I agree that is the basic question, except the question doesn't imply "tries" it implies "does". Which if you reread my last post you will see is a physical impossibility If that's true, then it also approximates a situation whereby you pull the nose of the jet up to a vertical cliff. In either case you are obviating the orginal premise, which clearly intends no resistance to horizontal movement of the plane other than through artificially changing the rotational speed of the wheels via a conveyor. (Additionaly, the original question clearly does not include factoring in wheel bearing resistance, tire overheating, etc. People are just confusing themselves with these things). Here is another way to look at this question: consider the plane sitting on the conveyor with the jet engines off. What happens if you then move the conveyor? The answer is the plane stays where it is, and the tires rotate. (Yes I know people are going to have trouble with this, but within the confines of this simple physics exercise, that is what would happen. F=MA, and there is no F being applied to the plane in this case). Simply stated: you can't affect the acceleration of the plane via modifying the rotation of the wheels. The original question posed assumes an impossible situation. You're just adding unnecessary confusion with this statement. The original question implies free-wheeling.
Visualise this, a radial from the center of the wheel to the front edge of the tire, parallel to the conveyer. When the tire rolls forward so that the radial is now tangent to the conveyer belt surface, it has rotated 90 degrees and moved 1/4 of the circumference. The distance the center moves forward is equal to the radial. Thus a complete circumference moves the center of the wheel by 4 radials or 2 diameters. NOT pi * diameter. If you take the sum of the vectors view, then it does resolve to the center of the wheel, which I also discussed. That is under this definition of the problem, the center of the wheel equals the plane's speed since they are fixed in relation to each other. If the conveyer matches the plane's speed, then some people answer differently than if the conveyer matches the tire rotational speed.
SMG2.....is this the 3rd or 4th wave of students you've had to take on? Unbelievable....keep up the fight!!!
seriously, 17 pages........ oh wow, just ephanized......... assuming you have essentially unlimited thrust, it is possible to take off. i say this because the wheel bearings would break before take off. here's how it goes: engine ignition, taxy to treadmill. throttle up (100%, for takeoff). thrust would cause foreward movement (ie. wheels rolling) wheels rolling = treadmill moving since thrust is independent from wheelspin, in miliseconds the airplane would start to move, and the wheels/treadmill would begin an run-away reactionary movement to try and stop the airplane from moving. since the engines are independent from the wheels, the airplane would begine to remove regardless, and (now the theoretical part) the treadmill and wheels would have a v -> infinity. now the real part. nothing can realistically go that fast, so im gonna error on the side of the plane (we'll say a standard 777) the wheel bearings would overheat, fail, and some crazy landing gear failure would appear as something solid hits the treadmill going infinity. now at this point, the plane would hit and the kinetic coeff fricytion would launch it back at some multiplier (the coeff) of the treadmill's speed. but say the treadmill in that nano second went from a friction to a frictionless surface (hey, we're already going infinity mph) the plane would eventually accelerate on the treadmill to takeoff speed and i guess take off. realistically, a treadmill of this size and performance metric could not be built, so........... basically, so
HA HA, this about as ridiculous as it gets. You are misunderstanding something so basic I'm not sure if you are kidding or not. read carefully: One rotation moves the center forward a distance equal to the circumfrence. BASIC BASIC BASIC. Go measure the circumference of your bicycle tire, move the bike forward one rotation of the tire, measure the distance the bike moved forward. GOOD, now you see they are EQUAL.
Okay.. I got owned! I thought about it some more and realized that my premise was entirely wrong to begin with! I assumed that the thrust and the wheel rotation are correlated. I assumed planes were like cars where the power comes from the engine to the wheels--I thought about it and realized that Bernoulli flow and Newton's 3rd law are the principles behind this problem. Therefore, the plane will take off as you mentioned correctly. To the person who stated something about rocket propulsion, it is the expanding gas (which comes from ignition of the fuel) that flows through the orifice in the back which propels the rocket forward--this is again Bernoulli flow and Newton's 3rd law in action.
OK, this is wrong after all, as pointed out. A point on the rim moves this amount in the x-axis during rotation, but the center follows the total path length. I'll just shut up now since I can't think straight any more.
I see where you've gone wrong, Ash. The question doesn't specify what speed the treadmill is matching. Is it matching the rotational speed of the wheel, or the directional speed of the wheel? It CANNOT match both, as they are independent of each other in this scenario. My first scenario that I quoted above is assuming the treadmill matches the rotational speed of the wheels. When the wheels stop rotating, the treadmill stops moving, even though the non-rotating wheels are sliding forward. If the treadmill matches the directional speed of the wheels (which, as long as the wheels are attached to the plane, should equal the plane speed in relation to the ground), then the plane will still take off, because the treadmill does not exert directional force. The plane will move forward at 100 MPH, and the treadmill will spin the wheels at an additional 100 MPH of rotational speed. This will in no way prevent the plane from moving. Get it?
Not the intent of the original question, but it works. You had better rethink this one. The car isn't going anywhere as it has ZERO kinetic energy. If you lock the brakes and the treadmill does the same, you still aren't going anywhere, you are sitting in exactly the same spot and the car still has zero kinetic energy. The car obviously won't accelerate from zero to 140 mph instantly as you suggest.
Hi Folks, Me again. First, thanks very much to Baasha for being so honest about 7 posts or so back. ylshih - also, thanks for agreeing with the wheel circumference part, and thanks to teak for reiterating the correct argument whilst I was asleep. RammsteinMatt - I'm with you all the way! I still maintain that the original problem statement is fine regarding both rotational speed of the wheels and forward speed of the conveyor... The diameter of the wheels doesn't matter. The wheels can (and do) have one speed in relation to the ground and a different speed in relation to the belt of the conveyor. It just so happens that the belt of the conveyor is moving backwards and the wheels spin faster. So the plane moves along, gets wind over its wings and takes off. Cheers! Rich.
My Gosh!! This one is still going!! So much dud math, so many clever thoughts while ignoring the big picture. THE PLANE WILL TAKE OFF AND FLY AWAY. A guy is on ice skates with a rocket pack on his back. The rocket fires, he goes forward. Once he starts moving the ice, as seen by an observer on the shore, starts shooting backwards. Given that the rocket is pushing him, and that he has no grip on the ice, his forward acceleration is unaltered. The key to this small problem is that the runway/conveyor belt has no grip on the planes motor or wings, they exist independently, and thus THE PLANE WILL TAKE OFF. There, now is everyone happy??
Hi Kram, I'm flamin' ecstatic!! It's what I've been saying all along. The conveyor belt can't 'grab' the plane because the wheels are freely spinning. I was thinking of posting and ice-skate type analogy, but in mine I was going to be dragging myself around the edge of the rink and pulling on the top of the wall at the edge cos I'm rubbish at skating! I look forward to the next, er, explanation of the plane staying still, or going backwards, or maybe sideways!! Just as an aside, I mentioned this to an engineer buddy of mine who has worked with aircraft carriers - it took virtually no time at all for him to come to the same conclusion as me, and mxblue, and smg2, and SteveR, and BMW.Williams, and many others - it doesn't make any £$%^&* difference what the wheels under the plane are doing - it pulls itself forwards against the air around it. Cheers! Rich.
OK I have found the answer ... its a stupid question! The question is more of a physcological test than a "puzzle" I guess. I agree that the belt has no purchase on the plane (we can assume the wheels are frictionless to simplify things) and that once motion begins the wheels and belt will rapidly reach a vmax of infinity, and the plane could take off. BUT, for airspeed to be attained the planes wheel speed (angular velocity at the tyres tread) would have to exceed the belts speed (or skid). So as the belt has no "purchase" on the wheels then the only way for this rule to be maintained is for the pilot to reduce thrust to stop the motion - plane does not take off. HOWEVER, if we theorise and say that once the conveyor and wheels hit vmax at infinity "which will happen quickly" that the only way the plane can then move is by pushing the "effectively" locked wheels against the conveyor. Which is the same as the plane powering against its locked brakes - is the torque of the engine sufficient to beat the locked brakes/tyres? If it is then plane takes off (with rather warm tyres) if not then it stays still (with rather warm engine) or it moves but cant reach take off speed. At the start of take off engines are usually throttled up and the plane is held with the brakes - but I imagine this isnt max throttle, so its a question of engine thrust - but I would hazard a guess that even if it moves it wont takeoff (unless the tyres overheat / pop etc but then we are talking silly things anyway, as I say, stupid question!). FINALLY, if we consider that "infinity" is ever increasing and we are not constrained by C then there will never be any resistance to motion and the plane will take off easily (with exceptionally warm wheel bearings)!
Man I need to see a video of this conveyor belt thing!!!!! Both argument seem to work, can someone go and just bulid a conveyor belt. Probably be quicker then sorting it out on here.
The question is a conundrum, therefore there is no absolute answer, other than there is no answer. It defines an impossible situation. Most of the people here now understand that generally a planes wheel rotation has nothing to do with the planes ability to accelerate and take off. That's pretty basic, but the postulate that the plane will fly within the strict confines of this specific question is flawed. As the question is stated the wheels, and the conveyor, must reach an infinite speed instantly. Since that can't happen, you really can't state the plane will fly.
but when you divide by zero, anything is possible. but seriously, speaking theoretically it is a conundrum. but speaking realistically (assuming you could build this machine) the plane would not take off. this is because the wheel bearings would overheat, fail, then sieze. and we would then have the same effect as stepping on to a treadmill going full speed
With all due respect, because it cant happen the plane will fly. Furthermore, (and now dealing from the bottom of the deck) I could point out that the headwind generated by this rushing conveyor belt will ensure the plane lifts off very smartly indeed.
A car would be still, but a plane is NOT dependent on tire contact with the ground in order to move. The tires would rotate twice as fast that's all. If you flip the situation, lets say if the plane is flying at 100knots an hour, and the head wind is 100knots an hour, the plane would indeed hover, but it's air speed would be 200knots.
Yep, that plane is still gonna fly. (Although I take RammsteinMatt's point that in the real-world the wheel bearings 'might' overheat and seize - although when you consider that Thrust SSC exceeded the sound barrier whilst driving on the gound then it shouldn't be too difficult to understand that it is possible for wheel bearing to be constructed that could do around about 360mph without seizing) And I disagree that the constraints of the problem are broken if the plane is propelled forward on the conveyor. There is no need for the pilot to have to keep the throttle low and stay in place, and there is no rule that says the conveyor will have to zoom up to infinite speed in order to satisfy the constraint. I've explained this already several times and given analogies of a guy pulling himself forwards under a moving truck on a skateboard, and also a guy walking along the aisle of a moving train, and I've explained that the sum of the vectors of a rotating wheel are zero and that the wheel speed IS the plane speed. As yet I haven't had any reasoned, believable argument as to how the constraint that the conveyor matches the speed of the wheels, but in the opposite direction has to be broken. Cheers! Rich. Will we get to twenty pages??
oops - forgot to ask in the last post RammsteinMatt - which album do you prefer, Rosenrot or Mutter? I have only Sennsucht and want to buy a pal of mine a Rammstein album for Xmas (he has Sennsucht too, bought it for him last year!! ) Regards, Rich Edit: ...and will I get arrested by an air-marshal for hijacking this 'plane thread mid-flight??
