The speed of the treadmill isn’t important. At all. What’s important is the force it applies to the plane. So why would moving the treadmill at twice the speed the wheels turn apply a force greater than the thrust of the plane’s engines?
But it doesn’t solve the problem that the treadmill doesn’t apply any force on the plane.
I wish someone would answer the question instead of saying “it doesn’t matter”. It may not matter, but I’d like an explanation on why not. The question being: how can the plane move forward when the wheel speed is always slower than the treadmill speed? In other words, if the plane is moving forwards referenced to ground, the plane’s wheels must be rotating faster than the treadmill is moving which is a violation of the premise. How do you get around this?
The plane moving forward is inconsistent with the wheel speed being slower than the treadmill speed. Therefore, the wheel speed is not slower than the treadmill speed.
“It doesn’t matter” is the answer to the question. we’ve answered you numerous times. your stubborn refusal to accept it doesn’t change anything.
because- as has been explained to you ad nauseum- the wheels do not control the plane’s acceleration. The plane does not use them to accelerate to takeoff speed. The plane does not care how fast the wheels are spinning. all the plane cares about is that the engines produce enough thrust to push it fast enough for the wings to generate lift. And all that depends on is that the velocity of airflow over the wings is great enough. the wheels have nothing to do with it. you keep obsessing over the wheels. Stop doing that. Fuck the wheels.
nate, I would suggest that you try an experiment. Buy a Hot Wheels toy car and take it to the gym. Place the car on a treadmill and roll it forwards and backwards to get a feel for how easily it moves. Then have somebody turn the treadmill on and progressively increase the speed. You’ll find that, even with the treadmill running at maximum, it’s just as easy to move the car. You won’t be struggling to push the car forward against the treadmill’s motion; the effort required will be trivial.
C’mon man. How many times do I have to state that I know that the propulsion of the plane is independent of the wheels/treadmill interaction. That’s not the point I’m debating.
Here it is again, please answer this: If I had a video camera focused on a wheel of a plane when it is on this treadmill, and in this video this plane is taking off as you guys imagine, what would I see?
Multiple choice:
(A) I see the treadmill moving faster than the wheel.
(B) I see the wheel moving faster than the treadmill.
But the question is whether or not the plane takes off. It’s not, “given that the plane takes off, what is the wheel speed compared to the treadmill speed”. It’s “can the plane take off if the treadmill speed is equal to (or greater than) the wheel speed?”. If you know the plane is moving forward, you know the wheel speed is greater than the treadmill speed. But the premise states that the treadmill speed is at least equal to the wheel speed. So what happens then?
I’m not sure why this question is so hard for people. The trivial answer is that a treadmill won’t stop a plane from taking off.
If we try and address the literal phrasing, the answer is that the treadmill and tires will turn at infinite speed in an infitesamally short time. The wheels and treadmill fly apart with the shrapnel killing the person who asked the question.
I think you’ve all missed the simplest solution; just have a short runway leading off the edge of a cliff face.
Costs nothing, and the only change you need to make to existing procedures is to make sickbags more widely available.
It would be a trivial uninteresting question if there wasn’t a caveat that the treadmill speed always tops the wheel speed. How the plane moves forward in this situation is what is interesting.
Ok, one more time.
B.
You can’t just invent a “caveat” that’s inconsistent with the way forces are applied in the setup. The wheels turns passively. Wheel speed is determined by the speed of the treadmill, not the other way around. If the speed of the plane is X and the speed of the treadmill is Y, the speed of the wheels is determined by the setup to be X+Y. However fast you turn the treadmill, the wheels always turn faster than the treadmill when the plane is moving forwards.
The original and fun problem is when the treadmill matches the speed of the plane. That’s interesting and fun, because it reminds people that planes are not the same as cars. A car couldn’t move forward with a speed-matching treadmill, but a plane can – interesting and tricky insight! For some values of interesting and tricky. Nate, do you agree that’s the original problem? I know you’ve described that as trivial, but it’s not trivial for people who don’t, for example, look around for physics modelling software to draw up a planetary gear system.
The physics-bending, and ultimately uninteresting problem is where it’s phrased such that the treadmill matches the speed of the wheels instead of the speed of the plane. That brings up paradoxes and infinities that Cecil addressed in the column and has also been addressed here.
You say you can address that issue by having the treadmill move faster than the wheels. Well, first of all, that’s different that either problem as stated. Second, I don’t see how that helps. If the plane is stationary, then the treadmill has to be moving. The plane is providing sufficient thrust to keep the plane stationary. The treadmill is moving backwards at 1 mph, let’s say, with the engines keeping the plane stationary. Well, if the treadmill is moving at 1 mph, so are the wheels, but you’ve said that the treadmill moves faster than the wheels, so it’s now at 2 mph, but so are the wheels, so now it’s at 3 mph, but so are the wheels, so now it’s at 4 mph, but so are the wheels… Do you see the problem?
In your idealized world of frictionless bearings and infinite friction treadmill surfaces, I say the plane takes off anyway. The plane doesn’t notice the treadmill moving, since the wheel bearings are frictionless and moves forward on its merry way, ignoring the treadmill and wheels chugging along infinitely fast.
Agreed?
And, with that, I predict this thread will die a natural death.
Wot? 'e says 'e’s not dead!
You’re asking “what happens when the treadmill is moving faster than the wheels”. We’re saying that nothing happens in that case, because the treadmill moving faster than the wheels is impossible. You might as well ask what happens when the airplane is made out of love, or when the treadmill moves straight up but the plane stays at the same height. No meaningful answer is possible, because the posited situation is impossible.
but what if it was possible?
Good morning ladies and gentlemen, welcome to Fubar airlines flight 206. Please remain seated as they strap the airplane to the conveyor belt and turn the belt towards St. Louis and launch us off to our 600 foot cruising altitude where we’ll fly at 45,000 MPH. Wait, 36,000 MPH, sorry. The smoking area is on the port wing for those who desire a break along the way. Please place your crying brat in the overhead and shut it until it locks securely.
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Thanks for being the only one brave enough to answer the multiple choice question. But as you’re aware, answer ‘B’ breaks the premise of the question, and in answer ‘A’ the plane never moves forward and therefore never takes off.
So you’re saying the wheel speed is determined by the speed of the treadmill and the speed of the plane relative to ground, W = X + Y, and my stipulation that Y >= W forces X to be negative or 0. Is this an impossible scenario? I don’t see why it would be. In a real world experiment it is definitely doable. You can imagine yourself standing on a giant treadmill wearing skates that gradually speeds up to 1e6 ft/s and your method of propulsion is a leaf blower. I think in this scenario you would have no trouble believing the treadmill will always move faster than your skate’s wheel speed because of your puny propulsion + rolling friction.
Please hold back your ire for a moment… I think it’s starting to make sense to me.
Nate, you might be under the mistaken impression that rolling friction increases with wheel speed. It doesn’t. Rolling friction is a constant. If your leaf blower has sufficient thrust to overcome the rolling resistance at 1 mph, it has sufficient thrust to overcome rolling resistance at 1,000,000 mph because the rolling resistance is the same at both wheel speeds.
Returning to the OP… putting aside the fact that it’s difficult to convince people that yes an airplane can take off from a treadmill, the treadmill is totally useless as a means of shortening the takeoff distance. Revving up the treadmill to make the airplane wheels spin fast does absolutely NOTHING to get wind moving across the wings. Your treadmill would have to be just as long as the runway it replaces.