Flight and the Conveyor Belt

Cecil's Columns/Staff Reports
141

It’s not our assumption. It’s in the original problem statement, that the belt matches the speed of the airplane.

This is true. If anyone doubts this, consider a disc-shaped object in free fall in space, with a single rocket motor attached to the edge so that the thrust is tangential to the circumference of the disc. Fire the rocket and the disc will experience both translation AND rotation… how much of each depending on the discs’ moment of inertia vs. its mass. If you want just rotation with no translation, you have to have another rocket on the opposite side.

Fine, but that’s only possible if the belt’s acceleration continues for as long as the airplane’s fuel lasts. And that is NOT consistent with the original problem statement, which says that the belt matches the airplane’s speed. If that’s the case, then the belt can’t very well be accelerating forever, unless the airplane is also… which means that the belt must not be very effective at nullifying the airplane’s forward movement! :smiley:

142

This whole disagreement comes from people who come to one conclusion by working with the problem as stated and the people who come to a different conclusion working with the problem as intended.

As intended, the plane has no motion relative to the ground, or more properly, the planes wings have no motion relative to th air. No airflow, No lift.

However, the problem as stated has the belt conveyor moving backwards onlyat the same speed that the plane moves forward. This will not produce enough friction to keep the plane stationary.

The plane, engines idle, sits on the conveyor, belts idle.
The planes engines start, producing however many thousands of pounds of thrust.
The plane accelerates to 1m/s.
The conveyor accelerates to -1m/s.
The conveyor imparts sliding friction to the plane’s tires.
The wheel and axle attached to the tires convert the sliding friction on the tires to a relatively miniscule amount of rolling friction.
The axel imparts this relatively miniscule amount of friction to the plane.
The plane’s engines continue to produce however many thousands of pounds of thrust… easily overcoming the tiny amount of counterforce generated by the conveyor.
The plane accelerates to 2m/s…
Lather Rinse Repeat.

Basically, the conveyor has to move at -$very_large_number m/s to generate enough backwards force on the axles (or skis or pontoons) to counteract the forward force of the engines for the velocity of the plane to be 0 m/s.

143

Here is an explanation that helped some people in another forum understand why the airplane will move forward on the belt and take off.

Here is the puzzle as stated at:

http://www.avweb.com/news/columns/191034-1.html

“On a day with absolutely calm wind, a plane is standing on a runway that can move (some sort of band conveyor). The plane moves in one direction, while the conveyor moves in the opposite direction. The conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction). Can the airplane ever take off?"

Let us break it down piece by piece.

“On a day with absolutely calm wind, a plane is standing on a runway that can move (some sort of band conveyor).”

Okay, everything is still right now. The plane is not moving. The conveyer belt is not moving and there is no wind so the atmosphere is not moving. Can we all agree on that?

“The plane moves in one direction,”

Now you have to decide just what it means for the plane to move. The way I interpret it for a plane to move is for it to move relative to the atmosphere because that is what the plane’s engines act on. That speed of movement is termed air speed. Since the atmosphere is calm air speed and ground speed will be equal.

“while the conveyor moves in the opposite direction.”

Now you have to decide what it means for the conveyer to move. I say that the conveyer is moving relative to the ground and atmosphere in the opposite direction of the plane.

“The conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction).”

Okay, now this is the part that gets so many people confused, but it is really very simple. You have to think about relative movement. In this case the ground and the atmosphere both serve as the stationary reference for all movement.

Ask yourself this question. Will the belt move if the plane does not move? The answer is no.

Now ask yourself this question. Will the belt move if the plane moves? The answer is yes. If the answer is yes then it is impossible under these conditions for the belt to hold the plane still relative to the ground and the atmosphere. Why? Because the plane will have to be moving forward relative to the ground and the atmosphere. If it is not moving forward then the belt will not be moving at all.

In other words, under the conditions stated, the belt is not even trying to hold the airplane still. It is only matching whatever the airplane’s speed is relative to the ground and atmosphere but in the opposite direction. If the plane is moving at 50 mph relative to the ground and atmosphere then be belt will be moving in the opposite direction at 50 mph relative to the ground and atmosphere. But if you measure the relative speed difference between the moving belt and the moving airplane it will be 100 mph.

Under these conditions the belt cannot even hold a car still. Think about it. If the belt is designed to exactly match a car’s forward motion relative to the ground the car can still move forward. If the car accelerates forward to 5 mph relative to the ground then the belt will accelerate backward to 5 mph relative to the ground, but the car’s speedometer will read 10 mph because it will be measuring speed relative to the backward moving belt.

Now, if the puzzle stated that the belt will move in the opposite of the direction that the plane is trying to move at whatever speed is necessary to hold the plane still, then that would be different. But that is not what the puzzle says. However, if the puzzle did say that then you run up against a practical impossibility because there is no moving belt in the world that could move fast enough to hold back an airplane through only the free spinning wheels. In fact it would be impossible to build a belt that could even come close to moving fast enough to hold the airplane still.

The airplane moves forward and reaches takeoff air speed no matter what.

I’m sure there will still be plenty of people that still don’t understand this explanation but this is a sincere attempt to enlighten, not argue.

Interestingly enough, I have read through this thread and other threads in other forums about this. I have seen several people convert from “it won’t fly” to “it will fly” but I have never seen anyone converted the other way. What does that tell you?

144

Agreed, but let’s face it, you’re in a PLANE. You’ve got a throttle, and a few hundred horses under the cowl. Hopefully, you didn’t buy your heap at Fast Freddy’s Fabulous Flyaways, and you can overcome the initial jerk of the conveyor, and the miniscule amount of friction from your wheels.

Gun it man, gun it !

145

ARGHH!!!
It is like bees in your head!

146

That would only be true if the car was attached to a rope anchored at a fixed spot, in which case the engine would be doing no work at all. If the treadmill is not accelerating, then from a relativistic standpoint there is no difference between a treadmill and a road.

147

This is a fascinating thread!

I started out thinking Cecil was wrong. Then the explanations got me thinking that there are two right answers, depending on how you define the question. Now, at long last, I see that there is NO WAY for the belt to cancel the plane’s forward motion.

Various analogies about mag-lev landing gear and tethering the tail finally helped me see that the landing gear is essentially independent of the plane. The wheels and the belt can match each other spin-for-spin until it ruptures the space-time continuum, but no force is holding the plane in place.

I’m usually a quick study (well, quick for my little place in the world – slow by Straight Dope standards), so it was interesting to feel my eyes slowly opening.

148

Here is a question for all of you. There is a car driving on that same theoretical treadmill. It’s speedometer reads 100 mph. How fast is it going?

If you happen to be standing on the treadmill, it’s going 100 mph right past you. If you’re on stationary ground just off the treadmill, it looks to be standing still. And if the treadmill is at the equator, and you’re at the north pole, the car is traveling a little over 1000 mph relative to you; and so is an airplane parked on the tarmac. Despite the blazing speed, that plane ain’t taking off.

Cecil answers the question poorly because he doesn’t clarify what is meant by speed. That’s why this is a trick question: there’s more than one way to answer, but you have to define the “speed” of the plane. However, is there a most appropriate definition of speed in this case? Maybe the speed of a plane on the ground should be considered to be the speed the plane travels relative to the ground it is sitting on. In other words, its ground speed. Cecil, on the other hand, presumes that the speed of the plane should be defined as its airspeed, even though it is still on the ground. Cecil must therefore believe that a plane parked on a (non-conveying) tarmac in a 30 mph wind is traveling 30 mph. His answer is correct when speed is defined as airspeed, but doesn’t ground speed seem more appropriate for a plane on the ground? The other sensible presumptions in this case are that the wind, or the airflow induced by the treadmill are negligible, and that this isn’t one of those few planes that can generate lift just by “pulling” air over its wings. Making these sensible presumptions, the plane doesn’t take off; any infinitessimal acceleration of the plane is matched by the conveyor belt until all wheels are spinning at infinite speed. If airspeed increases, then the conveyor belt speed isn’t keeping up with the (ground) speed of the plane, so the initial rules of the riddle are broken. Does this answer seem to make sense? No. Cecil’s seems to make more sense, and it’s what would happen in real life because you couldn’t create the friction-free world and magical conveyor belt that would allow the no-fly explanation to work.

So I ask you, Cecil, if there’s a plane being pushed north at 2 mph on a treadmill rotating south at 4 mp by the ground crew on an aircraft carrier traveling east at 15 mph in southerly currents of 5 knots and westerly winds of 20 knots at the equator, and you’re at the north pole, what is that plane’s speed?

149

HAHA you don’t know what a red herring is do you? It’s something that draws attention away from the central issue whether it exists or not.

Anyway, it’s all about the airspeed. Airspeed equals lift no matter what the ground speed or treadmill speed is. A parked plane will take off if the wind hits it at the right speed from head on (that’s part of how wind tunnels work). And a plane racing down the runway at 200MPH will not get any lift if the wind is matching 200MPH from behind it.

150

Like Cecil said the puzzle or question is flawed. It makes an false assumption.

It’s like asking a straight person if their mom knows they are gay.

There is no way the moving runway is gonna keep the aircraft stationary relative to the groud or the air since it is not driven by it’s wheels. The first thing you have to understand is if two identical planes are on paralell runways they are both gonna take off the same way even if one is on a treadmill runway. They will be neck and neck and take off together. Its not like the plane magically lifts from a relatively stationary position. They both go down the runway.

And it looks like my first posts are fudged because I misinterpreted what Cecil wrote.
Sorry Cecil.

151

One more try.

Let’s assume you have a toy airplane (or a toy car, for that matter; it doesn’t matter for the purposes of this thought experiment). You make it “taxi” by holding it with your hand and moving it along the surface of your mother’s brand-new Formica™ kitchen counter, while you provide your own exciting “vroom! vroom!” noises.

Now assume that you do the same thing, holding it on the conveyer belt in the shipping department of the factory where your father works, while you remain standing on the floor.

Gee! That conveyer belt doesn’t seem to make a difference, does it?

152

You mean - like an airplane on skis? Fixed landing gear, lots of contact with the ground, nothing to rotate…

And seaplanes.

There are small seaplanes and skiplanes that take-off quite nicely with less than 100 hp engines.

Yes, friction is a factor, but it’s not the only factor and once the airplane starts moving forward and the wings start generating lift that friction will diminish as less of the weight will be on the landing gear.

Propellor effects can also factor in here - I can lock the brakes of the Citabria I fly, preventing all forward motion, and if push the throttle forward the air passing back over the tail creates sufficient lift to raise the entire back end of the airplane up a couple feet. In fact, I have to use the stick to force the tail to stay on the ground during engine run-ups. So right there - if you have the correct model of airplane with which to test this theory the airplace can “cheat” by immediately lifting one of its wheels off the conveyor belt.

And finally, this winter (as many winters) I encountered ground ice while traveling about airports in winged vehicles. Probably the biggest indicator the airplane is rolling on ice (other than the failure of the brakes to work) is that the wheels stop turning (or significantly slow down) while the airplane continues forward. Why? They aren’t getting enough friction from the ground to cause them to rotate. Very strange to start a take off with the plane moving and the wheels not turning at all, but it can and does work.

Airplanes move because of props/jet thrust. They fly because air moves over their wings. Friction and what not on a conveyor belt might delay take-off, but if the plane can move forward it will take off.*

  • There are situations like foot-deep grass that might prevent take-off by dramatically increasing friction to the point the engine thrust can’t over come it, but “runway” and “conveyor belt” implies a normal surface for departure, that is, flat and free of tall weeds.
153

That is the whole point. You can’t build a conveyor belt to spin that way. The best description I have seen so far is the rollerskates on a treadmill. You stand on the treadmill with rollerskates, and what happens, if you do absolutely nothing but stand there? You’ll stay in the same place. Granted, enough friction can be built up against the wheels to propell you backwards.

What happens if you try to push against the ground, or surface of the treadmill to move forward?
Nothing. Your FORCE is against the treadmill, which is moving backwards. You’ll stay in the same place.

What happens if your force to move forward has no bearing on the ground itself? What happens if your “engine” is a rope attached to the wall and you pull yourself forward? You’re going to move forward regardless of what speed the treadmill is moving. Only, your wheels will be moving at the treadmill’s speed + the forward speed.

Everyone wants to talk about an outside stationary observer, which is a joke. This isn’t light speed we’re talking about. If I pull my self forward with a rope, which way will the observer see me pulling myself?

Even better, what happens if you move the treadmill outside, so the wind is at your back and you hold up a sail. Is the treadmill going to keep you from moving forward? No.

It doesn’t matter if you are being pushed by the wind or pulled by the engine. The force causing the plane to move forward has nothing to do with the force on the wheels.

154

No, he didn’t.

He describbed a force moving in one direction, causing the wheels to move in a counterroating direction which means the plane won’t move. It won’t fall off the edge, nor will it fly at 60 knots. The plane will stay put, and the wheels will SPIN in the opposite direction at 60 knots. Fire up the engine and the force of the engine pulls the plane through the air regardless of how fast the treadmill is moving.

155

Several posts have stated that putting a wheeled vehicle on a conveyor belt and then starting the conveyor will turn the wheels, leaving the vehicle stationary relative to a point off the conveyor. That’s just wrong. If I place any wheeled vehicle on a treadmill and turn on the treadmill, the wheels will not turn and the vehicle will be carried along until it reaches the end of the belt. (if the start of the belt is very sudden there may be some small initial movement of the wheels, but they will soon stop).
Another fallacy I’ve seen concerns a car on the belt. If the belt moves backward at 5MPH and you accelerate the cars engine to maintain the car’s position relative to a given point off the belt, the speedometer will read 5MPH, not 10MPH.
It does not matter what the source of power the vehicle has, whether it’s a engine driving the wheels, a propeller pulling an airplane, or your hand pushing a toy car, before the vehicle can move forward it must overcome the opposing speed of the belt.

156

I’m sorry, but you are incorrect. For any vehicle to move forward it must overcome the force, if any, opposing it. There is no force directly opposing the plane itself, only the freely rotating weels. start the engine and the plane will move forward.

Period.

157

Yep; you are correct. I was wrong.

I think that weakens my original argument, but doesn’t completely invalidate it. Cecil was comparing an automobile, whose engine outputs just enough power to overcome bearing friction and tire hysteresis with a plane whose engine outputs more than enough power to overcome bearing friction and tire hysteresis. Not a parallel situation.

158

Answer this. If the plane is sitting on a level treadmill, facing west, and the treadmill is started up moving east, what will happen to the plane?

159

Sorry, but unless you assume massless wheels, that’s simply wrong. Rotationally accelerating the wheels will impart a force on the plane. You can argue that force won’t be large enough (nothing wrong with that), but I should point out that the original problem doesn’t specify the size of the plane or the power of the engines.

But neither is that conveyor belt constructed to automatically match the wheel speed. Of course you can show different results if you alter the problem.

Yes there is. The force to rotationally accelerate the wheels will be transferred to the plane.

160

That is exactly true the way you stated it, but if you are referring to my comment from my post on the prevous page, that is not what I said.