Plane on a conveyor belt

[Janook]

Maybe you've heard this before...
I've got an answer, and I think it's right, but who knows?

Anyways:
A plane is on a runway.
Except the runway is a conveyor belt.
The conveyor belt is infinitely long.
The conveyor belt is spinning at a speed equal to whatever speed the plane would be moving forward if the conveyor belt were not there.
The conveyor belt is spinning backwards. (The plane is trying to move forwards.)
The plane's engines have infinite thrust/power, and the plane has infinite structural integrity.
The plane is on wheels which work properly.
There is no wind, and there are no forces acting on the plane besides the friction between the wheels and the conveyor belt, and the engines. Yes, there is gravity. my bad.
Edit: I'm not trying to be tricky with this statement. I'm just saying there is no force I haven't mentioned.
There is an infinite amount time.

The question:
Can the plane ever take off?

Edit:
More importantly, can the plane move forward?

Edit2: The engines are applying thrust to the air. (You're right on, moopanda.)
This is a typical commercial plane. You cannot adjust the direction of the engines. The engines are not mounted on the wheels.
The wheels do have breaks, since this is a normal plane, (normal planes have breaks on their wheels, right?) But I don't think that matters. I've never thought about it, in any case.

Soliloquy you're dead on too. (You're first paragraph, anyways.)
Does a pretty good job of summarizing my explanation.

[Gelsamel]

I don't think so, for the plane to take off there has to be lift, and without a breeze and without moving foward there is no way in hell it will get that lift.

[moopanda]

I don't know how planes get going, but here are two options:

The thrust is applied to the runway via the wheels. This seems unlikely since the wheels are below the plane and don't seem to be driven... I'm not really sure. In this case we would never get moving, assuming the speed matching of the conveyor belt is perfect.

The thrust is applied to the air by the turbines, and the wheels are simply there to reduce friction with the runway. (as opposed to dragging the plane on the ground) In this case, I think we *would* take off, and the wheels would just be spinning twice as fast as normal.

Does anyone know? I didn't really know if this was a spoiler or not, but better safe than sorry...

[Torn Apart By Dingos]

Do the wheels have breaks? In that case, going really fast and then breaking + inertia + noninfinite friction should move the plane forward.

If not, can you alter the direction of the thrust? If so, directing it sideways or down would be a good idea.

Finally, a smartass answer: you said no forces acted on the plane besides friction and the engines. That means no gravity! The slightest disturbance/asymmetry will make the plane leave the converyor belt.

[moopanda]

That means no gravity! The slightest disturbance/asymmetry will make the plane leave the converyor belt.

If there's *no* gravity, then the plane won't be exerting any weight force on the conveyor belt, and if I remember my high school physics, wasn't the friction force a fraction of the weight force? With no wind resistance we now have a totally frictionless and weightless system... which... is a bit too alien to contemplate for me I'm afraid.

[Soliloquy]

New to the forums, saw the link to the logic puzzles, and decided to sign up.

Here's how I see it, though I may be dead wrong:

the thrust is coming from the engine, not the wheels, and the force coming from the conveyor belt is applied to the wheels, which, being wheels, have the capacity to spin. this spinning of the wheels while the plane's engines are applying thrust would make it so that the force applied on the plane by the conveyor belt is less than the force applied on the plane through thrust, allowing the plane to accelerate, and thus move.

Even if what I just said is dead wrong, another thing to keep in mind is that the conveyor belt, presumably, has a finite coefficient of static friction. Thus, if the conveyor belt goes fast enough, the wheels will begin to slide as opposed to roll, becoming kinetic friction. Since forces of kinetic friction are less than those of static friction, the force of the conveyor belt on the plane would, at this point, be less than the force of the thrust on the plane, allowing for the plane to move once again.

Of course, since we're already dealing with infinites, I suppose it's possible that the belt does have an infinite coefficient of static friction, which would make my last point.

...Maybe the answer's just "no"

[cryonic]

The plane will take off. Assuming there is air which is acting normally.

The plane does not move by pushing against the conveyor belt it moves by thusting against the air.

The wheels will be spinning twice as fast as they would without the presence of the conveyor belt.

The ground only stops the plane dropping untill the lift generated by the wings exceeds the downward pull of gravity. - it does not interact with the thrust/lift generated by the engines. Except that there is some friction generated from types to conveyor belt. Wheels only exist to reduce friction between ground and plane to allow most efficient forward movement. The thrust is not transmitted to the ground via the wheels. If this was the case the plane would not be able to stay in the air when the wheels were not in contact witht he ground.

[Soliloquy]

The ground only stops the plane dropping untill the lift generated by the wings exceeds the downward pull of gravity.

The one problem with that statement is that, if the plane is stopped, it cannot generate lift, so if it's completely stopped until it generates lift, it will be completely stopped forever.

[moopanda]

The ground only stops the plane dropping untill the lift generated by the wings exceeds the downward pull of gravity.

The one problem with that statement is that, if the plane is stopped, it cannot generate lift, so if it's completely stopped until it generates lift, it will be completely stopped forever.

I think cryonic meant that the ground stops the plane in the vertical direction, not the horizontal direction. Ever played a buggy game (Farcry comes to mind) where the ground fails that objective?

[Gelsamel]

Actually yeh, I changed my mind now I thought about it more. With infinte thrust the amount of air being pulled under the wings from the turbine will be infinately enough air to generate infinately enough lift to make the plane fly.

[cryonic]

All planes start from stopped.

The situation is the same as saying the plane is on slippery ice and using skis or on water and using floats.

The surface it is on and the movement state of the surface is irrelevant so long as -

The accelleration of the plane caused thrust generated by the engines against the Air exceeds the decelleration caused by the friction of the wheels against the conveyor.

[Gelsamel]

All planes start from stopped.

The situation is the same as saying the plane is on slippery ice and using skis or on water and using floats.

The surface it is on and the movement state of the surface is irrelevant so long as -

The accelleration of the plane caused thrust generated by the engines against the Air exceeds the decelleration caused by the friction of the wheels against the conveyor.

I thought the conveyor belt pusing the plane back at which ever same speed the plane is going so it would always be stationary, but the thrust should be enough to cause the plane to lift.

[Shoofle]

I thought the conveyor belt pusing the plane back at which ever same speed the plane is going so it would always be stationary, but the thrust should be enough to cause the plane to lift.

The thrust alone doesn't create any movement - it's the movement of the air over the airfoils that creaes lift. The key is, however, that the conveyor belt is only turning the wheels - they turn relatively freely, so there's essentially no resistance from the belt. The plane accelerates because of the straight thrust of its engines.

[Gelsamel]

The thrust alone doesn't create any movement - it's the movement of the air over the airfoils that creaes lift.

I thought that was tacit.

[cryonic]

Your are thinking as if the plane is a car and depends on pushing against the ground to generate reactive force.

A car generates movement by having a wheel push against the ground. The ground also pushes against the wheel equally - causing the car to move.

Break it down.

Can a plane on a frictionless surface take off. - Well yes it would be easier as the thrust of the engines would move the plane more efficiently along the ground.

Can a plane on a frictionless surface take off if the surface is moving in the opposite direction the plane wants to fly in. - Well yes because the movement of the ground doesnt translate to the body of the plane.

So can a plane with a freely rotating wheel take off. - yes - happens every day - planes dont direct drive force though the wheels only braking force.

So can a plane with a freely rotating wheel take off. if the surface is moving in the opposite direction the plane wants to fly in - yes the wheel just spins faster.

Does the speed of the wheel affect the take off speed of the plane. - well duh - only through friction forces.

[Gelsamel]

I'm not even taking the frictionless surface into account.

If the turbines have infinate power then the turbines will be sucking in an infinate amount of air through the turbines, and hence under the wings. Wouldn't this produce an infinate amount of lift, and hence the place would fly?

[phlip]

All the motions of the plane are relative to the air - it is the engines pushing on the gases (with propellers or jets) that gives propulsion, and the air moving past the wings that gives lift. As cryonic put it, the ground is only there to support the plane until it has enough lift to support itself.

If you wanted a situation like the the mental image the OP tries to invoke, you would, instead of a conveyor belt, want a giant fan to create a headwind at exactly the same speed as that which the plane would otherwise be flying. In which case the props/jets wouldn't have any forward push on the plane, and it would stay stationary (well, they would have a forward push on the plane, exactly enough to counter the backward drag of the wind), but the movement of the air past the wings would still give lift, and the plane would rise vertically and then just hover (assuming the fan was tall enough, and was able to react quick enough to tweak the wind speed).

Incidentally, this sort of thing is why planes always take off and land into the wind - the air speed you need is the same either way, but it reduces the land speed so that you have a better chance of reaching the right speed or coming to a complete stop before you run out of runway.

[Ephphatha]

The plane will never take off. Even if the turbines can move an infinite amount of air, the engines don't generate lift. The engines generate thrust. The wings generate lift. If the plane never moves from its initial position, no air will ever pass over the wings in sufficiant quantity, speed, or even in the right direction to generate any lift at all.

Edit: assuming of course that the plane never moves from it's initial position. I'm revising my answer.

The plane will never take off, because it'll run out of fuel before reaching take off speed. Plane wheels aren't exactly that refined, being designed to support that immense weight. They'll probably blow very quickly leaving the plane sliding along the conveyer belt on it's belly. Either way, it's still going to be stuck there.

[shimavak]

I agree with the assertions put forward by Ephphatha; however, I disagree on the result.

I propose the following assertion:

It has been stated that the jet must, by the definition of the scenario, never move with respect to the immobile ground, and therefor the air attached to the ground (assuming no wind).

The jet must intake air to sustain the turbine engine, and the amount of thrust generated is directly related to the amount of air taken in. Since we have infinite thrust able to be generated by the engine, it must take in an infinite amount of air over some unit of time.

This air must come from in front of the jet, and as their is a significantly greater amount of air ABOVE the turbines than between the turbines and the ground, it will eventually neccesarily take in more air above than from below.

The flow of the mass of this displaced air will eventually be the same as the mass of the jet, thus the jet has pushed down its own weight. This is the definition of displacement.

Eventually the jet will not so much fly as float!

[phlip]

The thing is that the problem states that there is no wind, just the conveyor belt moving backwards at the speed that the plane is trying to move forwards. To imply from this that the plane is stationary even when the engines are going is wrong, as said by many people above.

[shimavak]

To imply from this that the plane is stationary even when the engines are going is wrong, as said by many people above.

I believe that we are interpereting the problem in two different ways. I am imagining that the plane is being held in a specific horizontal position by the conveyor belt.

I suppose one thing to consider is whether or not the wheels themselves have friction. If they are assumed to be massless, frictionless wheels, no thrust would be needed to keep the plane in the same position and it would be trivial, as any thrust added could not be countered by the belt.

If the wheels had mass (and hence rotational inertia) they would initially resist the motion by the belt, but once an equalibrium was reached, no thrust would be required to maintain the initial conditions (plane stationary).

If we had friction, the plane would have to generate some thrust by moving air in order to maintain the conditions of the experiment. This is assuming we are not dealing in the trivial case of all of the velocities being zero.


With the final case, the thrust must be generated by air being drawn in, which must be expelled in order to produce a thrust. To continue to draw in air, it must be replaced from in front of the plane. The air above and in front will be replaced more readily than that above and below (more steradians of available influx), so there would be a tendancy for the turbine to "pull" in that direction. After this, it is just a matter of providing enough thrust to equalize the gravitational attaction of the plane.

Once that is done, as soon as the wheels leave the ground, the plane will recieve quite a large acceleration (no more friction on the wheels).

After that, it may or may not be able to maintain any altitude, as the pressure imbalance will be corrected by the plane moving to a new position forward. I could envision a situation where the plane was in a somewhat harmonic motion, bouncing off the belt.

Edit: Fixed the quote.

[phlip]

Stuff about friction

There is no wind, and there are no forces acting on the plane besides the friction between the wheels and the conveyor belt, and the engines. Yes, there is gravity. my bad.

There's no friction between the wheels and the plane.

[Curris]

Quick idea with no support to back it up, but what if you accelerated the plane (and thus, the conveyor belt as well), and then activated the brakes and shutdown the engines. As the engines are operating, the plane stays in position relative to an external reference. But afterwards, the plane accerates backwards, being pushed by the conveyor belt. Let's say at a near infinite speed. Now I know most wings are not designed to be flown in reverse, but with enough speed, most surfaces of any shape will generate lift, and so the plane flys, in reverse.

Discuss?

[cryonic]

Curris the shape of a wing generates lift in forward motion - the same shape in reverse motion causes negative lift. this principle keeps formula one cars on the ground during sharp high gee turns.

[hythloday]

The plane will take off. As a restatement of the question, consider the situation where the plane is already in-flight, but then a cosmic entity stops it in place (and removes its inertia, to prevent the passengers slamming into the seats in front of them at close to the speed of sound), and then allows it to move only in the local z axis (ie forwards or back) until it has enough lift to equal gravity. It's intuitively obvious that it will be able to do so.

I believe a normal plane would be able to do this - you don't need to assume infinite thrust.

[TwoBuy]

I'm not a huge aerodynamics person, but I saw this "there are no forces acting on the plane besides the friction between the wheels and the conveyor belt, and the engines. Yes, there is gravity. my bad."

I thought airfoils relied on the friction of air on the wing to generate lift, if that force doesn't exist, wouldn't it not be able to take off even if it could move forward?

[shimavak]

Stuff about friction

There is no wind, and there are no forces acting on the plane besides the friction between the wheels and the conveyor belt, and the engines. Yes, there is gravity. my bad.

There's no friction between the wheels and the plane.

Is there any inertia to the wheels?

If there isn't, then any thrust applied by the turbines will have to be countered by an acceleration of the belt. If this is the case, it will still be a race condition between the air above the belt and the air above the turbine; eventually, the turbine will take in more air from above it than from below it.

If there is inertia, each additional bit of thrust from the turbine will be countered by an acceleration from the belt. Still, we can eventually reach an intake velocity such that the air above the turbine is being drawn from more so than the air below, and then lift will be generated.


If we neglect all friction, as another poster points out, the turbines themselves won't work. They generally rely on being able to force air into the opening, and without friction, they cannot push the air. It becomes an exceedingly non-physical ideal.

[phlip]

But don't forget the line

The conveyor belt is spinning at a speed equal to whatever speed the plane would be moving forward if the conveyor belt were not there.

Let's just say, for the sake of having a number, that our hypothetical plane needs an airspeed of 200kph to lift off, a number I pulled entirely out of thin air and is probably not even in the right order of magnitude, but who cares, it's a number.

Now, consider a plane completely stationary, but on a treadmill going at 200kph. I doubt the friction from 200kph of spin on the wheels is enough to overcome the power of the engines... mainly because if it did, the plane would never be able to reach that speed and take off on a real runway.

Certainly you could make the conveyor belt run many many times faster to get enough friction to counter the engines, but that's not what the problem specifies.

[shimavak]

But I believe you said the key, it requires an airspeed of some certain value. We have a zero ground speed due to the belt, and no wind, so therefor we have a zero airspeed.

The plane will no longer take off due to any sort of aerodynamics of the wing shape. Rather, the only thing which would make it lift off now would be a bouyancy effect, which would eventually happen as the turbine sucks more air from in front of it than the world can replace. Eventually, this will create a low enough pressure in front of and above the plane to have it lift off the ground. After this point, the belt cannot prevent it from gaining ground speed and thus airspeed.

After this point, it will be able to fly, but only if it can get off the ground by some other means than having air flow over the wings.

[LE4dGOLEM]

A plane... takes off by shifting air hitting the wings going over the wings FASTER than it goes under (aerofoil) - pressure diffrence means partial vacuum means upwards movement

Relative to the ground, yeah, that plane's going like a bat out of hell because of the engines.

Plane's being held in place by the conveyor belt, so no movement relative to the air.

If there's no wind, then the plane has to move itself relative to the air to create that requires partial vacuum, and because it can't move relative to the ground it can't move, and so it can never take off.

EDIT: Of course the real question is, will all the snakes weigh it down? (sorry)

(hey look, this is my first ever post here on XKCD. blame the frontpage link on the comic)

[ulnevets]

if the plane can generate infinite thrust, then it can travel at the speed of light.

if it travels at the speed of light, time freezes.

and then...?

[phlip]

We have a zero ground speed due to the belt

I still claim this is wrong, and you haven't said anything to dissuade me from that. The only mechanism you've suggested is that the conveyor belt is going fast enough that the friction on the wheels overcomes the thrust of the engines... but if that was the case then the friction of the wheels would overcome the thrust of the engines when the plane is trying to take off on a normal runway (when there's no headwind).

Planes take off from airports when there's no/negligible headwind all the time, so the conveyor belt clearly isn't going to slow it down enough to keep it with a zero ground speed.

[LE4dGOLEM]

[...the friction on the wheels overcomes the thrust of the engines... but if that was the case then the friction of the wheels would overcome the thrust of the engines when the plane is trying to take off on a normal runway...

Wheels don't provide the power, nor does their friction do anything when the plane's taking off as it's using regular internal combustion/push by output method, pushing the plane forward, generating relative wind resistance (like you said) thus the aerofoil works, with relative air pressure.

Just a moment,

The question:
Can the plane ever take off?

No.

whee. question answered

[TwoBuy]

I'm going to agree with phlip here. Since the wheels aren't powering the forward motion, the conveyer belt is irrelevant.

Since there is no friction in the wheels (only between the ground and the wheels) all friction force from the conveyer belt will be applied only as torque in the wheels and will not counter-act the forward force of the plane.

This means the plane will take off as if there were no conveyer belt. It is completely irrelevant to the force diagram.

[Mo®]

(I suck)

[phlip]

The plane is traveling forward at zero miles per hour.

But it isn't. That's the whole point. The conveyor belt has no (or very little) effect, and the plane will still be moving forward relative to the ground or the air at sufficient speeds for liftoff.

[Mo®]

The plane is traveling forward at zero miles per hour.

But it isn't. That's the whole point. The conveyor belt has no (or very little) effect, and the plane will still be moving forward relative to the ground or the air at sufficient speeds for liftoff.

Yes it is. If it were moving forward relative to the ground, it would drive off the front of the conveyor belt.

If I'm on a treadmill and I'm moving forward relative to the ground, I'm going to run off the font of the treadmill because the treadmill is not moving.

[phlip]

If I'm on a treadmill and I'm moving forward relative to the ground, I'm going to run off the font of the treadmill because the treadmill is not moving.

Well, the puzzle stipulates an infinitely long treadmill...

But the running-on-a-treadmill analogy is flawed - when you run, you move forward by pushing on the ground. A plane builds up speed by pushing on the air. A better analogy would be holding wearing rollerskates, and pulling yourself along a horizontal pole, and then doing the same thing on a treadmill - the treadmill won't really make you any slower, it'll just make the wheels on the rollerskates spin faster.

[Mo®]

Well, the puzzle stipulates an infinitely long treadmill...

The conveyor belt is spinning at a speed equal to whatever speed the plane would be moving forward if the conveyor belt were not there.

You can make the treadmill or belt as long as you want. The puzzle stipulates that the plane isn't going anywhere.

[phlip]

Just because the conveyor belt is moving, doesn't mean the plane won't be going forward anyway... look at my pole/rollerskates analogy...
Pulling yourself along a horizontal pole while wearing rollerskates you could probably get about half a metre a second. If you set up a treadmill to go at half a metre a second, put on rollerskates and pull yourself along a horizontal pole, you'll still get half a metre a second. The wheels will spin as if you were going 1 metre a second, but that's besides the point... you're still moving forward.

Replace "rollerskates" with "landing gear", "pole" with "air" and "pulling" with "propeller/jet engine" and you get a plane.