A formulation of the airplane/conveyor belt problem that "works"?

Cecil's Columns/Staff Reports
1

Links to original columns:

http://www.straightdope.com/columns/060203.html
http://www.straightdope.com/columns/060303.html

As I read the various incredibly long threads on this question, I found myself wondering if there was a formulation that would “work” in the following ways:

  • The phrasing is not ambiguous
  • It’s deceptive
  • There is a clear and demonstrable answer
  • It illustrates something interesting about flaws in most folks’ intuition about airplanes, conveyor belts, physics, or all three.

To my mind, the question is designed to trap people who know (or think they know) how planes fly. The idea is to get such people to reason, “well, the plane is stationary relative to the air, and air has to flow across the wings to get lift, so it can’t take off.” Among newbies to the message board this is by far the most common “wrong” answer. And of course the trick answer is that the conveyor can’t keep the plane stationary (I’m ignoring BR#2, which is to my way of thinking an interesting side problem).

It’s very much like the Monty Hall question, which is clearly designed to trick people who know (or think they know) how to calculate probability. After Monty opens a door we’re left with two options, so our instinct is that each door has a 1/2 probability of being the winner. The prior knowledge that Monty always opens a door, can’t open the originally selected door, and can’t open the winning door shifts the probabilities in a way most people aren’t familiar with. That makes it an interesting problem that illustrates a subtlety of probability.

So can we formulate a version of the airplane problem that essentially limits us to BR#1 as the correct answer yet isn’t completely trivial? Here’s my stab at it:

"Imagine a light plane is sitting at one end of a long conveyor belt the length and width of a runway. The plane is pointed toward the other end. The plane has a top speed of around 120 knots and a takeoff speed of 75 knots. It’s a clear, calm day with no wind. The conveyor belt can spin in either direction and has a top speed (relative to the ground around it) of 300 knots. The conveyor is computer controlled, and is designed to spin as fast as necessary to keep the airplane stationary relative to the ground around the conveyor belt. So if the plane starts to move forward down the conveyor/runway, the conveyor will immediately spin in the opposite direction in order to keep the plane in place. The conveyor has very quick acceleration and can change speed effectively instantly.

Can the plane take off?"

This pretty clearly removes the possibility of the BR#2 interpretation, because the conveyor can’t go that fast. It also makes it clear that the conveyor is trying to keep the plane stationary. I still think it’s a fairly deceptive problem with an interesting answer.

What do other folks think? Is this an improvement on the original problem as given in Cecil’s column? Anyone have a better way?

Don

2

I just thought of one or two angles you probably want to cover that could sidetrack the issue and keep the plane from taking off:

· Area wide power failure keeps conveyor from working.
· Computer crashes.
· Computer gets auto-update and reboots while plane takes off.
· Computer hibernates waiting for plane to take off.
· Cheaply designed conveyor belt flaw and shreds itself to pieces.
· Plane has no fuel.
· Plane has flat tires.
· Plane design so flawed it cannot fly.
· Person driven mad by over discussion of the question destroys plane.
· Asteroid destroys plane.
· Another plane crash lands and destroys plane.
· Major earthquake swallows plane.
· Nuclear winter destroys plane.
· God suddenly destroys all planes on grounds we were never meant to fly.
· Plane becomes sentient and refuses to takeoff.
· Pilot becomes sullen and refuses to take off.
· Pilot has cocaine induced heart attack and is unable to takeoff.
· Air traffic control refuses to let plane takeoff.
· FAA refuses to let plane takeoff because it is insuficiently substantiated.
· DEA arrests pilot on grounds pilot might have cocaine
· FBI arrests pilot on grounds DEA might beat them to it.
· IRS arrests pilot on grounds pilot has never filed a return.
· Laws of physics suddenly change during takeoff.
· The Matrix is unplugged.

I have this sneaking suspicion that I might have missed one or two issues. Anyone else see anything to add?

3

Yeah, that’s in the extended dance remix version.

4

I’m sorry if I’m unable to grasp a simple concept (and I’m sorry if my question has been beaten to death in other discussions of this problem – I haven’t searched the net for them). But a plane takes off because it is propelling itself through the medium of the air at a sufficient speed for the air to create lift on the wings. Making the ground move backwards won’t cause the entire medium the plane is sitting in to move too, so the lift is created. The backwards-moving runway will cause a certain amount of air movement below the plane, but not nearly enough to cause the lift (which depends on air moving across both the top and the bottom of the wing). I think the entire mass of air around the plane would have to move backwards at the same speed as the runway in order for Cecil’s answer to be correct.

5

<SIGH!/> Now we’re back to the original problem.

A backward-moving runway will not pull an airplane backward, because the landing gear is free-wheeling. The propellors and/or jet will move the plane forward, pretty much ignoring the stupid treadmill, and the plane will take off normally.

However, the original question is badly worded. You can, if you like, assume that the runway is moving backward so fast that the friction in the free-wheeling landing gear is enough to hold back the airplane. In practice, this requires the treadmill to be moving at thousands of miles per hour. So, if you do this, you can say either that the plane will not take off (because it is being held back), or that the entire system will melt down, burst, or otherwise fail (because of the ridiculous stresses involved), or that the air being moved by proximity to the thousands-of-miles-per-hour treadmill will be enough to launch the plane. Without exact details of what the landing gear is made of, the power of the engines, the friction in the bearings, what the treadmill is made of, etc., etc., which of the three will happen is impossible to determine.

6

This is my point exactly - in the version of the question I posted, I think you can unambiguously say that Chrisk33 is wrong (no offense to Chrisk33 at all), because there is no way for the conveyor belt in that version to keep the airplane from taking off.

So the fact that Chrisk33 still doesn’t understand the correct answer is perfectly fine - it’s a trick question, after all. I’m not trying to provide a better answer, I’m trying to provide a better question, one that doesn’t lead to people quibbling that the treadmill really could keep the plane still if it was allowed to have infinite acceleration and limitless speed, etc., etc.

I’m assuming that parabolis feels like this is arguing about how many angels can dance on the head of a pin. And hell, maybe it’s the ambiguities in the problem that are what keep it going. If the answer was cut-and-dried, maybe it wouldn’t be nearly as popular because there’s less to argue about.

I tried the version in my first post on a friend the other day. He once wrote a book about flying, so he certainly knows how planes work. He picked the wrong answer right away for the usual reasons (a plane that isn’t moving through the air has no lift), but didn’t have any trouble understanding why that answer is wrong once I explained it to him, and he thought it was an interesting problem that illustrated a flaw in people’s intuition.

Take it for what it’s worth; really, the original question as posed by Cecil is fine. I understood what it meant and I thought the answer was interesting when I first encountered it. I just wondered if there was a way to shut out the whole “continuously accelerating conveyor” line of thinking.

Don

7

I like it. And believe it fairly represents what the sky jockies who first dreamed up the problem had in mind.

8

OK, I’m perfectly willing to be proven wrong, but I still don’t see how the fact that the plane’s wheels are free-wheeling makes it possible for the plane to take off.

Imagine that the whole system is in an enclosed box the length of the runway. This runway is long enough for the plane to take off from under normal circumstance. The box has air in it. If the plane can take off it’ll hit the ceiling of the box, but that’s neither here nor there for purposes of the discussion.

The runway, which is inside the box, spins backwards and the plane is able, with the thrust of its engines, to remain at the beginning of the runway. Sure, its wheels are spinning, they must to let the plane stay in the same spot in relation to the length of the runway.

Now, will you accept that the air in the box is the medium through which the plane must move in order to achieve lift? And that the plane has to move through most of the length of the runway in order to push hard enough into the air to create lift? Think of the air as a fluid (which it is) and the box as sort of an aquarium. I argue that unless the plane accelerates through the air contained in the length of the box, it cannot create lift. The fact that it is moving forward enough to keep pace with the runway is interesting but immaterial to the creation of lift. The entire box is not moving backwards at the speed of the runway, so the air around the plane is not moving backwards against the plane to create lift.

Heck, put on a pair of Icarus wings and stand on a treadmill. No matter how fast that treadmill goes and you run to keep up with it, your wings are not going to get any lift. Want to prove it? Put a whirlygig on your head and see whether running on a treadmill makes it spin.

9

With all due respect, you’re approaching the problem from the wrong perspective. Instead of trying to prove that the plane can’t take off, try to understand why it does. Because in the form of the problem posited in the OP, there is no doubt that it can, for the reasons stated in Cecil’s first column.

10

I so do not want to rehash the whole discussion, so let’s just say this–the wheel+conveyor system provides negligible force on the aircraft, either braking or propulsive, because the wheels are allowed to turn freely on their axles. The entire propulsive force is generated by the propellor+air interaction, which is unaffected by the wheels. Picture this in your head–take a toy plane, put it on a conveyor, and pull it forward by the propellor. The wheels spin faster than on a usual takeoff, but that conveyor isn’t gonna put any appreciable drag on it unless you have the brakes on. Makes sense?

11

I think you’ve almost got it, but you should add in the statement “The wheels of the plane are very strong, have little internal friction, and do not fail at speeds up to 450 knots.” That way some wiseguy can’t say that the landing gear shreds once the conveyor belt reaches top speed.

Oh, and Chrisk33, your Icarus wings example is flawed. The human does not take off, but that’s because he is trying to run forward with his legs to gain speed and thus lift, instead of using his rocket backpack and roller skates, like a plane does. Maybe this will help you: a normal car with wings strapped to it on a conveyor belt does not take off. A car with two jet engines and wings that is in neutral does take off. See the difference?

12

I admit I was “tricked” into the wrong answer when originally pondering this problem.

The best way to explain it out for me is:

Start with a normal fixed runway, and say the distance for the plane to lift is L

Then take the same runway and start the reverse treadmill. Same thrust by the plane, maybe takes off in .9xL … Tweek up the treadmill some more… Now plane takes off in .5xL. Tweek more… Tweek more. Now .001xL. One more tweek, and you have lift off at 0 x L.

Put it on the middle of the treadway, tweek some more, and maybe you have takeoff at -.5 x L (???)

13

dmunsil, you’re falling prey to one of the basic assumptions which flawed the original question.

If you believe that the conveyor belt can put enough backward force on the plane as a whole via friction through the wheel bearings to counteract the engine’s thrust, then yes, the conveyor can keep the plane from moving. So, that’s BR#2, I believe.

14

I specified the top speed of the conveyor at 300 knots. So it tries to keep the airplane stationary, but fails.

The point of reformulating the problem was to eliminate the possibility of the conveyor actually stopping the plane from moving, while keeping it deceptive. So the wording needs to subtly imply that the conveyor is going to stop the plane from moving, but the trick is that no matter how hard it tries, it can’t do it.

I see your point that the phrase “as fast as necessary” seems at odds with the earlier given maximum speed.

Perhaps:

“The conveyor is computer controlled, and is designed to spin as fast as necessary (up to its max of 300 knots) to keep the airplane stationary relative to the ground around the conveyor belt.”

To me, that extra emphasis on the 300 knots weakens the deceptiveness. The person hearing it or reading it wonders “why so much emphasis on the top speed of the conveyor?”

But maybe that’s just me. Can you suggest a better phrasing?

Don

15

I think you may still be confused. No matter how fast the conveyor spins (within reasonable limits), the plane still takes off in roughly the same distance relative to the ground. If anything, the extra friction from the wheels could increase the takeoff distance by reducing slightly the plane’s maximum acceleration.

If you put a huge wind machine at the end of the runway instead of using a conveyor belt, you could tweak things to get the plane to take off without moving relative to the ground, at least until the plane lifted out of the airstream from the wind machine. But the point of this problem is that the conveyor has essentially no effect on the plane. The plane just pulls forward and takes off as though the conveyor wasn’t moving.

The “trick,” simply put, is this: the conveyor has no appreciable effect on the problem. Whether you spin it forward or reverse at just about any speed has no effect on whether or how fast the plane takes off.

All of the above requires, of course, the disclaimer that if you imagine a conveyor that can achieve any speed or any acceleration then all bets are off - it could conceivably slow the plane down. And if you slow the plane down so it can’t move forward relative to the air around it, it can’t take off. That’s BR#2, in Cecil parlance.

But limiting ourselves to a conveyor that can go, say, two or three times as fast as the maximum speed of the plane makes the answer clear. Which is that the conveyor does close to nothing. The plane’s wheels spin like mad, but it otherwise takes off normally.

Don

16

I think that I finally have my head around this one.

Imagine a plane suspended in the air, with no forward velocity. Start the engines and let go. The plane will drop (thanks to gravity) but accelerate forwards. Forget about the fact that the plane would fall nose down and assume it is so evenly balanced that it would fall parallel to the ground. As the plane’s speed increases it will eventually pass through the air at a sufficient forward velocity to create enough lift to stop itself from dropping, then level out and finally gain altitude.

Now introduce the runway (conveyor). Its purpose is simply to avoid the drop part. ie the plane remains in contact with the ground until it gains enough speed through the air to create enough lift to take off.

The point about the conveyor moving is the trick.

If the plane has air speed then it must be moving forwards (relative to the ground), which means that it is not actually stationary. Granted as the plane speed increases so does the belt but this simply means that the wheels are spinning incredibly fast - this is irrelevant though as all they are doing is creating the contact between plane and ground - nothing else.

17

The original problem is interesting because it demonstrates how easily people are misled by red herrings.

Everybody knows that a plane is propelled through the air by jet or propellor engines, but in this problem they connect the speed through the air with the wheels, because they are misled by the presence of the conveyor belt.

The problem is not interesting in a mathematics or physics sense and any responses that mention specific speeds or dimensions or introduce equations seems to entirely miss the point. When you understand the answer, it’s not like you slap your forehead and exclaim:

DUDE - I TOTALLY FORGOT about the ANGULAR MOMENTUM of the WHEELS!!

It’s just isn’t that complicated.

Given that there is no primary connection between the wheels and the airspeed, I don’t see what is interesting in trying to fiddle with the problem to amplify or create secondary effects. Even if you can do it, so what? Is it just to be able to say “see I am too smart!” after falling for the red herring in the initial problem?

You may or may not be smart, but I assume it is obvious (just look around you) that the human brain can be pretty average at problem solving :slight_smile:

Just keep swimming.

18

I’m still not getting it. Permit to add an element to the scenario which will illustrate my confusion.

Let us imagine the plane on conveyor belt scenario. Let us say the airspeed needed for the plane to rise is 75 mph. I am on the side of the conveyor belt scenario, standing still. Suddenly I remember that I have left my beloved stuffed dog Muffie inside the plane and I must have it immediately. I find a ladder nearby, open the cabin door put one end of the ladder on the ground beside the conveyor belt and the other end in the doorway, then climb inside the cabin. (Because of course it would be very painful to make contact with the conveyor belt since it is travelling at 75 mph). Inside the plane, I find Muffie.

Now, if airspeed has to be 75 mph how was I able to get inside the plane since it’s somehow travelling fast enough to get airborne, when I am travelling at 1 mph at best? Isn’t it the case that all the energy created by the centrifuge is simply being absorbed by the plane’s wheels, which have wonderful frictionless that allow them to spin quite freely?

19

Butzi64, this formulation was introduced in response to several protracted threads (each about a dozen pages long, IIRC) converting what some of us thought was a clear enough problem (see column 1) into a much murkier and interesting one (see column 2). If you prefer to use some formulation of the original question without the OP’s tweaks, that’s okay. But, some people are going to exploit the ambiguities.

20

Evil Captor, I’m afraid I don’t understand your scenario, much less how it would aid understanding the original problem. What is the speed of the plane relative to the ground? (Take-off speed of 75 mph would be relative to the air.) If zero, there’s nothing to explain. If one mph, it would be just like what would happen with an ordinary runway and an airplane taxiing at that speed. The ladder would tip over, be dragged along the ground or bend, depending on how firmly it is attached to the plane and how much friction it experiences viv-a-vis the ground.