for different vehicles is rather narrow-minded.

107

This is where your example falls apart. the car will show 10mph on the speedo but sit stationary to an outside observer not on the conveyor

but the plane will thrust forward and show 10mph on its AIRSPEED indicator and continue to accelerate and WILL move forward to the outside observer

Once again the airplane is thrust against the atmosphere NOT against the conveyor so it moves regardless of what the conveyor does

the CAR on the other hands ROTATES ITS WHEELS to attain motion so its like being in a hampster wheel it just spin its wheels in place because the conveyor matches its speed.

Understand ?

Here is another way to understand this. 2 car on a conveyor 10mph conveyor matching 0mph right ? good

now you in the first car toss a baseball to me in the second car I am standing up and I take a whack at it.

2 things happen. first the ball DOES sucessfully travel from you to me and assuming I am good enough and hit it it WILL travel the other way when I hit it with the bat.

but its on the conveyor ? thats because its NOT deriving its locomotion from anything relevantly connected to the conveyor. the conveyor is irrelevant to the baseball.

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

108

Bob, we all understand that. The difference between the car and the plane is this: the car moves by pushing against the ground, whereas the plane moves by pushing against the air. If the ground is moving backwards (as on a treadmill), then, I agree, the car will not move forward—the car can’t get any purchase on ground that moves out from under it as fast as its wheels spin, and thus can’t move forward. But if the ground is moving beneath the plane, there is not much effect, because the plane’s engines are pushing against the air, which is not moving backwards. The treadmill will roll beneath the plane’s freewheeling wheels, and won’t add much to the situation (unless you’re thinking like Zut and are proposing a super-speed treadmill that generates enough friction to counteract the engines—I’m not talking about that case, and neither are you, it seems). It doesn’t matter much at all whether the plane is on wheels or hovering a few inches above the treadmill on some sort of magnetic cushion, like a bullet train—the motion of the treadmill just doesn’t exert enough force on the plane to overcome the force of the engines, which will do what they do regardless of what the ground is doing.

The situation with the plane is nothing like the situation with the car. It’s more like trying to stop a freight train by applying a belt sander to the side of the engine—since the train is still able to push against the rails to move, the belt sander won’t slow it down much at all. Or, as Cecil pointed out, if you’re on roller blades standing on a treadmill, with some other force moving you forward (like a rope you can pull on, or a rocket strapped to your back), it doesn’t matter how much the treadmill moves backward; you’re still going to move forward. (If the treadmill moves so fast that the force of friction on your roller blades is enough to overcome the force pushing you forward, then you won’t, but the wheels will fail long before that happens.)

If you eliminate the treadmill and replace it with a big lake of frictionless ice, the situation is still pretty much the same: a car would be unable to move (since its wheels can’t find purchase on the ice), but a plane would be fine (since the engines are still engaging with the air, and the wheels or skids are pretty much irrelevant). If you strapped rockets to the top of the car, it too would move forward while on the treadmill.

109

Another way to look at this: the airplane’s “speedometer” is actually an airspeed indicator. In the situation you describe, the car’s speedometer will say 10mph, and the plane’s airspeed indicator will say 0mph, since there’s no air moving past the plane. If you tell the pilot to go 10mph forward, he’ll run the engines until the plane is moving 10mph forward through the air. (From the plane’s point of view, this would feel like a 10mph breeze, even though it’s traveling 20mph with respect to the treadmill.) At this point, he will obviously leave the car behind, since the car is still moving 0mph with respect to the air (and 10mph with respect to the treadmill).

The wheels are largely irrelevant to the plane, but they’re everything to the car. Let’s look at your example again, but this time have one monkey walking on the treadmill and another dangling from monkey bars that aren’t connected to the treadmill—although it is wearing roller skates that just touch the ground. We tell both of these monkeys, “Move forward at 5mph!” The one on the treadmill will stand still (from an observer’s POV), since the treadmill will move backwards at -5mph, but the one hanging from above will move forward, unaffected by the treadmill. This might confuse the monkeys, but it makes perfect physical sense; the hanging monkey will see his buddy moving backward at -5mph and say, “Hey, he’s on a treadmill!” and the walking monkey will see his buddy moving forward and say “Hey, no fair, those monkey bars are moving at 5mph and carrying him along, so his net speed is 10mph!”

110

I think the big problem people are having with this is that the premise upon which the question is based is faulty. “This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same [speed] (but in the opposite direction).” The question (as has been hinted at) involves what SPEED we are talking about. The conveyor might well be able to match the plane’s speed as measured against the moving belt (which is moving), and the plane’s wheels might spin backwards rather swiftly, but that is irrelevant. What is important is the plane’s speed as measured against the AIR (which, in this example, is not). THAT speed is obtained by thrust exerted against the surrounding air, NOT against the conveyor belt.

If the conveyor belt moved the surrounding air, and moved it to exactly equal the engine thrust (like a wind tunnel), then the plane would not move relative to the air, and would not take off. So the premise of the question, stated with more clarity, is “This conveyor has a control system which tracks the plane’s THRUST against the air, and by spinning the plane’s WHEELS is able to counteract that THRUST.” As may be seen, this is, in fact, a logical and mechanical impossibility.

The example of a car on the belt may help. As has been noted, since a car DOES move by exerting thrust through its wheels to the roadway, a moving belt would keep the car from moving relative to the conveyor. However, if a tow truck was parked at the end of the belt, and used a winch to pull the car to the truck, the car would move no matter how fast the conveyor belt (and the car’s wheels) turned, because the thrust would then be independent of the belt.

I hope this helps to clarify things.

111

I have been following this thread with great fascination, not of the physics involved, but rather to the great leaps and twists of logic involved. I had initially intended to stay out of the discussion, yet here I am. I am particularly bewildered by those who have not yet come to Basic Realization #1. Much of what I will write here has been discussed before, though I hope that my perspective will make the difference.

I am a pilot. I pilot small airplanes (e.g. Cessna Skyhawk, Piper Cherokee). I am frequently in and around airplanes. This does not make me an expert in aerodynamics or physics, but I do know something about how airplanes fly, and move about in general. Based upon my experience with the Cessna Skyhawk:

BR#1, The airplane will take off.

I agree. Why?

  1. Airplanes move through the atmosphere. An airplane rolling down the runway is simply an airplane moving through air while, coincidentally, in contact with the ground. The friction with the runway does impart some drag to the aircraft, but because the airplane has wheels, that friction/drag is not enough to prevent the aircraft from reaching or maintaining its climb velocity. If you remove the atmosphere (yet still provide combustion air to the engine) the Cessna Skyhawk will not move forward. Conversely, if you remove the ground and not the atmosphere, the airplane will accelerate and move forward.

  2. The wheels on an airplane turn because the aircraft is moving, not the other way round. The speed at which the wheels revolve is not always directly relative to the ground speed of the aircraft (e.g. wheels can revolve slower on slick surfaces or faster on … um … a treadmill.)

  3. The “speed” of the airplane in this thought experiment is either its ground speed or its air speed, not the speed at which the wheels are revolving. To state otherwise is simply an attempt to derail the experiment. You may as well say that the “speed” is the speed at which the propeller rotates or the speed at which the pilot chews his gum. Again, see #2. Airplane “speed” is always measured in Air Speed (true or indicated) or Ground Speed (the speed at which the airplane moves from point A to point B). The one relevant to take-off is Indicated Air Speed.

  4. If one insists that the “speed” is measured in the revolutions of the wheel, the experiment then contains a feedback loop as the “speed” of the wheel is increased by the speed of the treadmill which is increased by the acceleration of the wheel and so on. The wheel is also being pulled forward by the airplane and so will always be accelerating, which in turn the treadmill keeps accelerating, etc. etc… The only limit on the rate of the acceleration is the treadmill, which has no limitations as framed in the question; “it can be built” which I assume means to any specification. So in the “wheel speed” scenario, on GO, the gear fails and the plane is thrown off the back of the treadmill as the treadmill instantly accelerates to astronomical speeds. I believe Cecil covered this. I also find the assertion that the “speed” is the relative difference between the treadmill and the fuselage specious for the same reason.

So, ignoring #4 as it is a specious definition of “plane speed”, the (Cessna Skyhawk) airplane moves forward through the air at 67kts, the conveyor moves in the opposite direction at 67kts, the wheels spin freely at ~1850 RPM, assuming a 14” diameter tire. The treadmill to tire or wheel to axle friction at these speeds is IMO not great enough to impart drag equal to or greater than the thrust generated by the Skyhawk. In order to stop the airplane, drag must be greater than or equal to thrust. If you can prove that these speeds would indeed create enough friction to stop the aircraft or impede its accelleration, I will concede.

Is that tire spinning fast enough to disintegrate? I don’t know, but 1850 RPM is not all that fast so I don’t think so. If it does, I would suggest one get better tires. Perhaps one should assemble the Airplane axles, wheels and tires from the same stuff used in the treadmill (treadmills, I believe, drive with wheels) as we have already established that those parts do exist and are of infinite strength and durability. Of course doing so will also change the outcome of the “wheel speed” scenario.

BR#2 Is there a way to set up the conveyor so that it overcomes the thrust of the engines and the plane remains stationary and doesn’t take off?

Stationary, no. By definition, the plane must move before the conveyor reacts. Stop the plane from taking off? Sure, but not by matching the speed of the plane.

112

Practical, yes. Logical, no.

Not really. Different versions of this problem exist, containing different wording. See, for example, the end of Cecil’s original column on the subject. Different wording tends to imply different interpretations of the problem. There’s nothing particularly wrong with your interpretation, but to characterize other interpretations as “attempting to derail the problem” is pretty narrow-minded.

Eh. Cecil said something like this in his original problem, but he was simply wrong. Or, more charitably, he didn’t adequately cover what his assumptions were. He corrected himself in the more recent problem, which leads to Basic Realization 2, and the discussion thereof.

113

Please excuse my poor formatting, I’m still trying to figure this out. Never posted on a message board before.

Arguing BR#1; In both of Cecil’s columns, http://www.straightdope.com/columns/060303.html and http://www.straightdope.com/columns/060203.html , the question is posed such that the conveyor tracks the “plane speed”. I argue that this must mean ground speed or air speed. The term “plane speed” is IMO a narrow term that refers to the moving mass of the airplane and not the rotation of the wheel. I agree that different wording changes the question. I’m simply replying to the question asked.

BR#2 is based upon countering the Force of the Airplane, not the speed of the airplane. The force is counteracted through extreme accelleration of the wheel thus creating enough drag. In practical terms this would disitigrate the wheel assembly. I do concede that IF the wheel assembly can minimally withstand the forces required, then the plane will remain stationary.

I do not see how BR#2 contradicts the “wheel speed” paradox.

114

Damn, I got scared and thought I had posted something that I hadn’t remembered…

Well, I haven’t, and that isn’t me.

115

D’oh! (Imagine my head banging on the table repeatedly!)

The plane takes off. It took the right combinations of words to explain things!

Thanks all.

116

Nope, sure isn’t. Sorry 'bout that.

117

Welcome to my world.

118

You have the right idea but your incorrect. if the conveyor moved the AIR to match the speed of the airplane the airplane would indeed take off as soon as the air the conveyor was moving exceeded the airplanes take off speed. what would not have is the plane would not be able to “make progress” over the GROUND since its GROUND SPEED would always be zero. It would very easily take off though and as soon as it left the area of influence of the conveyors wind the pilot would be in big trouble :slight_smile: since his airspeed would immediately drop to zero.

but the question says the air is calm so we might be ok with ignoring the the air the conveyor moves.

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

119

Even if the treadmill is to match and oppose the speed of the planes wheels and not the speed of the plane the plane will still take off since the treadmill will never go faster than the wheels are going.

You see if the wheels are going 10 and the treadmill is going reverse 10 at that point the situation is satisfied. you say wait thats 20mph so now the treamill must go 20 NO

that is your MEASURED speed from the reference point of the wheels. the actual speed that is used to determine treadmill speed is 10mph. the infinite speed issues is a falacy based on the false assumption that you use the percieved rotational speed and not the actual speed to configure the treadmill. IE a math error not a reality error. the treadmill will go faster as the plane goes faster and become irrelevant once the planes reaches and airspeed sufficient for takeoff.

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

120

Yes.

Although there are certain hazards inherent to doing that sort of take-off.

I have hovered a fixed-wing aircraft at altitude. I have even managed to fly one backwards relative to the ground (relative to the air, of course, it was still going forward)

I’m sick of this “debate.” Here’s the deal: I have access to airplanes. I’ll even pay to rent one (why not? I do that just about every weekend anyway). Get a suitable treadmill - it will need to support about 2,000 lbs (I’m putting in a safety factor here - I’m not insane). It will need to be something that can be assembled and powered on either the end of a runway (I suggest a rural airport for this use, one with little traffic) or in a field (with property owner’s permission). We’ll need considerable clearance for the wings, which will be a minimum 30 foot span (I’ll use a high wing, to aid in clearing any machinery). We might also need to speak with the local FSDO to keep things legal, but crazier things have been approved.

You provide the treadmill, I’ll fly the damn airplane. We’ll settle it once and for all.