

Problems that ruined my world - sdfx
http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/

======
DanielStraight
I think the simple answer to the airplane on a treadmill problem is that once
you understand what the problem is REALLY saying, the problem becomes
uninteresting and trivial. If the problem wasn't intended to give the
impression of a stationary (with relation to the surrounding air) plane taking
off, no one would've cared in the first place and there would be no
discussion. No one is getting confused by math or aerodynamics or physics or
anything. They're getting confused by a problem intended to be confusing.

~~~
Retric
Physics is not intuitive for most people. The aircraft problem works for the
same reason you can yank a table cloth from under a wineglass without knocking
it over. But watch someone do that and it will seem like a magic trick.

Edit: Perhaps a better explication might be how a tail wind increases a modern
cars top speed at almost a 1 to 1 ratio.

------
rottencupcakes
There's an obvious intuition that the author seems to be missing with the 'two
aces' paradox.

If you have a hand with two aces, you have two chances of getting the "Ace of
Spades." Having a hand with two aces doesn't make it any more likely to have
"an ace" than having a hand with one ace does.

This pretty much explains the phenomenon.

~~~
Retric
Does that logic somehow change when someone says:

    
    
      "I have an Ace."
      (You calculate the probability.)
     "Then they say I have an Ace of harts."
      (You calculate the probability.)
    

PS: I suspect the actual answer relates to how you talk about your hand in
bridge.

------
run4yourlives
Airplane on the treadmill will take off huh?

You know those big giant things sticking out from the side of the fuselage?
They're called wings. Air flowing over the and under the wings creates
different pressure; hence, lift. There is a magic speed for each aircraft that
over which the lift will be sufficient to move the entire aircraft vertically.
For a 747, this is 180 mph/290 km/h at 80,000 lbs. That's pretty fucking fast.

Any movement relative to the ground that is under that speed will not result
in flight. This is true even if the aircraft is actually in the air.

If, as the problem states, the treadmill can equal the forward thrust of the
engines, no lift will be produced by the wings, ergo, it will stay right where
it is.

This is why aircraft have an airspeed and a ground speed. It's also why
stalling is a major concern.

EDIT: the fact that finding a treadmill that can act in a manner as suggested
by the problem is impossible is another issue completely.

EDIT 2: Think people: They have breaks on aircraft wheels do they not? (They
do) You can sit on a runway with the engine revved equal to your break ability
and _not move_. If the engine in the problem can outperform the treadmill,
you've solved a different problem.

~~~
TrevorJ
Airspeed over the airfoil, not groundspeed is the primary factor involved in
lift. The logical fallacy in your argument is thinking that the wheels provide
the forward momentum, but they don't, it is the propeller that does.

That being the case, the plane will be pulled forward down the runway
irrespective of how fast the treadmill under it is moving because the speed
that the wheels rotate has nothing to do with how much airspeed is generated,
other than the negligible friction that is transmitted to the airframe through
the wheel bearings.

Said another way: a car traveling on the treadmill will be slowed becasue its
forward momentum is transferred through the wheels. An airplane will not be
slowed because the forward momentum is generated by the propeller and has
nothing to do with how fast the wheels are spinning.

~~~
run4yourlives
>Airspeed is the only factor involved in lift.

The only way to achieve airspeed required for flight when you are sitting on
the ground is to move the wheels. (ie, roll forward).

Your argument suggests that if the aircraft was attached to the ground with
steel beams it would somehow break free into glorious flight. It would not.

The treadmill removes the ability of the aircraft to achieve correct airspeed,
regardless of whether it is powered by prop, jet or ion engine.

Any assumption that the aircraft can move forward is predicated on the fact
that the treadmill cannot match the engine's output. This is accurate, but
like I said, outside of the scope of the problem.

Try this:

1\. Aircraft has floats, not wheels. 2\. Aircraft's normal take off speed is
100 kts. Let's say the engine can produce a take off speed of 100kts at
2000rpm. 3\. Aircraft is on a river moving the opposite direction at 50kts.

Q: Will the engine be able to achieve the required 100kts airspeed at 2000rpm?
If not, if I can somehow match the river to the airspeed, is it not sound
theory that I can prevent the aircraft from _ever_ reaching that 100kts take
off speed?

~~~
TrevorJ
The treadmill does not remove the ability of the plane to move forward. If it
did, you are absolutely correct, the plane could not fly.

The wheels on an aircraft are not attached to any drive mechanism, they are
free spinning. In the scenario described they would act as a bearing and
mitigate nearly all of force applied by the treadmill. There simply isn't
enough friction transmitted to the airframe to counteract the force created by
the propeller.

An airplane with no airspeed sitting on a treadmill would not fly, as you say.
An airplane generating enough airspeed will. The treadmill does not prevent
the airplane from moving forward with sufficient airspeed.

~~~
run4yourlives
The wheels not being powered is irrelevant. The engine produces forward power.
The plane moving forward causes the wheels to move in relation to the forward
power of the engine.

By your logic, putting breaks on aircraft wheels would be irrelevant. But
guess what, they can and do stop the aircraft from rolling forward.

If you counter that movement - of the wheels - you counter the forward
movement of the aircraft, because the movement at this point is against the
ground, not the air.

Ergo, no flight.

~~~
edu
The plane will fly, and the wheels will spin at the double of the airspeed.
The engines 'push' against the air, not the ground.

------
Locke1689
My issue with this problem was that it was presented rather poorly to me the
first time. Instead of saying that the treadmill matched the negative of the
velocity of the plane relative to the ground, the questioner instead stated
that the treadmill "matched the speed of the wheel." I took this to mean that
the treadmill matched velocity of the wheel as would be given by a
conventional speedometer.

Of course, this is prima facie ridiculous. Consider, at any nonzero speed, the
speed of both the treadmill and the wheel would quickly approach infinity.
Given that the speedometer speed would essentially be the velocity of the
wheel + the tangential velocity of the wheel touching the treadmill, the
entire thing would just blow up (remember, non-slip == velocity of the wheel
== velocity of treadmill). Therefore, I basically answered c) it's a stupid
problem and the plane wouldn't move.

Of course, if you define it this way, than it is obvious that the plane would
move.

------
lisper
Geez, people, for the love of Pete, even if you can't wrap your brains around
why the plane would take off at least learn to spell "brakes" correctly!

------
nopassrecover
Awesome, I'd love to see more links like this if anyone has them.

~~~
andreyf
[http://en.wikipedia.org/wiki/Category:Probability_theory_par...](http://en.wikipedia.org/wiki/Category:Probability_theory_paradoxes)

and more generally -
<http://en.wikipedia.org/wiki/Category:Mathematics_paradoxes>

Apologies to anyone who was expecting you to get work done today.

~~~
nopassrecover
Haha thanks!

