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  1.    #1  
    Every time I've seen this problem on message boards, it's triggered heated debates. And so I figured this would be great for TC. Both sides are always absolutely certain they are correct and that the others are complete idiots. Here it is:

    A plane equipped with fixed horizontal engines and wheel landing gear is placed on a huge treadmill runway. The treadmill has a clever design and always matches the speed of the plane, but runs in the opposite direction. Will the plane take off and fly or not?
  2. #2  
    Hmmm... if the plane is on a treadmill that's matching its speed in the opposite direction, is the plane standing still? It would be standing still if the wheels of the plane were providing the forward thrust, but they're not...

    It's the jet engines providing the forward thrust, not the wheels, so the jet would take off...

    As a test, put your roller blades on and stand on a treadmill. Using a 20 foot rope stretched out and tied to the wall in front of you, turn on the treadmill and immediately start pulling your self forward using the rope. Does the treadmill matching the speed of the wheels on your blades stop you from pulling yourself forward and off the treadmill?

    I say the jet plane moves forward and takes off.
  3. #3  
    If the plane does not move, no lift would be provided by the wings. Even with the jet engine involved, the force required to lift is dependent on the wings, at least for planes.
  4. Hanuman's Avatar
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    #4  
    Plane stays grounded. Wind needs to pass over the wings to provide lift. The design of the wings make the air above the wings travel faster then the air below the wings. This difference in airspeed caused an upward pressure or "lift". No fast wind, plane no go up.
    Palm IIIe -> Visor Deluxe -> Palm 505 -> Clie tj37 -> Treo 600 -> Treo 700P
  5. #5  
    This problem was posted on airliners.net and got well over 200 responses. Debate was lively. Turns out the question was worded wrong and implied that the conveyor belt matched the speed of the -wheels- but in the opposite direction. (If that was the case, the plane could never even begin to move.) Once the wording was corrected, consensus was reached very quickly.

    The original question as posted elsewhere has the conveyor matching the speed of the -airplane- but in the opposite direction, which would only mean the tires of the plane would be spinning at twice their normal speed during the takeoff run. In that case, as long as the airplane doesn't exceed the tires' speed rating, the plane would be able to take off normally. The engines don't power the wheels; they push the airframe. As long as the brakes aren't applied, the plane will accelerate and take off.
    "Yeah, he can talk. It's gettin' him to shut up that's the trick!"
    -Shrek
  6. #6  
    Quote Originally Posted by Tom LaPrise View Post
    The original question as posted elsewhere has the conveyor matching the speed of the -airplane- but in the opposite direction, which would only mean the tires of the plane would be spinning at twice their normal speed during the takeoff run. In that case, as long as the airplane doesn't exceed the tires' speed rating, the plane would be able to take off normally. The engines don't power the wheels; they push the airframe. As long as the brakes aren't applied, the plane will accelerate and take off.
    That's not true IMHO.

    Speeds are always relative. What does "the treadmill matches the speed of the airplane" mean? In the case of a car, I guess the meaning would be that the treadmill goes in the opposite direction of the car, at the speed the car WOULD make on normal ground. So at a speed of 60 mph the car would drive 60 mph relative to the surface of the treadmill but 0 mph relative to the ground around the treadmill.

    The case of the plane seems different at first glance because there the jets push the plane forward, not the wheels. However, this is not really relevant in my view. The airplane could just as well roll forward like a car by moving the wheels/using a motor. For the speed of the wheels or the speed of the airplane relative to the ground, this makes no difference.

    Therefore, when the plane starts moving on the runway, the airplane is basically identical to a car: the wheels roll forward, but the treadmill compensates that movement relative to the surrounding ground, and - more importantly - relative to the surrounding air. Suppose the pilot increases forward thrust - the treadmill will just go faster as well, the plane still stands still relative to the air. Because of this, no air flows over the wings, no upward lift is produced, the plane remains on the ground and continues to behaves just like a car would - it stands still relative to the ground/air and does NOT take off.

    I know that there are lots of wrong solutions to this problem to be found on the Internet, but I am quite confident that the above is correct. If you disagree, please state which part of my reasoning does not work.

    I guess the misunderstanding stems from the fact that the situation of a plane in the air is different. After takeoff, the movement of the plane is independent of the movement of the wheels. But this is not the case as long as the airplane is rolling on the ground. There, it cannot gain speed relative to the air because of the speed-compensating treadmill and stays grounded.
    “Reality is that which, when you stop believing in it, doesn't go away.” (Philip K. ****)
  7. #7  
    I guess the plane could fly, if the the treadmill produced enough drag to move the air above it (the treadmill) to a point (i.e. sufficient air velocity) where this movement (of the air, across the wing) would produce lift.

    Since we are talking hypothetically, I guess this is as hypothetical as it gets.
  8. #8  
    Sigh. A knowledge of physics is a good thing, but clearly not something widely held. Tom LaPrise has it exactly right. The speed of the tires has nothing to do with the speed of the airplane.

    Clulup: your reasoning is wrong because your basic assumption is wrong. There is a fundamental difference between the engines driving the wheels to move the plane and the wheels spinning freely as the plane moves. In fact (jumping to your last statement) the movement of the plane on the ground IS independent of the speed of the wheels. There's no difference in the forces that move a plane on the ground vs. in the air. Consider the case of a sea plane, on pontoons: no wheels spinning at all, but the plane takes off. In this case the speed of the plan on the ground [water] is clearly independent of the speed of the objects in contact with the ground. Or think of this: take the wheels off the plane, replace them with skids, and have the plan take off from an ice covered field. Again, the speed of the parts in contact with the ground are irrelevent to the speed of the plan.

    Look at Big-OTRs post again: This is not a bad example. you pulling on the rope is an external force, just like a jet engine. Even if the treadmill speeds up as you move forward, you still keep moving forward.

    Let's expand on this. The plane is like you on roller blades: No force directly turning the wheels. Say the treadmill's moving at 5 mph. Now mount a big fan just behind you that generates enought thrust to move you 5 mph, and you let go of the rope. What happens? Do you stand still? Or do you get pushed forward into the front of the treadmill?

    The wind from the fan is a force completely external to the treadmill, just like your muscles pulling on the rope. The fact that the treadmill is spinning backward doesn't reduce the force of the wind pushing on your back. If that wind would push you forward on solid ground, it will push you forward on the treadmill. The wheels on your skates would turn faster, because THEY are moving at a speed of 10mph relative to the belt (the 5mph speed of the belt + the 5 mph speed of you). The only difference in speed would be caused by the extra friction in the skate's bearings, caused by the extra heat generated by the spinning wheels.

    But the jet engine isn't mounted on the ground, you say? Doesn't matter. It's physics, again. You could mount the fan on your head pointing backwards, and (assuming your neck is strong enough), the forces involved are exactly the same. "For every action there is an equal and opposite reaction." The fan blowing air backwards pushes you forward.

    I suppose this won't convince a lot of people either. But the laws of physics will prevail, regardless of what anyone believes. (That's the nice thing about science, as opposed to politics: facts are facts, and don't change just because someone doesn't like them.
    Bob Meyer
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  9. #9  
    Quote Originally Posted by meyerweb View Post
    Sigh. A knowledge of physics is a good thing, but clearly not something widely held.
    I was aware of the things you wrote. I think it all depends on what "the treadmill matches the speed of the airplane" means. As I wrote above, my understanding is that the treadmill goes in the opposite direction of the plane, at the speed the plane WOULD make on normal ground.

    So suppose the airplane is in a state in which it would - under normal conditions - roll on the runway at a speed of 275 km/h or 170 mph (takeoff speed of an A320). Due to the conditons given in the original question (according to my understanding), this means that the treadmill rolls at 275 km/h into the other direction, resulting at speed zero relative to the air. No?

    The more thrust you give, the faster the treadmill goes into the other direction... or doesn't it?
    “Reality is that which, when you stop believing in it, doesn't go away.” (Philip K. ****)
  10. #10  
    Sod that, get yourself a Harrier jump jet. End of problem. Next!
    Animo et Fide
  11.    #11  
    Quote Originally Posted by clulup View Post
    I was aware of the things you wrote. I think it all depends on what "the treadmill matches the speed of the airplane" means. As I wrote above, my understanding is that the treadmill goes in the opposite direction of the plane, at the speed the plane WOULD make on normal ground.

    So suppose the airplane is in a state in which it would - under normal conditions - roll on the runway at a speed of 275 km/h or 170 mph (takeoff speed of an A320). Due to the conditons given in the original question (according to my understanding), this means that the treadmill rolls at 275 km/h into the other direction, resulting at speed zero relative to the air. No?

    The more thrust you give, the faster the treadmill goes into the other direction... or doesn't it?
    Yes, but since the wheels of a plane spin freely, the treadmill wouldn't slow down the plane by much. The plane will go 275 km/h in one direction, and the treadmill will be zipping 275 km/h in the other direction. And the wheels will be spinning at the equivalent of 550km/h.

    Interpreting the speed of the plane as being relative to the moving surface of the treadmill doesn't work. When the plane's engines start, the plane will begin moving; the treadmill is not capable of offsetting the force of the engines. And then you'll be trying to set (the speed of the treadmill) = (the speed of the plane relative to the treadmill). But that equals (the speed of the treadmill) + (the speed of the plane relative to the ground).
    a=a+b, where b>0
    Can't happen.
  12. #12  
    Quote Originally Posted by samkim View Post
    Yes, but since the wheels of a plane spin freely, the treadmill wouldn't slow down the plane by much.
    That's not true. The wheels of a plane may spin freely once the plane is in the air, but before that, they are subject to friction exactly like the tires of a car.

    A knowledge of physics is a good thing, but without knowledge of the problem at hand, it can lead astray... As it said in the first post, "the treadmill has a clever design and always matches the speed of the plane, but runs in the opposite direction."

    If that means the treadmill will always run at the speed the plane would have on normal ground, but into the opposite direction, the plane will not move relative to the air and not take off. I think this is clear and not disputed.

    "Taking off" simply means "reaching takeoff speed" (about 275 km/h or 170 mph for an A320), meaning speed relative to the surrounding air. If we forget what the original question was and and just say "a plane stands on a treadmill which accelerates in some way until it reaches 275 km/h, while the plane accelerates into the opposite direction of the treadmill. Can the plane take off? Obviously it can, as long as it can go at least twice the 275 km/h and reach takeoff speed relative to the surrounding air. Relative to the treadmill, the plane would then roll at 550 km/h. However, in this example, the behaviour of the treadmill is different from the one in the original question (it just rolls at 275 km/h and does not continue to accelerate as the plane accelerates).

    From this thought experiment, one can also see that the whole question has nothing to do with whether it is a car or an airplane. Think of a car standing on a treadmill going 50 mph. Can the car reach 50 mph relative to the ground/air around the treadmill? Obviously it can, as long as it is able to drive at a 100 mph in the opposite direction on the treadmill. The only difference is that 100 mph is not takeoff speed and the car doesn't have wings.

    Both you and meyerweb seem to confuse the treadmill question with some sort of "zero friction" situation. With zero friction on the wheels, indeed a plane could move in the opposite direction of the treadmill, while a car could not (a plane gains thrust because it is pushed by the exhaust fumes in the jets, or pulled by the propeller, while the car needs the friction on the tires, which is zero in this example, so it cannot move).

    However, in the original problem nobody ever said there is no friction, so all the talk about differences between a jet and wheels and planes and cars may be impressive, but irrelevant to the present case.
    “Reality is that which, when you stop believing in it, doesn't go away.” (Philip K. ****)
  13. Hanuman's Avatar
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    #13  
    Wow all that goobly gook just to say what we've all been saying

    Plane stays on the ground (or in this case the magic treadmill)
    Palm IIIe -> Visor Deluxe -> Palm 505 -> Clie tj37 -> Treo 600 -> Treo 700P
  14. #14  
    Nope. Plane takes off unless somehow, there can be enough friction in the wheel assemblies to equal (in the opposite direction) the thrust of the engines. The only way that's going to happen is if the brakes are applied or the wheels reach insane speeds and begin to seize their bearings. Depending on the airplane, it might only have to be going 65 MPH (maybe 160 MPH for large jets), so the tires will be spinning as if the light plane was doing 130 MPH or the big jet is doing 320 MPH. The wheels will not generate anywhere near enough friction to resist the plane's acceleration even at that point unless the brakes are on.

    In the incorrectly-worded version (conveyor matches the speed of the plane's wheels rather than the speed of the airplane), the airplane cannot accelerate because the wording of the problem wouldn't allow it. For the plane to move forward relative to the air around it, the wheels would have to move faster than the conveyor belt.

    In the correctly worded version, though, the plane will take off with the wheels spinning at twice normal RPM, provided the tires don't blow apart.
    "Yeah, he can talk. It's gettin' him to shut up that's the trick!"
    -Shrek
  15. #15  
    the treadmill matches the speed of the airplane: What is the reference frame in which the speed is being measured?

    The most common interpretation: the reference frame is attached to the ground => Plane will take off

    Plane moving 200 mph relative to the ground, treadmill top surface moving 200 mph relative to the ground in the opposite direction. Plane takes off because it has a relative ground-speed and air flow over it's wings. Wheels spin at twice the rpm it normally would have.
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  16. #16  
    Quote Originally Posted by Tom LaPrise View Post
    In the correctly worded version, though, the plane will take off with the wheels spinning at twice normal RPM, provided the tires don't blow apart.
    correct, the engines work relative to the air not to the ground, as long as the plane can get enough airspeed it will take of regardless of the ground speed.
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  17. #17  
    From a physics standpoint, it's no different than (for example) charging down the runway in a Cessna 172 at 120 knots with a 60-knot tailwind. The 172 will take off at 60 knots indicated (with pilot and two passengers--been there and done that--though not with a 60-kt tailwind!), and wheel friction at 120 knots is nowhere near enough to keep the plane from accelerating.
    "Yeah, he can talk. It's gettin' him to shut up that's the trick!"
    -Shrek
  18. #18  
    Quote Originally Posted by ToolkiT View Post
    correct, the engines work relative to the air not to the ground, as long as the plane can get enough airspeed it will take of regardless of the ground speed.
    That is true, but for gaining speed on the treadmill it makes no difference that the plane is powered by jet engines and not via the wheels. Acceleration works just as well with wheels powered like those of a car - at least until takeoff, then the wheels would be no good anymore and the plane would come down again.
    “Reality is that which, when you stop believing in it, doesn't go away.” (Philip K. ****)
  19. #19  
    Why is the plane on the treadmill?
    Is it trying to lose wieght?
  20. #20  
    Quote Originally Posted by clulup View Post
    I was aware of the things you wrote. I think it all depends on what "the treadmill matches the speed of the airplane" means. As I wrote above, my understanding is that the treadmill goes in the opposite direction of the plane, at the speed the plane WOULD make on normal ground.

    So suppose the airplane is in a state in which it would - under normal conditions - roll on the runway at a speed of 275 km/h or 170 mph (takeoff speed of an A320). Due to the conditons given in the original question (according to my understanding), this means that the treadmill rolls at 275 km/h into the other direction, resulting at speed zero relative to the air. No?

    The more thrust you give, the faster the treadmill goes into the other direction... or doesn't it?
    To use my previous example, it would be no different than if your A320 barrels down a long runway at 550 km/h with a 275 km/h tailwind. The indicated airspeed will be 275 km/h, so the plane will take off. The only way the plane could NOT take off is if a) it runs out of runway, which would not happen in the treadmill scenario, or b) friction within the landing gear is such that the plane cannot reach 275 km/h in the first place (wheels are spinning fast enough for 550 km/h, remember). I'd bet the tires would blow before that, but assuming they don't, I doubt bearing friction could be that high.

    The speed of the treadmill isn't reacting to the thrust you apply; it's reacting to how fast the airplane is moving relative to the rest of the Earth. The plane will accelerate normally, but the belt will speed up to match (in opposite direction) the plane's speed relative to Earth, so the wheels will just have to spin twice as fast as normal.

    (To clarify: The plane accelerates to 275 km/h "relative to Earth," the surface of the belt accelerates to 275 km/h in the opposite direction "relative to Earth," and the wheels accelerate to sufficient RPM for the difference in the velocities--or the sum of the speeds, same thing--of 550 km/h.)
    Last edited by Tom LaPrise; 12/08/2006 at 06:55 AM.
    "Yeah, he can talk. It's gettin' him to shut up that's the trick!"
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