this post was submitted on 26 Jun 2024
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My favorite is planes on a treadmill.
Mostly because fans still argue about it and it’s hit the point they had to ban PoaT comments.
Which is insane as it’s not that difficult to understand. When a plane is on the ground, its gear/wheels will roll at ground speed, but the wings provide lift at airspeed.
If the ground is being moved under the plane (as on a treadmill,) the wheels will just roll faster.
Sure they’re not zero friction and some of that needs to be overcome; but this is something encountered on a daily basis all across the world- or rather, the opposite.
If the wind is coming from ahead, its airspeed is increased and the plane needs a lower ground speed to get into the air where if the wind is coming from behind, then they need more.
(This is why carriers set course into the wind when launching jets,)
At no point is ground speed and airspeed necessarily the same (i suppose you could have a calm day, but most days, the wind is blowing at least some.)
I found it hard to understand because neither they nor any of the other sources I've seen explaining this even attempted to answer what I thought was an incredibly obvious question: at what point does this become true? A stationary aeroplane on a treadmill will obviously move with the treadmill. I assume an aeroplane moving at like 1 km/h still gets pulled backward by the treadmill. At what point does the transition occur, and what does that transition process look like? Why can't a treadmill prevent the plane from taking off by pulling it backwards by never letting it start getting forward motion? Where does the lift come from?
I can understand how a treadmill doesn't stop a plane that's already moving, but how does it get lift if it is prevented from accelerating from 0 to 1 km/h of ground speed (relative to the real ground—relative to the ground it experiences, it is moving forward at the same speed as the treadmill is moving backward), since until it starts getting lift, airspeed and ground speed are surely effectively equal (wind being too small of a factor)?
That’s the thing - it is not prevented from accelerating. The wheels are functionally frictionless. That’s why planes have brakes. The plane pushes on the air to move, & the treadmill could accelerate backwards until the plane’s tires explode.
so, every wheel or ball or any other kind of rolling-thing has rolling resistance, which is how we sum up the total drag on the system. A steel ball bearing on a steel plate will have a significantly lower rolling resistance than, say, a steel cube on that same plate. Tires have some- but not a lot- of rolling resistance.
You can see that in a car, just put it into neutral and watch as you slow down, even on flat ground. Plane wheels also have rolling resistance. it's just the way our world works. But it's generally ignored because it's hard to model perfectly and in any case pretty negligible relative to the amount of acceleration being put out by modern aircraft engines.
A treadmill will only push an aircraft or whatever else along, with an acceleration that is equal to, or lower, than the rolling resistance. If you try to accelerate the plane faster, it'll 'slip', and the plane will remain largely stationary- like the dishes in the tablecloth trick (if you want to try that at home... make sure the tablecloth doesn't have a hem, heh.)
But, keep in mind you're thinking about the plane relative to either the ground, or the treadmill's belt.
the plane's wings and it's engines are 'thinking' about the plane relative to the air it's moving through. It's the airspeed that generates the lift, and the engine isn't coupled to the wheels, they're just rolling along doing their thing. (aircraft engines work by taking a volume of air and accelerating it. newton's equal-and-opposite does the rest.)
Oh wow thank you. This is genuinely excellent and immensely helpful. I think this bit:
As well as this video that I found where a pilot explains how under specific but unrealistic conditions you could construct a treadmill that does indeed prevent an aeroplane from taking off,
Really helped solidify my understanding of the problem. So you end up with a situation where the wheels are going to be slipping, just like the slippage created when your hand pushes a toy car on a treadmill.
Thanks!
So, another way to think about it is with Kites.
The air flows around it the same way it would any other kind of aircraft, though they have effectively zero ground speed.
They do differ in that, being tethered, they’re pulled through the air, with the wind providing the energy to stay up.
But they’re still moving through the air, and the airfoils are inducing drag to convert some of that energy into lift.
In both cases, the important speed is relative to the air, not the ground and not the treadmill. The wheels might impart some drag while they’re on the ground, but they’re never going to impart enough to overpower the engines- 747s typically take off at about 75% of their rated take off power, which means a longer take off roll, but less wear and tear.
The key insight is that the force a plane uses to move is independent of the ground, because planes push on the air, not the ground.
Imagine you put a ball on a treadmill and turn it on, what happens? The ball starts to spin and move with the treadmill. Now take your hand and push the ball backwards against the motion of the treadmill, and the ball easily moves in that direction. The force your hand put on the ball is exactly what planes do, since they push on something other than the ground (the treadmill) they have no problem moving, no matter how fast the treadmill is moving.
The tricky bit is that the air within a few millimeters of the treadmill will move with the treadmill. The air slightly above that will be slightly disturbed and also move a bit in the direction of the treadmill. If you had an extremely long and extremely wide treadmill (say the length and width of a runway) it's possible that the air at the height of the propeller would be moving along with the treadmill, rather than staying still, or moving with prevailing winds.
But, even in that case, the plane could still take off. All the plane needs to do is move the body of the plane through the air at enough speed to allow the wings to start generating lift. If the air at propeller-height is moving with a treadmill that is moving at take-off speed, the plane might take off with zero forward speed relative to the non-treadmill ground. But, as long as you're not somehow preventing the propeller from moving the plane through the air, the plane will always be able to take off.
There are videos of planes taking off by themselves in high wind, and videos of VSTOL (very short take-off and landing) planes taking off and landing using only a few metres of runway.
The point it occurs at is when the plane uses the air to propel itself.
It's always true.
What do you mean? The plane has its parking brakes on and moves with the treadmill surface? If you don't have parking brakes engaged and start up a treadmill under a plane, the plane's wheels will spin and the plane will stay pretty much in one place. Because the wheels are free to spin, initially that's all that will happen. The inertia of the plane will keep it in place while the wheels spin. Over time, the plane will start to drift in the direction the treadmill is moving, but it will never move as fast as the treadmill because there's also friction from the air, and that's going to be a much bigger factor.
Moving at 1 km/h relative to what? The surface of the treadmill or the "world frame"? A plane on a moving treadmill will be pulled by the treadmill -- there will be friction in the wheels, but it will also feel a force from the air. As soon as the pilot fires up the engine, the force from the engine will be much higher than any tiny amount of friction in the wheels from the treadmill.
It isn't prevented from accelerating from 0 to 1 km/h of ground speed. The wheels are spinning furiously, but they're relatively frictionless. If the pilot didn't start up the propeller, the plane would start to move in the direction the treadmill is pulling, but would never quite reach the speed of the treadmill due to air resistance. But, as soon as the pilot fires up the propeller, it works basically as normal. A little bit of the air will be moving backwards due to the treadmill, but most of the air will still be relatively stationary, so it's easy to move the plane through the air quicker and quicker until it reaches take-off speed.
Plane on a treadmill is really interesting because if you understand how planes work its so obvious what will happen you don't need to test it. Planes move on the ground by running their engines, which push against the air, the wheels provide zero motive force. It's also why planes need tugs to move away from the gate, you can't run the engines in reverse. Planes are not cars, but people tend to assume the thing they don't understand works like the thing they do understand, and refuse to believe their hasty assumption is wrong even when told directly their hasty assumption is wrong.
You actually can run the engines in reverse. They have thrust reversers. There's very good reasons that they do not reverse the plane from the stand using the engines, but it is possible.
my criticism of PoaT actually has to do with the scale model they used to prove it.
scale aircraft have ridiculous power-to-weight ratios
but that's just me being a stickler.
I think the confusion is that the conveyor belt is running at a fixed speed, which is the aircraft's takeoff speed. That just dictates how fast the wheels spin, but since the plane generates thrust with its propeller, the wheels just end up having to spin at double takeoff speed. Since they're relatively frictionless, that's easy.
The more confusing myth is the one where the speed of the conveyor belt is variable, and it always moves at the same speed as the wheels. So, at the beginning the conveyor belt isn't moving, but as soon as the plane starts to move, and its wheels start to spin, the conveyor belt movies in the opposite direction. In that case, the plane can't take off. That's basically like attaching an anchor to the plane's frame, so no matter how fast the propeller spins, the airplane can't move.
Except it’s not like attaching an anchor. The plane isn’t physically attached.
The wheels will just roll double whatever the current ground speed is. If the plane has enough thrust to take off with the treadmill moving an inverse of its take off speed, then it has enough force to start rolling, too.
At most, the force applied by the treadmill would be sufficient over enough time to lengthen the take off roll, but given enough space to do so, the plane will take off.
To keep the plane from rolling forward; the treadmill would have to be able to apply an equal force as the engines, it can’t do that through the wheels- the wheels can only apply a force equal to their rolling resistance and friction in its mechanics.
If the conveyor moves at the same speed as the wheels, it is exactly like attaching an anchor. That isn't the myth they were testing, but it's a more interesting myth.
It can do that if it can spin the wheels fast enough. Picture the ultra-light airplane from the episode with big, bouncy wheels and a relatively weak propeller. If the treadmill was moving 1000 km/h backwards, that little propeller could never match the force due to rolling resistance from the wheels.
Just to clarify; you understand that because the engines are pushing on the plane itself and not the wheels, by the time the wheels start moving, the plane is already moving relative to ground and air alike.
Which, said another way, this thought problem appears confusing because it’s being considered from otherwise irrelevant reference frames.
An anchor sufficient to keep the plane from rolling forward is different because the force it is apply is significantly greater.
Sure, you can deflate the tires and increase the rate of spin on the wheels. But at that point, you might as well ask “can we creat a scenario where planes can’t take off”
To which the answer is definitely “Yes”,
And as a side note, if we assume the wheels are indestructible, which I’d argue is only fair, then even if what you’re saying is true and we ramp up the drag induced by the wheels sufficient to counter the engines… then the wind generated by the rolling treadmill would be producing a sufficient headwind for the plane to take off. (Remember, the air resistance of the treadmill’s belt moving will accelerate the air some.)
But again, the wheels have almost zero drag to begin with, the speed at which the roll is independent of both the actual groundspeed and the airspeed of the airplane.
If it has the thrust to over come friction at take off speeds, and at standing, then it has enough power to get to take off velocity eventually.
On the other hand, this entire conversation assumes the thrust to weight ratio is less than 1. If it’s more than one, well they just…. Go straight up.
The wheels are attached to the plane so they move at the same time as the plane. But, I get what you're trying to say, that the wheels are effectively being dragged by the plane, they're not powering the movement. But, what you need to think about is that if you oppose that dragging by moving the conveyor belt in the opposite direction you can prevent the plane from moving at all. Yes, the wheels are merely dragging and there isn't a lot of friction there, but friction increases with speed. And, if you move the conveyor belt fast enough, you can stop the plane from moving relative to the ground, which can stop it from moving relative to the air, which can prevent it from taking off.
No, by definition it's the same. The conveyor moves with however much speed is necessary to stop the forward motion of the plane. The conveyor would eventually go so fast that it generated enough force to stop the plane from moving, so it's indistinguishable from an anchor.
You don't need to deflate the tires, you merely need to increase the speed at which the conveyor moves to match the speed of the wheels.
That seems like an unfair assumption because you're assuming that the conveyor belt has second-order effects on the air (i.e. generating a "wind" over the wings of the plane), while ignoring the second-order effects the conveyor would have on the wheels (massive heat from friction leading to failure).
I mean, the discussion is of a plane, not a helicopter or a rocket. Even jet fighters with a thrust-to-weight ratio of more than 1 typically have engines that only have that ratio once they're at high speed, not from a standing start. That's why even fighter jets on carriers need a catapult-assisted takeoff. A VTOL aircraft like a Harrier wouldn't need that, but then its takeoff speed is zero, and the myth isn't very interesting when the conveyor belt doesn't move.
no. I'm saying that by the time the wheel is rolling, the plane's is already moving forward, the engines have already overcome the drag in the wheels. the treadmill is locked to the wheels, not the plane. The plane would continue accelerating even as the wheels reported weird rates of turning.
As for the (very brief) time delay, that's a function of the plane's gear's suspension that is quite well sprung.
the rate of roll on the tire is, effectively, decoupled from the airspeed (and groundspeed) of the plane. which makes this:
... entirely different. an affixed anchor does not allow the free motion that a wheel would.
And one of a few things happen. Either the plane has enough engine thrust to overcome the acceleration induced by the wheels, and therefore takes off, or it does not.
In the case that it does not, the wheels would continue spinning in increasing RPM until the plane begins moving backwards. because, again, the airspeed of the airplane is not dependent on the wheel's RPM. Assuming the airplane doesn't crash from suddenly becoming incredibly difficult to control.. eventually it would take off anyhow. because the airflow over the wings would still generate lift. (though they would become horribly inefficient.) and therefore take off.
this is of course ignoring the whole "can a pilot actually control that and manage a take off like that" thing. If you don't want to grant godlike piloting skills, we could then just make the treadmill irrelevant and leave the brakes on.
The wheels are attached to the plane, so they move at the same time. There's going to be slight flex due to rubber and metal not being insanely stiff, but essentially as soon as the plane starts moving forward through the air, the wheels start rolling forward along the ground. Since the conveyor belt cancels the forward movement of the wheels, the movement of the plane ceases too.
Initially, for a few tenths of a second, or a few seconds sure. But, during that time, the conveyor belt would be moving faster and faster as it matched the speed of the wheels. The faster the conveyor moved, the more friction there would be, and the more drag there would be from that friction. Eventually you'd reach an equilibrium where the drag from the wheels was equal to the thrust from the engine, and the plane would cease moving forward. It would be exactly like the plane being anchored to the ground, except instead of a stationary anchor, the anchor would be a spinning treadmill in contact with a spinning wheel. In a world without a magic conveyor belt that could instantly adjust to the speed of the wheels, there would be some slight forward and backward movement of the plane, but that's just like being attached to an anchor with a bungee rather than a rigid rope.
The wheel doesn't have free motion. By definition, the conveyor is moving at the same speed as the wheel, so the wheel is locked in place. With a real conveyor belt there would of course be some lag as the motors of the conveyor accelerated the belt, but using the hypothetical as defined, the axle of the wheel couldn't ever move because every rotation of the wheel would be matched by a movement of the conveyor belt.
The thrust would have to be infinite because, by definition, the conveyor is always going to match the velocity of the wheels. If the wheels were truly frictionless, then the conveyor belt would have no effect at all. But, any real wheel will have some friction that will increase with speed, so there will always be some speed where the force backwards from the friction of the spinning wheels matches the force of the engine.
As an aside, my guess is that most real airplane wheels would probably fail pretty quickly at just double the normal takeoff / landing speed. The centripetal force acting on the spinning parts of the wheel and tire increase with the square of the velocity, so 2x as fast means 4x as much force. 3x as fast and 9x as much force. So, if you did this with a real wheel, you'd destroy the wheel pretty quickly. Of course, the same applies to the conveyor belt, but I'm going to assume that it's specially engineered to survive this challenge.
The plane wouldn't move backwards because if the wheels slowed down, the conveyor belt would slow down too. Of course, that's in a world where the conveyor belt could adjust its velocity instantaneously, but for this thought-experiment you can say that if the pilot cuts the engine or something, the wheels don't spin as fast, so the conveyor belt slows down, and the plane remains in one spot.
In the thought-experiment world, there wouldn't be any airflow over the wings because the plane would be stationary. In reality, there would be some airflow due to the movement of the conveyor belt, but the wheels would probably melt long before that was enough air to give the plane lift while stationary relative to the world around.
Plane on a treadmill always seems so obvious to me. Planes don't have power connected to their wheels. Put a plane on a dynamometer and crank the engine up as fast as it will go, and the wheels will still not spin. At the same time, water planes use pontoons and are still able to take off just fine.
The question I have is, can a plane take off with a tailwind that matches the speed that the propeller is pushing out.