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A system could never be developed where a conveyor could prevent a plane from taking off. As soon as the planes thrust overcomes friction in the wheels the conveyor doesn’t matter.

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If the friction between the wheel and the conveyor is greater than the friction of the wheel bearings, wouldn't the turning of the wheel prevent the plane from overcoming the friction between the wheel and the conveyor?

That has been my assumption all along, that the rotation of the wheel and the conveyor would prevent the plane's wheels from sliding along the conveyor.

Let's assume

• the maximum passenger load the airplane can carry is 150 passengers weighing 250 pounds each, but it is actually loaded with Japan's 300 largest sumo wrestlers, and
  
• NASA's big machine used to transport space shuttles to the launch pad is sitting on the far end of the conveyor belt, and
  
• the brakes to the aircraft are frozen shut, and
  
• there is a chain attached to the tail of the aircraft attached to a million ton weight sitting off the conveyor belt behind it, and
  
• an asteroid bigger than any that have struck the earth in the last ten million years is headed right for the aircraft, and
  
• the airport is right next to the beach with a half mile high tsunami approaching, and
  
• the pilot is J. Fred Muggs, and
  
• the fuel tanks are completely empty.

Will it then be able to take off?

I say no.

interesting...if the fuel tanks had some fuel in it, I think it could still fly. Naturally the chain holding onto the 1 ton weight would break and the sumo wrestlers would add their own form of thrust into the mix....

It really depends on you interpret what "completely empty" is....I choose to read it as saying there is still some fuel left. Naturally the plane would then still be able to take off.

under Eric's set of assumptions, I think the plane would have to use some of the magic dust Santa uses on his reindeer. Since we're in about the same plane of reality, why not?

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you could build a conveyor that could accelerate its speed to infinity

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Lets go ahead with this absurd assumption (for the record, the plane takes off and this assumption is in fact idiotic, but lets look at the other side of the argument)

We have a belt going at infinite speeds. This belt will begin to affect the air currents above it (since this assumption obviously necessitates friction). The air speed above the belt will eventually reach speeds capable of generating sufficient lift to allow the airplane to fly.
Now the air will have a velocity profile that is most likely parabolic (fluid dynamics). The plane will begin to rise and as the wind speed reduces, the plane will begin its forward motion and proceed to its destination.

So anyway you cut it, the freaking plane takes off.

but anyway...

here is a force diagram to help the real argument

no fly, no lift

It would never take off.... Yes, there is thrust provided by the engines like many have said, but that thrust force has nothing to do with lift. Lift is provided by air moving around the wings, and if the plane is sitting in the same place, it just won't happen unless you have a headwind that is equal to the speed needed to cause enough lift (but then again when is the last time you have seen a SUSTAINED 200+ mph wind). All the engines are really doing is pushing the plane forward so as to reach a speed where the lift force on the wings is enough to lift the plane, not actually pushing the plane upward. Only way it would work is with a vertical take off plane such as the Harrier Jump Jet which actually directs its engines’ thrust directly downward (lifting the plane directly up without ever moving down the runway, never using the lift provided by wings) as compared to a normal plane which all of its engines thrust is directed straight back (in order to push it forward to have air move around the wings). Although in the case of the Harrier the conveyer would never have to move anyway to keep the same speed of the wheels because they never would have to move to take off. The reason that all planes don’t use this concept of directing thrust downward is because of the fact that this is very inefficient and uses a far larger amount of fuel as compared to the traditional aircraft and its long takeoff dependent on the airspeed over its wings.

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So, by your reasoning, if you have a plane on a conveyor, and the plane's engines are off, when you start the conveyor, the plane will not move. Rather the wheels will just start spinning backwards to match the conveyor's speed. Does that make any sense to you?

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Yes...if the wheels actually reduced the rolling resistance to 0 the plane would stay standing while the wheels rolled backward to compensate.

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And don't be silly about wheels being required for takeoff. Show me a plane that can start off on the ground with it's landing gear up and takeoff.

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Good a time as any to burn my first post, I guess...

The spirit of the problem as stated is that the speed of the conveyer is always equal and opposite the speed of the airplane relative to the conveyer surface, or the speed of the airplane's wheels. If this is true, then by definition the airplane's speed through the air is zero. Thus, no takeoff.

However...there is a problem:

Start with F = ma.

There are two forces acting on the aircraft: a thrust force and a frictional force. The frictional force is simply some dimensionless coefficient c multiplied by a Normal force
N=mg, where g is the gravity constant.

So far we have T - cmg = ma
=> a = (T-cmg) / m.
T, c, m and g are all constant, therefore, a is constant.

First, lets look at an airplane on a normal runway. The velocity of the wheels, Vw, is simply equal to acceleration multiplied by time:

Vw = at

Introduce an opposite moving conveyer belt velocity, Vc, and the equation becomes

Vw = at + Vc

In the problem statement, Vw = Vc, but this would lead to:

Vc = at + Vc

Which, after t > 0, is mathematically impossible unless

a = 0
=> T = cmg

which we know is not true (if it were, then the plane would not be able to take off at all).

So I guess the real answer to the original problem is that there is no answer; the conditions imposed are impossible, not just practically, but theoretically.

Vw = at + Vc?

This equation is wrong. Tell me why the velocity of the conveyor has anything to do with the velocity of the plane? You are trying to do conservation of velocity, which just doesn't happen.

Vw is the velocity of the airplane WRT to the surface of the conveyor. If that surface is moving at a velocity Vc, then of course Vw and Vc are related.

the plane is not relying on friction created by the tires to push it forward!! I don't know how much clearer that can get.

The airplane and the conveyor have absolutely no mechanical connection. If the conveyor matches the forward speed of the plane in the opposite direction, the wheels will simply spin at double the speed they normally would.

Ok, suppose you are right.

Vc = -Vw, not Vc = Vw

Vw = at + Vc

So then 2*Vw = at

But in reality the equation should only read
Vw = at

If you reverse the sign in one equation then you have to reverse the sign in the other equation as well.

Vc = -Vw
Vw + Vc = at

This would still work out to

at = 0

Which, again is only possible if the Thrust produced is exactly equal to the frictional force, which we know can not be true of an airplane attempting to take off.

Looking back at your posts, I think we are actually on the same page here. Look at the original equation I came up with.

Vw = at + Vc

If we introduce Va, which is the velocity of the airplane WRT to the air, then we can easily see that this velocity is equal to the velocity of the airplane WRT the conveyor, Vw, minus the velocity of the conveyor, Vc:

Va = Vw - Vc

It then follows that

Va = at

Which means that the airplane's velocity WRT the air is not affected at all by the velocity of the conveyor. In this we are in agreement

But that is not what I said in my first post. What I said is that for an airplane attempting to take off, its thrust is much greater than any frictional force opposing it. Therefore, it is physically impossible to enforce the condition that the speed of the conveyor is equal and opposite to that of the speed of the airplane relative to the conveyor.

again, not true...

the conveyor in the problem attempts to match the speed of the plane WRT the air only in the opposite direction.

if the conveyor tried to match the speed of the plane WRT the conveyor, the conveyor would accelerate exponentially.

I agree with your first statement; If the conveyor speed WRT the air is equal and opposite the airplane airspeed, then yes the airplane will take off, and the wheels will have an angular velocity twice that of the airplane airspeed. But this statement admits that there is motion relative to the air, so of course the plane would take off.

As far as your second statement, I stand by my original conclusion: The velocity of the conveyor WRT the air can never equal the angular velocity of the wheels. Accually I think I agree with your assesment that any real conveyor that attempts to keep up will accelerate exponentially. I'm just saying that the conveyor will never actually catch up.

They conveyor belt is moot.

The plane will take off.

hth

conveyor belt just spins the wheels faster, nerds.

flies.

If there is no friction between the wheels and the landing gear legs, what holds the wheels in place so they can roll? You need friction for them to stay in place (IE, conveyor moving, plane still, wheels roll).

If there is no friction between the tires and the conveyor how do the wheels/tires even roll? (you need friction to roll the tires, no friction means tires don't roll).

Some of you guys are missing the point...

No matter how fast the conveyor belt moved it would not keep the plane from moving forward. The plane would NOT sit still.

Someone mentioned the ski lessons at Sun & Ski Sports as an example...

Picture this... ever see the tow ropes they use on the bunny and tubing hills? That tow rope is like the thrust of a jet engine. As long as you are holding onto the tow rope the ground can be going 1000 mph in the other direction and you will continue traveling the direction of the tow rope.

or a poma lift at a ski resort. Friction between your skis and the ground and even gravity working against you, but you still go up. Man its hilarious when some newbie eats it on the poma lift!

If a person was running full speed on a treadmill would they be able to throw a paper airplane hard enough to make it fly?

it flies.

[This message has been edited by 1876er (edited 12/13/2006 1:02p).]

yes, but they would be thrown off-balance, fall down, and get thrown off the treadmill and that would also be hilarious. But that's a one-car accident, whereas the poma lift incident is more than likely to take down a few beginners behind the person that fell. That's funnier in my opinion.

The conveyor at sun & ski isn't designed to match your velocity at all times as the conveyor in this example. I haven't seen anybody draw a FBD showing why the plane would fly, but I've seen a couple that show why it WOULDNT.

1876 - now pull on the rope.

1876... exactly.. pull on the rope. Is it hard to move the skateboard forward? The rope is the thrust it doesn't matter how fast the treadmill is going the skateboard is alway just as easy to pull forward.

If I pulled on the rope, the treadmill would no longer be matching the speed of the skateboard. Isn't that one of the limitations of the problem. Obviously if the plane was actually moving, it would generate lift.

Well, if the speed of the skateboard was zero, the treadmill wouldn't be moving at all. However, the treadmill can speed up all it wants to match the speed of the skateboard. You can still pull it forward almost as easily as if the treadmill wasn't there.

Jeez, I'm about to give up.

Set the treadmill on the slowest speed and pull the rope. Then put the treadmill on the fastest speed and pull the rope.

Did you have to pull the rope harder on the faster speed to make the skateboard move forward?

it flies.

[This message has been edited by 1876er (edited 12/13/2006 1:02p).]

It flies.

[This message has been edited by 1876er (edited 12/13/2006 1:01p).]

1876er, if I were you I'd stop posting for a while. You need to think about what you are saying and maybe you will understand how ridiculous your statements are.

nm, it flies.

[This message has been edited by 1876er (edited 12/13/2006 1:00p).]