7 ms·

Problems that ruined my world

30 points by sdfx 17y ago

nopassrecover 17y ago
Awesome, I'd love to see more links like this if anyone has them.
- andreyf 17y ago
  http://en.wikipedia.org/wiki/Category:Probability_theory_paradoxes http://en.wikipedia.org/wiki/Category:Probability_theory_par... and more generally - http://en.wikipedia.org/wiki/Category:Mathematics_paradoxes http://en.wikipedia.org/wiki/Category:Mathematics_paradoxes Apologies to anyone who was expecting you to get work done today.
  - nopassrecover 17y ago
    Haha thanks!
- TrevorJ 17y ago
  Here is a video of an airplane taking off on a treadmill. http://www.youtube.com/watch?v=IbRcg3ji_Pc http://www.youtube.com/watch?v=IbRcg3ji_Pc The crux of the issue is that no matter how fast the wheels spin in the opposite direction they exert almost no force on the airframe, thus failing to counteract the forward pull of the propeller.
- slackenerny 17y ago
  Last year Vaughan Pratt (of automata theory and Knuth-Morris-Pratt algorithm) asked on a hardcore theoretical physics group blog how does a light mill (device allegedly proving corpuscular nature of light, http://en.wikipedia.org/wiki/Light_mill http://en.wikipedia.org/wiki/Light_mill) really work. To my knowledge this question after 150 years still awaits consensus over a satisfactory anwser (while having far too plenty seemingly convincing ones). http://golem.ph.utexas.edu/category/2008/07/light_mills.html http://golem.ph.utexas.edu/category/2008/07/light_mills.html A toy simulation: http://www.gregegan.net/SCIENCE/LightMill/LightMill.html http://www.gregegan.net/SCIENCE/LightMill/LightMill.html
run4yourlives 17y ago
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 17y ago
  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 17y ago
    >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?
deleted 17y ago
[deleted]
DanielStraight 17y ago
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 17y ago
  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.
Locke1689 17y ago
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.
rottencupcakes 17y ago
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.
- TheSOB88 17y ago
  Thank you so much! I really need to add "reversing the problem" to my repository of problem-solving techniques, but I just can't seem to find it.
- Retric 17y ago
  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.
lisper 17y ago
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!

30 points by sdfx 17y ago

nopassrecover 17y ago

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

andreyf 17y ago
http://en.wikipedia.org/wiki/Category:Probability_theory_paradoxes http://en.wikipedia.org/wiki/Category:Probability_theory_par... and more generally - http://en.wikipedia.org/wiki/Category:Mathematics_paradoxes http://en.wikipedia.org/wiki/Category:Mathematics_paradoxes Apologies to anyone who was expecting you to get work done today.
- nopassrecover 17y ago
  Haha thanks!
TrevorJ 17y ago
Here is a video of an airplane taking off on a treadmill. http://www.youtube.com/watch?v=IbRcg3ji_Pc http://www.youtube.com/watch?v=IbRcg3ji_Pc The crux of the issue is that no matter how fast the wheels spin in the opposite direction they exert almost no force on the airframe, thus failing to counteract the forward pull of the propeller.
slackenerny 17y ago
Last year Vaughan Pratt (of automata theory and Knuth-Morris-Pratt algorithm) asked on a hardcore theoretical physics group blog how does a light mill (device allegedly proving corpuscular nature of light, http://en.wikipedia.org/wiki/Light_mill http://en.wikipedia.org/wiki/Light_mill) really work. To my knowledge this question after 150 years still awaits consensus over a satisfactory anwser (while having far too plenty seemingly convincing ones). http://golem.ph.utexas.edu/category/2008/07/light_mills.html http://golem.ph.utexas.edu/category/2008/07/light_mills.html A toy simulation: http://www.gregegan.net/SCIENCE/LightMill/LightMill.html http://www.gregegan.net/SCIENCE/LightMill/LightMill.html

run4yourlives 17y ago

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 17y ago
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 17y ago
  >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?

deleted 17y ago

[deleted]

DanielStraight 17y ago

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 17y ago
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.

Locke1689 17y ago

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.

rottencupcakes 17y ago

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.

TheSOB88 17y ago
Thank you so much! I really need to add "reversing the problem" to my repository of problem-solving techniques, but I just can't seem to find it.
Retric 17y ago
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.

lisper 17y ago

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!