A harmless little riddle

JimFiore wrote:

This has been extraordinarily entertaining. Some of the answers just make me giggle. Apparently, though, some people have not determined that there are multiple ways of interpreting the original question, and some are simply impossible to answer because they are nonsensical. It's kind of like asking "What happens when an irresistible force meets an immovable object?" or "If god is all-powerful, can he make a rock so large that even he can't lift it?"

Very entertaining, indeed.

I agree Jim. Great entertainment. And like I said a page back, or so, depending on how you interpret the question, and what assumptions you make about the plane and the treadmill, the answer could be virtually anything. It would be a very interesting essay question. Well, essentially, that is what it has turned into.

I go away for 4 days. This thread goes from on page 25 to page $!41$!!!!!!! PAGE 41!

I haven't read all the new ones, but the plane still flies!

Heh, and I find it entertaining that people still don't realise that assumptions are different than interpretations.

There are no conditions of the treadmill which would prevent the plane taking off, no stated conditions of the atmosphere which would prevent the plane from taking off, and no conditions of a plane (which could otherwise take off on a stationary runway) which would prevent the plane from taking off.

Any movement of the treadmill is entirely irrelevant, because, as stated many times before (and the whole point of this little exercise) all that matters is that the plane has something to support itself on before it can generate enough airspeed to lift off - whether that surface is moving or not doesn't matter. No other way to interpret it.

the magic breath of life wrote:

Well I registered my username and this crap won't be happening again.

Are you saying you are the original "the riddler"?

Did your "magic breath of life" id get highjacked earlier?

No, there are conditions where the plane would not fly- If the engine were not powerful enough to move a heavily loaded (overloaded perhaps) plane, then the friction from the wheel and treadmill could indeed be enough to keep the plane from flying.

Ah, but that's an assumption and not an interpretation.

See below for a post I made earlier regarding the alternative interpretation...

However, in order to fly, the plane must be moving relative to the ground, correct? And if the plane is moving relative to the ground, then the TANGENTIAL velocity of the wheels is GREATER than the TANGENTIAL velocity of the treadmill. With me so far? While this is perfectly feasible and possible and the plane will no doubt take off under this scenerio, IF YOU INTERPRET THE RIDDLE AS CONSTRAINING THE TANGENTIAL SPEED OF THE WHEELS TO BE EQUAL TO THE TANGENTIAL SPEED OF THE TREADMILL, THEN THIS SCENERIO VIOLATES THE CONDITIONS OF THE ORIGINAL RIDDLE!

Ok, try this one, which I've repeated a billion times, it feels like:

The speed of the airplane is measured as wheel speed.

No. If the treadmill runs in the opposite direction at the same speed as the airplane's wheels, the airplane will not move forward and not take off. That is an interpretation where the plane doesn't fly. Of course, with that interpretation comes a lot of assumptions that seem pretty far fetched. But the riddle itself is pretty far fetched.

And I am ignoring the one guy that claimed because of a boundary layer of wind created by the movement of the treadmill, it would still take off. This is argumentative, as we don't know how thick that boundary layer of wind is, and I haven't seen any evidence that suggests it would be the required ~20 feet high for a typical jetliner.

yessiree wrote:However, in order to fly, the plane must be moving relative to the ground, correct?

Wrong. A plane flies on airspeed, not groundspeed. A plane generates its own airspeed regardless of what the wheels or treadmill are doing.

Nobody said anything about wheel speed - this is an arbitrary made-up additional factor which isn't part of the original riddle. Sure, I could make things up too. The plane is able to flap its wings like a hummingbird. It takes off no matter what. The plane is powered by Jedis and can do whatever it wants. Sure...go ahead, make things up.

The riddle states that the treadmill matches the plane's speed, not the wheel's speed. If the treadmill is moving, this must be relative to the ground in which it is sitting on. But, it doesn't matter - the treadmill is supporting the plane the same as a normal runway would. If takeoff speed is 100mph, the treadmill accelerates to 100mph in the opposite direction as the plane's (airspeed OR groundspeed - same thing here) takeoff velocity of 100mph is achieved.

There is no condition of the wheels or treadmill which prevents the plane from taking off, unless it's a bunch of made-up bullsh*t like excessive and entirely unrealistic amounts of friction.

KnowItAll wrote:

And I am ignoring the one guy that claimed because of a boundary layer of wind created by the movement of the treadmill, it would still take off. This is argumentative, as we don't know how thick that boundary layer of wind is, and I haven't seen any evidence that suggests it would be the required ~20 feet high for a typical jetliner.

Maybe it's a very rough surfaced treadmill and the plane is assumed to have 4x4 off-road capability, although the turbulence in that boundary layer would probably make it difficult to tell when you've actually lifted off that rough surface.

Look , the premise of a runway length treadmill is kind of entirely unrealistic in itself, and then taking the wording of the question to mean that the treadmill will match the speed of the plane no matter what (however unreal seeming), is the theoretical premise (accepted on faith of the wording of the question) upon which a lot of people base their response that the plane cannot take off.

wail wrote:

Wrong. A plane flies on airspeed, not groundspeed. A plane generates its own airspeed regardless of what the wheels or treadmill are doing.

Nobody said anything about wheel speed -

Nobody said anything about air speed either. You're making things up, just as much as someone interpreting it as wheel speed.

Completely regardless of where the power comes from, you can measure the speed of a plane 3 ways. Air speed, Earth ground speed, and treadmill ground speed. All three measurements could be correct interpretations of "speed".

So when answering the riddle, you should really spell out all 3 of them, and indicate what would happen in all 3 cases. That would be the most "correct" way of answering the riddle. You're not adding anything, you're just taking a loosely worded sentence and clarifying your interpretation.

yessiree wrote:

IF YOU INTERPRET THE RIDDLE AS CONSTRAINING THE TANGENTIAL SPEED OF THE WHEELS TO BE EQUAL TO THE TANGENTIAL SPEED OF THE TREADMILL, THEN THIS SCENERIO VIOLATES THE CONDITIONS OF THE ORIGINAL RIDDLE!

And how do you get the "TANGENTIAL SPEED OF THE WHEELS" out of "the same speed the plane is moving"? That is not an interpretation, that's a misinterpretation. Give it up, man.

admire the persistance anyway wrote:

And how do you get the "TANGENTIAL SPEED OF THE WHEELS" out of "the same speed the plane is moving"? That is not an interpretation, that's a misinterpretation. Give it up, man.

Ignoring how a car is propelled, wouldn't someone be talking about tangential speed of the wheels if they said they got a car up to 60 MPH on the dyno? What was the car's speed on the dyno? 60 MPH. What was the runner's speed on the treadmill? 10 MPH. Speed is often expressed in regards to the circumstances. If you said, what was the plane's speed going on a giant treadmill? The first thing that someone would say, is probably in reference to the treadmill belt. So they would give the wheel speed. But, if you were a pilot in the plane, and couldn't see the belt speeding underneath you, you'd probably look at your airspeed read out and see that you were going 0 MPH, even if your wheels were spinning like mad on the treadmill. It's all about perspective. And either answer is correct.

wail:

draw a force diagram. when the treadmill speeds up, in order for the wheels to spin faster, the treadmill must exert a torque on the wheels. but a torque must be considered both as a force and as a torque. so if the treadmill accelerates fast enough, then the torque on the wheels will be large, and so the force on the wheels (and on the plane as a whole) will be large.

of course, this requires an extremely large friction force between the wheels and the treadmill, but yeah, whatever.

admire the persistance anyway wrote:

And how do you get the "TANGENTIAL SPEED OF THE WHEELS" out of "the same speed the plane is moving"? That is not an interpretation, that's a misinterpretation. Give it up, man.

It's not my fault you don't know what tangential speed means. It's a perfectly rational interpretation, as KnowItAll has pointed out.

"Tangential speed of the wheels" is just another way of saying "speed of the plane relative to the treadmill." If I reworded "speed of the plane relative to the earth" in the same way, it might sound something like "tangential speed of the wheels plus tangetial speed of the tangential speed of the treadmill." Not wrong, just a different interpretation.

Bitch.

KnowItAll wrote: So when answering the riddle, you should really spell out all 3 of them, and indicate what would happen in all 3 cases. That would be the most "correct" way of answering the riddle. You're not adding anything, you're just taking a loosely worded sentence and clarifying your interpretation.

Nobody said anything about air speed? What? That's how a plane takes off. When you are solving a problem such as this, surely you need to assume but one thing: all other factors are equal ie the wind isn't doing something crazy like blowing 1000mph in the same direction as the plane.

I say that all other conditions not stated in the riddle need to be equal - ie the air is still. Making things up by adding any other condition about the plane (ie some condition of the wheels other than the fact that they function normally as airplane wheels shoud) isn't following this logic.

Let's assume an arbitrary takeoff velocity of the plane as being 100mph, but the engines can generate much more thrust than this (given that the crusing speed of a jet airliner is 500-600 mph but this could easily be an f-18 which can go 2000+ mph).

1. The plane is still, the air is still, the treadmill is still all relative to the ground upon which the treadmill sits. The plane starts its engines, generating thrust. The plane's air and ground speed increase until the plane is going 100mph relative to both the air and ground, and the treadmill speed increases until the treadmill's groundspeed is 100mph in the opposite direction. The plane takes off.

2. The plane is still, the air is blowing at the plane at 50 mph, the treadmill is still all relative to the ground upon which the treadmill sits. The plane starts its engines, generating thrust. The plane's air and ground speed increase until the plane is going 100mph relative to the air and 50mph relative to the ground, and the treadmill speed increases until the treadmill's groundspeed is 50mph in the opposite direction. The plane takes off.

3. The plane is still, the air is blowing from behind the plane at 50 mph, the treadmill is still all relative to the ground upon which the treadmill sits. The plane starts its engines, generating thrust. The plane's air and ground speed increase until the plane is going 100mph relative to the air and 150mph relative to the ground, and the treadmill speed increases until the treadmill's groundspeed is 150mph in the opposite direction. The plane takes off.

The riddle says nothing about wheel condition (friction) but, really, the only way that the plane could not take off is if the plane's wheels were too restrictive to travel, in the last scenario, at 300mph. Given that sportscars can travel well over 200mph these days, it's not unrealistic to think that a wheel could withstand that, and that the thrust of the plane could overcome the friction of the wheels.

I'll say it again: there is no condition stated where the plane *doesn't* take off, and no correct interpretation of the riddle has the plane remaining on the treadmill.

wail wrote:

Nobody said anything about air speed? What? That's how a plane takes off. When you are solving a problem such as this, surely you need to assume but one thing: all other factors are equal ie the wind isn't doing something crazy like blowing 1000mph in the same direction as the plane.

I say that all other conditions not stated in the riddle need to be equal - ie the air is still. Making things up by adding any other condition about the plane (ie some condition of the wheels other than the fact that they function normally as airplane wheels shoud) isn't following this logic.

...

I'll say it again: there is no condition stated where the plane *doesn't* take off, and no correct interpretation of the riddle has the plane remaining on the treadmill.

Jesus. Yes, nobody said the word "air" anywhere in the riddle. They did not specify what "speed" they were talking about. We all know that airspeed is what determines liftoff, but it's not stated in the riddle and "speed" could be determined as "wheel speed relative to the treadmill belt".

Now I'm friggin keeping all other conditions completely equal, not making anything up or adding anything to the riddle. I'm just making an assumption that "speed" could mean "wheel speed relative to the treadmill belt". Are you still with me?

The engines start and at the same rate that the airplane's wheels begin to accelerate, the treadmill begins to accelerate in order to keep their speeds exactly equal, just like the riddle says. The treadmill is matching the "speed" of the plane. Are you still with me? Since the acceleration of the treadmill is matching the acceleration of the wheels, the plane has no airspeed and no speed relative to the fixed ground.

As the forces of the engines and the treadmill pump kinetic energy into the wheels, and heat and sound energy via the internal friction of the wheels' bearings, the wheels spin up rapidly, absorbing that energy and keeping the plane stationary with respect to the air.

Now one of two things happens. You assume the wheels are rated to an infinite speed, so they just keep accelerating indefinitely. Or the wheels breakdown, instantly increasing the drag between the treadmill and the airplane's ruined wheels. This is pretty irrelevant though, as the treadmill will still speed up or slow down to match the acceleration of the plane and keep it still with respect to the air.

There is damn scenario of how the plane will not take off and I didn't add anything to it. I just interpreted the speed of the airplane to be the ground speed with respect to the treadmill belt.

I agree that there are two possible interpretations of the riddle. That does not mean that the two interpretations hold equal value. If a reference frame is not stated in a physics problem, usually it is assumed that the reference frame is the earth. Because of this, I think that "the plane will take off" is the more correct answer.