A jet plane on a large treadmill

[DJSapp]

New question, new crazy. Since this time the wheels and treadmill must accelerate instantly in order for the plane to move, the force exerted to move the plane will be essentially infinite. So you can't ignore bearing or rolling friction or the wheels' moment of inertia. Until the thrust blows up the wheels (allowing them--and, not incidentally, the treadmill--to stop) the plane's velocity remains literally infinitesimal. The very brief wind offered by the treadmill won't have enough time to lift the plane because the elevators are too high up to get the full force of the blast (the air doesn't accelerate as fast as the wheels) for long enough to lift the nose and give the wings enough angle of attack to overcome the fact that the air below them is moving (briefly) faster than the air above. Or maybe the air takes a while to slow down and the pilot could time it just right as the boundary layer suddenly stops? This seems unlikely.

On the thrust vs. static friction question, it seems like you calculated that the engines could overcome friction up to u=0.83 and then stated that the tires have u=0.85. So the plane would actually not quite move with the parking brake on. That's the kind of thing that really bears looking again to make sure it's right every decade or so.

In this case it might not matter much--the real question is what is the friction coefficient of whatever part of the landing gear the plane rests on after the wheels are burned off?

FROM PAGE 7: THE 747 CAN TAKE OFF WITH IT'S WHEELS LOCKED (provided the ensuing fire doesn't create it's own problems)

Ok, so I was too busy tuning my skis last night to write the proof, but here's the just of it:

Originally Posted by Newton's laws
1. An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

2. Net force equals mass of the object times acceleration of the object F=ma

3. For every action, there is an equal and opposite reaction i.e. you push on a table, the table pushes back

Motion of the aircraft as dictated by newton's laws:

F of engine thrust ACTING UPON STATIONARY AIR = 95,000 lbs/engine
Total thrust = 95,000 * 4 = 380,000 lbs
F of rolling friction from the conveyor acting on wheels = u/radius of wheel * m*g = u/2 ft * 910,000 lbs
(assuming 4' diameter wheels, couldn't find actual size in google)

NOTE THAT ROLLING FRICTION IS INDEPENDENT OF WHEEL SPEED

If thrust is greater than rolling friction, the plane will have an unbalanced force (read: plane will be accelerating reltative to air) acting upon it, causing it to move forward down the runway, giving it airflow and lift. It is simple to verify that for the plane to be held still

u/2 * 910,000 lbs = 380,000 lbs

u = .835164 !!!!!

That is 83% of the force applied to a 747 is lost to rolling friction. Imagine being on a bike and only getting 17% of your energy put into forward progress. (A side note, sliding friction of rubber tires is close to .85, the 747 could in theory take off on a regular runway provided it's long enough, with it's wheels completely locked up!) In the real world, rolling friction losses are closer to 0.5%. The plane will accelerate reltative to the stationary air due to an unbalanced force between the engine thrust from the plane acting on the air and the force of friction due to the conveyor moving at any speed.

The conveyor cannot stop the 747. Bow to its might.

Still true. Even moreso for the lightweight 787 dreamliner and 737 max as they have better thrust to weight ratios. The thrust generated by the engines can overcome the static friction force of the wheels. Then you're using kinetic friction coefficients (which are lower than static friction) for rubber on dry pavement, until the wheels shred. Then you're using steel on pavement kinetic friction coefficients which are even lower. So as the wheels are destroyed and it is skidding on its rims, the plane accelerates FASTER. (and then the fire catches up with it).

[DJSapp]

Simplified explanation:

I'm riding my bike on roller trainers, going 20mph, and stationary on the rollers. I pedal faster, the rollers go faster, and I never move.

You enter the room, stand behind me, and push me forward off the rollers. The speed of the rollers can't stop me, because the energy pushing me forward is completely independent of the wheel speed.

You're the jet's engines, pushing the plane forward.

Your example sucks.

Now if I could find that drawing that someone did of a guy on a treadmill wearing rollerblades, being pulled by a motorcycle, that would be more fitting.

[jono]

I agree about the eventual result, but your calculation says the opposite of what you conclude: the maximum coefficient which the 380,000 lbf thrust could overcome would be 0.835 according to that, and the actual (as you stated) is 0.85. So the plane can't move with the wheels locked.

So even in the new scenario the tires stay stuck to the treadmill until they start to fail, at which point slippage allows the plane to start accelerating and the treadmill to stop short of either c or truly infinite acceleration. Until the tires slip the force of the treadmill is basically equal to the thrust at any given moment and the inertia of the wheels defines their rate of acceleration.

[Rip'nStick]

Yesss it's back

[Dantheman]

Now this is where the paradox begins

Our problem INCORRECTLY assumes the wheels drive the aircraft. They do not. To solve using the above information we agree on:

Airspeed - Wheelspeed = 0 = Wheelspeed + Treadmill speed

Consolidate terms:

Airspeed - 2x Wheelspeed = Treadmill speed.

Further simplification

Airspeed - 2x treadmill speed = Treadmill speed.

This equation is only true for treadmill speeds of zero and infinity. Per physics, the airplane (and therefore the wheels and treadmill) will begin moving as the engines do not act upon the treadmill. This causes the wheels to move. As the equation is only solvable for values of zero and infinity, the treadmill will continue to accelerate to infinity.

I'm skeptical of the "infinite acceleration" paradox. I can't quite put my finger on an equation that proves it, but I think it comes down to the fact that the equation "Airspeed - Wheelspeed = 0 = Wheelspeed + Treadmill speed" is not balanced. Wheelspeed + Treadmill speed will always equal zero, but Airspeed - Wheelspeed does not always equal zero. At any airspeed >0 airspeed-wheelspeed /= 0 by definition. Infinite acceleration of the treadmill should require infinite acceleration of the plane. I can't shake a gut feeling that for any speed >0 Treadmill speed = -Wheelspeed = 2x Airspeed. Fuck, this is going to bug the shit out of me.

[jono]

I'm skeptical of the "infinite acceleration" paradox.

You are right about that. It's not infinite acceleration, it's actually the force couple on the wheels divided by their moment of inertia. T = I x alpha, alpha = T/I. The force is the thrust (divided by the number of wheels) because anything higher would cause the plane to move. The thrust meets the treadmill force at the wheel bearings, primarily as a radial force (and a little tangential as friction). Neglecting friction the torque would be thrust force times the radius of the tire.

The plane does not accelerate, so the sum of the horizontal forces on the plane is zero. Same for the wheels. But the wheels are free to spin up according to T=I x alpha...until they explode or start slipping and change the treadmill conditions.

[Svengali]

[stfu&gbtw]

Jesus christ, you guys can complicate some simple shit.

If the wheels were the plane's source of thrust, and the treadmill were moving fast enough to cancel out the forward motion the wheels would create, then the plane would not take off.

Since the wheels on a plane are NOT the source of thrust, and since they do not impart significant drag on the plane, their speed and direction of motion have no bearing on the source of thrust, nor on the speed and direction of the plane. The treadmill could be spinning so quickly that the wheels are spinning backwards and the plane would still take off.

[mcski]

Still waiting for KITT to get off the fucking treadmill

[Mustonen]

Doesn't this just make wheelspeed irrelevant? If treadmill speed = wheelspeed, the plane has to overcome static friction of wheels to treadmill whether the wheels are spinning at infinity or at zero. Throw out bearing considerations, wheels blowing up, etc., because those all end up at approximately the same place.

Might as well assume the wheels are locked up. That's actually the only way the question makes any sense in the real world. Instantaneous acceleration to infinity isn't a real thing.

It's an unnecessary and illogical complication to imagine that the treadmill is designed to stop the plane. It isn't. It just matches wheelspeed. You've lost your frictionless platform, that's all.

[Flyoverland Captive]

Yes, unless the treadmill ramps up to insane speeds instantly, destroying the tires, wheels, and landing gear before the plane can make enough forward progress to get airborne.

[Mustonen]

Yes, unless the treadmill ramps up to insane speeds instantly, destroying the tires, wheels, and landing gear before the plane can make enough forward progress to get airborne.

Yeah, talked about that in my edit. Instantaneous acceleration to infinity isn't real (would that even properly be termed acceleration, technically? Aren't you missing a component?) That's why it's a mind bender. Like time travel (in the go back in time to meet your younger self sense). Again, it's not about overcoming the plane's forward acceleration, it's just loss of a frictionless platform.

[jono]

The KITT issue is similar to the new airplane question in that recognizing the inertia of the wheels is clarifying. The car comes up to the ramp going just a little faster than the ramp, the tires chirp a little as they slow down almost to ramp speed, and the difference in kinetic energy (say, 5 mph for the car plus whatever the wheels had at, say, 50 mph) gets converted to potential energy as it climbs the ramp.

Since the wheels are light compared to the car the fact that they have to slow down a lot more than the car keeps things manageable. When each set of tires hit the ramp the car speeds up just slightly, but only by like 0.1 mph.

[jono]

Yeah, talked about that in my edit. Instantaneous acceleration to infinity isn't real (would that even properly be termed acceleration, technically? Aren't you missing a component?) That's why it's a mind bender. Like time travel. Again, it's not about overcoming the plane's forward acceleration, it's just loss of a frictionless platform.

In the original question where treadmill speed = plane speed that's not a concern because the treadmill only needs to go about 2x stall speed. In the version where the wheel speed is matched the mass of the wheels keeps infinite acceleration at bay and also "holds" the plane still until force exceeds traction. At which point the plane accelerates and takes off--but first it blows up the wheels in a most spectacular misuse of energy. Then it lands in the Hudson.

[DJSapp]

I agree about the eventual result, but your calculation says the opposite of what you conclude: the maximum coefficient which the 380,000 lbf thrust could overcome would be 0.835 according to that, and the actual (as you stated) is 0.85. So the plane can't move with the wheels locked.

So even in the new scenario the tires stay stuck to the treadmill until they start to fail, at which point slippage allows the plane to start accelerating and the treadmill to stop short of either c or truly infinite acceleration. Until the tires slip the force of the treadmill is basically equal to the thrust at any given moment and the inertia of the wheels defines their rate of acceleration.

I'm skeptical of the "infinite acceleration" paradox. I can't quite put my finger on an equation that proves it, but I think it comes down to the fact that the equation "Airspeed - Wheelspeed = 0 = Wheelspeed + Treadmill speed" is not balanced. Wheelspeed + Treadmill speed will always equal zero, but Airspeed - Wheelspeed does not always equal zero. At any airspeed >0 airspeed-wheelspeed /= 0 by definition. Infinite acceleration of the treadmill should require infinite acceleration of the plane. I can't shake a gut feeling that for any speed >0 Treadmill speed = -Wheelspeed = 2x Airspeed. Fuck, this is going to bug the shit out of me.

Reread all of my posts from pages 1-7. They're still relevant to the new question.

Both of you are stuck on the same part of the paradox. The airplane cannot have wheelspeed without having airspeed, because creating airspeed is the SOLE method an airplane can move itself. It is the nature of the motion of an aircraft. If the plane has airspeed, the treadmill cannot match the wheelspeed. Wheelspeed = airspeed + treadmill speed. This is not a logical equation, and it is why your head hurts. Because it is a physical impossibility.

The only conditions which the equation wheelspeed = airspeed + treadmill speed are 0 and infinity. So back to jono's point: if the airplane is stopped, nothing is stopping it from rolling and having a 0.05 coefficient of rolling friction. Once it is moving, you have to use kinetic friction. And another out is the frictional coefficient between the ceramics in the brakes is actually lower than the rubber and the pavement.

So the revised question can be summed down to: if an airplane is sitting on a runway and starts moving, and once it moves a brick wall magically appears in front of it, will it take off?

[Mustonen]

Reread all of my posts from pages 1-7. They're still relevant to the new question.

Both of you are stuck on the same part of the paradox. The airplane cannot have wheelspeed without having airspeed, because creating airspeed is the SOLE method an airplane can move itself. It is the nature of the motion of an aircraft. If the plane has airspeed, the treadmill cannot match the wheelspeed. Wheelspeed = airspeed + treadmill speed. This is not a logical equation, and it is why your head hurts. Because it is a physical impossibility.

The only conditions which the equation wheelspeed = airspeed + treadmill speed are 0 and infinity. So back to jono's point: if the airplane is stopped, nothing is stopping it from rolling and having a 0.05 coefficient of rolling friction. Once it is moving, you have to use kinetic friction. And another out is the frictional coefficient between the ceramics in the brakes is actually lower than the rubber and the pavement.

So the revised question can be summed down to: if an airplane is sitting on a runway and starts moving, and once it moves a brick wall magically appears in front of it, will it take off?

Fuck. Just...no. Jeezus.

Who the duck said anything about airspeed? Nobody. You keep adding that silly bullshit.

Wheelspeed = treadmill speed at infinity and zero, sure.... but that means the only relevant part of the mechanism is the friction between the wheels and the treadmill (or the attachment of the wheels to the plane).

So, to reframe, the wheels are locked out. Does the plane take off? Interesting question that's been analyzed earlier in this thread (quite competently by you yourself, IIRC) but not exactly a mind bender.

[Mustonen]

In the original question where treadmill speed = plane speed that's not a concern because the treadmill only needs to go about 2x stall speed. In the version where the wheel speed is matched the mass of the wheels keeps infinite acceleration at bay and also "holds" the plane still until force exceeds traction. At which point the plane accelerates and takes off--but first it blows up the wheels in a most spectacular misuse of energy. Then it lands in the Hudson.

Yeah, that's what I said way back when (I might have still been focus...). I agree with the rest, but the whole instant infinity thing is far enough outside of our ability to reasonably understand that I think it's most instructive just to set the wheels at 0. That's basically what's happening, here.

[mcski]

You guys can do whatever the fuck you want to the wheels, but that Bitch is still gonna roll right off the treadmill and take off...and when it lands it will be on the back of a truck right next to KITT

[jono]

The only conditions which the equation wheelspeed = airspeed + treadmill speed are 0 and infinity.

Most of the original stuff remains relevant, but this point here is only valid when the wheels are tracking the treadmill. They will only do that if the force exerted by the treadmill is low enough that the traction of the tires is not exceeded. Since the treadmill matches the wheel speed it will accelerate slower once the wheels start to slip and "slow down." Since velocity and force are not discontinuous, there is no way to jump to infinity, but once slippage starts acceleration ensues. There's no paradox or singularity if the only magic is in the treadmill, everything else conforms to normal laws of physics, just maybe not the oversimplified versions they offer in most physics classes.

But yes, the plane stays still until the tires slip or the wheels explode, because that's the definition offered. The only way that happens is if the treadmill accelerates very fast. Work the load paths through from the thrust to the treadmill and you'll have what I outlined above where the radial force in the bearings is the major force stopping the plane. This isn't a statics problem so while the forces sum to zero at that instant the torque does not and thus the treadmill and wheels accelerate according to T=I x alpha. If you draw out the diagrams and force summations this has a solution (you probably have to estimate I and r for the wheels) and there's no discontinuity involved--you just wind up calculating how long it takes the tires to explode.

[jono]

Oooh, DJ I think I see what's causing the issue: in the new version the treadmill is somehow controlled to match the wheel speed, but the wheel speed is not forced to do anything but react to conditions. So the wheels control the treadmill speed and the thrust controls its force (since airspeed must remain zero as you've shown). And since force and speed are related by T = I x alpha, you can derive the wheels' speed as a function of time up to the slip point.

[jono]

You guys can do whatever the fuck you want to the wheels, but that Bitch is still gonna roll right off the treadmill and take off...and when it lands it will be on the back of a truck right next to KITT

Technically it's going to slide before it takes off, but landing in KITT's trailer is a solid plan!

[east or bust]

My understanding is that the plane does not take off under ideal conditions… as in the wheel bearings are frictionless and the tires don't explode… The only way it could is if the air speed resulting from the friction between the treadmill and air exceeds that required to provide the necessary amount of lift. Is this correct so far? Disclaimer, I didn't read any of this thread.

[Mustonen]

My understanding is that the plane does not take off under ideal conditions… as in the wheel bearings are frictionless and the tires don't explode… The only way it could is if the air speed resulting from the friction between the treadmill and air exceeds that required to provide the necessary amount of lift. Is this correct so far? Disclaimer, I didn't read any of this thread.

The treadmill doesn't act upon the plane to keep it stationary, only to match the speed of the wheels. The plane will move and take off if the force of the jet engines can overcome the friction between the wheels and whatever the treadmill is made of.

[Mustonen]

...So the wheels control the treadmill speed and the thrust controls its force (since airspeed must remain zero as you've shown).

False premise. The system isn't designed to control airspeed. Wheelspeed = -treadmill speed. That's the equation. Nothing else has been defined. Thrust doesn't control the force of the treadmill!

If wheelspeed cancels out treadmill speed, then the problem becomes only whether the engines can overcome the friction between the wheels and the treadmill.

[iceman]

Plane goes back in time.