well, if you’re going to get that far into the ideal “spherical cow” world, then you have to treat the bearings and tires as frictionless, at which point the belt/conveyer can’t apply any force to the aircraft at all.
missed the edit window:
sigh
there’s no need for math.
in the idealized situation, you have a conveyor which can instantaneously accelerate to match the rotational speed of the wheels. which means you have to also treat the wheels/tires/bearings of the plane as frictionless with no rolling resistance. therefore the conveyor can’t act on the aircraft with any force to impede its forward motion, and it takes off.
in the real world, you can’t have a conveyor which can instantaneously accelerate to match the aircraft’s wheels. the aircraft’s engines push it forward via thrust, and it takes off with maybe a bit more distance before rotation and takeoff.
if you want to have a half-ideal, half-real-world situation such as a ideal conveyor which can actually accelerate instantaneously but have an aircraft with real-world characteristics (friction in the wheel bearings and tires with rolling resistance) then the tires will explode and/or the bearings will burn up from overspeed.
I don’t understand why people have so much trouble with this.
I cannot tell if this is another plane on the conveyor belt thread (guys, we’ve thrashed this one around endlessly; the straight dope on it is able to be found by searching old threads…), or a twist on the plane on the conveyor belt question (where the belt goes the same direction as the plane). Either way, I think it is relatively silly to conceive of using conveyor belts rather than catapults to launch planes.
I promise I’m not trying to be stubborn and maybe someone can explain it in a way that I have an “a ha!” moment but that hasn’t happened yet. Can someone explain this:
Given that the plane moves ahead 1’ wrt ground and the plane’s wheels have a circumference of 1 ft and so with one rotation the belt moves backwards 1 ft. Where does the 1’ advancement come from? At some point the wheel would have to turn faster than the belt moves, right? This is where I get hung up.
If the planes engines are running, thrust from them pushes the plane forward. There’s nothing (or essentially nothing since there is some friction) the treadmill can do to counteract that. The wheels of the plane aren’t powered; they spin however fast they need to spin as the plane accelerates forward.
So it’s impossible for a treadmill to keep a plane in place - the thrust of the jet engines isn’t acting on the treadmill through the wheels. The plane accelerates and takes off regardless of what happens between the wheels and the treadmill.
With a conveyer belt, the ground moves under the plane instead of the plane moving over the ground. The concept behind this is similar to you on a treadmill. The treadmill goes backwards & your walking/running at the same speed as the treadmill keeps you in the same relative place in the room even though you are moving. Therefore, if we design a treadmill big enough & fast enough, a plane could take off in much less space than a runway uses. However, planes fly because of the shape of the wing & airflow over the wing. A plane on a treadmill isn’t getting airflow over the wings in the same way that moving down a runway would.
Oooh, now there’s an idea: Tow the plane, on skis, to the top of a mountain & let it go down the run, similar to a ski jumper.
The problem is that you can’t land this way & there’s no longer a long runway for it to land on (remember we wanted to build housing on all of that 'wasted" space at the airport.) Ooohhh, I’ve got it, if we fly the plane into the back side of said launch mountain, it will stop in a short distance. :rolleyes:
There are many runways in difficult to reach places without much room that are angled so planes get a boost during takeoff. They also land uphill for the same reason.
the thrust from the engines (or propeller.)
Forget about the wheels. Ignore them completely. it’s that simple. all the wheels do is let the plane roll. they’re not like a car. As I posted above:
Imagine a small toy plane on a treadmill. One with an engine in it.
Hold the plane in place while you turn the treadmill on. You see that the treadmill is now moving under the plane, the wheels are spinning. It doesn’t require much effort at all to keep the plane still.
Now, turn on the engine. You will feel the plane start to pull forward. If your treadmill is long enough, when you let the plane go, it will take off pretty much exactly as it would have if the treadmill were stationary.
The only thing the treadmill adds is wheel speed. If the speed of the treadmill + forward movement is too much, then the tires may blow up, or the bearings seize, but outside of mechanical problems (which would be a bigger issue with giant treadmill moving over 100 mph), it doesn’t do anything.
If the plane is moving forward at 1 m/sec, and the treadmill surface is moving backwards at 1 m/sec, the wheels are spinning at 2 m/sec.
Or, another analogy: The plane is sitting on an ordinary runway, trying to take off. But there are a bunch of people standing around the runway preventing it from moving forward by whistling. If the plane starts to move forward anyway, then the people whistle louder, so that they’re always whistling loud enough to stop the plane from moving. What happens?
Of course, the plane takes off anyway, because you can’t stop it just by whistling, no matter how loud you are.
But at some point, if the plane is moving forward wrt ground, the linear wheel speed is greater than the conveyor belt speed in any of your explanations. The thought experiment doesn’t allow for that, right?
The thought experiment is about how fast the wheels are spinning, it’s about comparing a plane vs a car on a treadmill. A plane will go forward because it’s not being propelled by it’s wheels, a car will stay stationary because it is. Now whenever this discussion pops up, it gets twisted around with friction and infinitely accelerating treadmills and so on, but that’s not really the point of the question.
Ok. I get that. It makes this a much less interesting thought experiment though.
You can’t have a treadmill that exactly matches the speed of the wheels on an airplane moving forward, because that’s impossible.
You could have something like this. You put the plane on the treadmill. Engine off, treadmill off. You start the engine, but just barely, not enough to move forward. Now you start the treadmill, but just barely, so the plane is barely moving backwards. Now you put on the gas to the engine and the treadmill, so that the plane’s engine doesn’t pull it forward relative to the ground. If the plane was on the ground, not a treadmill, it would be just taxiing forward, not fast enough to generate any appreciable lift.
In this case the forward motion of the engines is just enough to counteract the backwards friction of the treadmill, and so you could have the plane sitting still on the backwards treadmill, just like a car driving forwards on a backwards treadmill.
But increase the gas to the engine, and the plane moves forward. Because it isn’t moving forward by pushing the treadmill backwards, it is moving forward by pushing the air above the treadmill backwards.
So a system where every time the wheels move forward 1 meter the treadmill moves backwards one meter is only possible by carefully calibrating the engine to very low thrust. Or so I can imagine.
But there’s no system where you turn the engines on full and the plane moves ahead by 1 m/sec and the treadmill moves backwards 1 m/sec and the wheels turn by 1 m/sec. That can’t happen, because 1 + 1 != 1.
Think of it another way. An airplane could take off on a frozen lake covered with extremely slippery wet ice. You turn on the engines, the engines push air backwards and the plane moves forward. Yet the wheels on the airplane, due to the slippery wet ice, don’t turn at all, or maybe only a few times. The wheels don’t have to turn for the airplane to take off. The plane is moving forward at 10 m/s, but the wheels aren’t turning at 10 m/s, they’re just sliding along the icy lake, not spinning. They’re acting like skis, not wheels.
A normal runway isn’t made of slippery ice, and so the wheels are present to reduce the friction instead. And so the wheels, in contact with the rough tarmac spin as the plane moves forward. But if the wheels hit a patch of slippery oil on the tarmac they might skid. That won’t matter though, because the wheels spinning isn’t what moves the airplane forward. What moves the airplane forward is pushing air backwards, not pushing the tarmac backwards.
I think it’s trivial to see that the plane would take off on a treadmill that couldn’t instantly match the speed of the wheels. I was under the impression that the belt had to match the speed of the wheels rotation and could the plane ever move forward under this ideal. That’s where I was getting confused. I’d still like the know the answer to my scenario though.
The point you’re missing is that the treadmill does nothing to hold back the plane. The wheels spin freely, no matter how fast they are going it won’t affect the plane’s forward speed.
Basically, there’s no way the scenario can play out the way it’s described in the problem. The treadmill can spin as fast as you can imagine but it doesn’t matter. The plane, propelled by the thrust of the engines on the air, moves forward.
It could work for a car, since the wheels push the car forward. But not a plane, where the wheels don’t do anything.
So how does the plane ever move forward? The only way is for the wheel’s circumference to cover more “ground” per unit time then the conveyor belt’s backward speed. So if the wheel circumference is c, and the distance the belt moves is d, and d/sec always equals -c/sec. This, in its ideal form, keeps the plane from moving forward w.r.t. ground. The only way the plane moves wrt ground is if d/sec != c/sec. If this is wrong, I’d like to see why. This is the more interesting problem, in my opinion. I’d like to know what the imaginary plane would do in this scenario even though it obviously can not be created in the real world.
If I stand on a moving treadmill with roller skates, I can easily move myself forward and backwards w.r.t ground with my arms pulling or pushing the handrails. I understand that. I don’t know what would happen in the case that the treadmill magically matches the speed of the wheels. I have the feeling I wouldn’t be able to move my skates w.r.t. ground. But I’d love to know why I could.
Maybe a better example is that the treadmill always moves faster than the plane’s wheels. Would the plane takeoff then? How would the plane ever move forward relative to ground? In my roller skate example, if my wheels were limited to 5 ft/s and the treadmill was moving at 6 ft/s, I’d quickly fall off the treadmill regardless of my arms providing the power of my movement.
In order for the treadmill to hold the plane back there would have to be some way for the force of the treadmill to be imparted to the plane through the wheels. There isn’t. Therefore there’s nothing the treadmill can do to hold the plane back.
The point of the problem is to illustrate the fact that the situation cannot occur in real life. The magical notion of the treadmill always accelerating to hold the plane back ignores the fact that there’s no way for the treadmill to restrain the plane.