Mess with your minds: A wind powered craft going faster than a tail wind speed.

[electrodacus]

But the power doesn't have any directionality.  That is one of your fundamental misunderstandings.  The backwards force on the wheels is less than the forward force of the prop.  Net force: forward.  The power input at the wheels is less than the power consumed by the prop.  Net power: positive.  The fact that the force at the wheels is backwards is *irrelevant*

Power has directionality if that is what I chose. Think about power getting in or out of a battery and if you do not use a sign how will you know the state of charge of that. Same here in order to know what direction the vehicle has moved you will want to have a sign (your choice of convention).
You need to understand that force at the wheel is not force action on vehicle body. The wheels are connected together and if gear ratio will be 1:1 then yes the delta between those forces will the the force acting on the vehicle but there forces will be equal assuming ideal case or slightly higher force (breaking force) on the generator wheel in real case so vehicle will move backwards (again a chosen convention in my case right to left).
There is no gear ratio you can select that will make the vehicle in my diagram move from left to right.  You will get confused if you try to use forces as you need to understand the relation between the two wheels as one wheel power's the other.

The power input at the wheels is less than the power consumed by the prop.

Your sentence makes no sense. Wheels powers the propeller so in an ideal system the two can be at best equal and in a real system power at the propeller will be lower than power generated at the wheels as wheels provide power to propeller and nothing else.

   

Kinetic energy is not a vector and has no direction.  No equation in units of energy or power tells you anything about direction of motion.  You need to look at force or velocity.  I worked out your treadmill problem above.  You haven't addressed it.  Can you find a *mathematical* flaw in it?  Or you can actually do the experiment.  You obviously haven't done it.

As mentioned direction can be added to both power and kinetic energy as you start with a stationary vehicle co zero kinetic energy and if you want to know the direction you can decide to use a sign to indicate the direction of travel.
The experiment can only show that vehicle will move from right to left and I do not need to do the experiment since is super obvious to me. You can do the experiment and you will be surprised that you will not be able to build a vehicle that move from left to right if you respect my diagram.
As I offered to others here if you can prove a vehicle as shown in my diagram moves from left to right I will pay for your expenses related to the experiment.

[Brumby]

I'm beginning to think we are being trolled.

[electrodacus]

Indeed you should get the same result if you use two different methods to determine the same thing, but the trick is in figuring out which one is most appropriate given the problem.  I cannot fathom why you don't understand Newton's laws of motion, or at least are ignoring them.  Do you actually not believe that the car is going to go in whatever direction the net force is?

Any method will be appropriate and all will give the same result is applied correctly. In this particular case I think using power is the simplest and simple means less chance to mess up the calculation.
Yes vehicle will go in the direction the force on the vehicle body points to (force at the wheel not the same with force on the vehicle body).

Maybe if you add a delay you will be able to better understand.
So say you have a 2:1 gear ratio that means G generator wheel will do 2 revolutions for each 1 revolution at the motor wheel.
Now say for one second a breaking force of 2N is applied to the generator wheel and say speed is 10m/s vehicle moving from right to left a distance X
Then say that energy was stored in a perfect battery and now it is available fully to Motor wheel that will have 4N of force but speed will be half 5m/s so total distance traveled in 1 second this time from left to right will still be X thus vehicle did not moved in any direction even if there was a force imbalance if that is all you look at.
Is it not much easier to just look at the Power and get to the same conclusion ?

[electrodacus]

I'm beginning to think we are being trolled.

I'm the one possibly being trolled. If not it seems you just need to run the experiment as you can not properly calculate the result.

[bdunham7]

In this particular case I think using power is the simplest and simple means less chance to mess up the calculation.

Well let's try power again and let's not use any tricky delays, storage or anything else, just the generator and the motor as you originally suggested.  Lets see what we can agree on so we can find out where we diverge?

So for the starting position, lets say the car is not moving, the belt is moving from right to left at a given speed, I think you mentioned 1m/s, but it doesn't matter.  We load the generator wheel to produce power, which creates a force that acts on the wheel and is transmitted to the belt, which provides the usual equal but opposite reaction and since the wheel is held to the body of the car by an axle, that force is transferred to the car.  Simultaneously we use M to hold the car in place by preventing M from turning.  The force from G through the body of the car is transmitted to M and then to the fixed platform, which provides and equal but opposite reaction.  You don't need to worry about following or balancing the forces since you want to jump straight to energy, but I'm stating it now so there's no confusion later. So now we are generating energy, but not using it anywhere and the car is not moving.  Any problems so far?

[Brumby]

.... it seems you just need to run the experiment as you can not properly calculate the result.

Sage advice.  You should try following it.

[electrodacus]

Well let's try power again and let's not use any tricky delays, storage or anything else, just the generator and the motor as you originally suggested.  Lets see what we can agree on so we can find out where we diverge?

So for the starting position, lets say the car is not moving, the belt is moving from right to left at a given speed, I think you mentioned 1m/s, but it doesn't matter.  We load the generator wheel to produce power, which creates a force that acts on the wheel and is transmitted to the belt, which provides the usual equal but opposite reaction and since the wheel is held to the body of the car by an axle, that force is transferred to the car.  Simultaneously we use M to hold the car in place by preventing M from turning.  The force from G through the body of the car is transmitted to M and then to the fixed platform, which provides and equal but opposite reaction.  You don't need to worry about following or balancing the forces since you want to jump straight to energy, but I'm stating it now so there's no confusion later. So now we are generating energy, but not using it anywhere and the car is not moving.  Any problems so far?

I use the delay so you can better understand what happens (It seems it did not worked).
What G transmits to M is not a force but a power and if vehicle is ideal then vehicle will just stay in place the power from G is all transferred to M so since one opposes the other result will be zero (again ideal case).
In a real system only one thing can happen no matter the gear ratio and that is that power available to M is lower than what generated by G (multiple losses) thus vehicle will be moving from right to left.
So energy is transferred in this case from treadmill to vehicle and some of that will be lost as heat trough friction and some will end up as kinetic energy.   

[bdunham7]

if vehicle is ideal then vehicle will just stay in place the power from G is all transferred to M so since one opposes the other result will be zero

OK, lets just focus on that so I understand your position.  If the car is staying in place and G is generating the power, how does M use the power in an ideal machine?

[electrodacus]

if vehicle is ideal then vehicle will just stay in place the power from G is all transferred to M so since one opposes the other result will be zero

OK, lets just focus on that so I understand your position.  If the car is staying in place and G is generating the power, how does M use the power in an ideal machine?

As an ideal machine you will call this a perpetuum mobile as generator will power the motor and then motor turn the generator it is just a particular case of that since vehicle is stationary. Obviously that can not be build in real life so vehicle will move from right to left no matter how good (low friction) your vehicle is.
So in ideal case no power is generated or used as vehicle will remain stationary as that was the starting point.  G wheel will just rotate with exact same speed as the treadmill no friction and no force so zero power and wheel M will just stay as it is no rotation zero speed so zero power.
Any small friction in the real case will result in vehicle moving from right to left pushed by the treadmill. Since you can not have more power available at M than what you can generate at G no matter you decision in gear ratio vehicle will just move from right to left.

Your explanation involves larger force at wheel M but that is irrelevant as long as power is lower than on wheel G. While this may sound non intuitive for you it is the reality and there is nothing you can do about that.
The same sort of flawed explanation is give to the Blackbird vehicle when the reality is just simple energy storage and so vehicle is nothing special as it will charged during initial phase from the wind power and then quite a bit before it even gets to wind power it starts to use the stored energy and ass soon as that is used up at most a few minutes the vehicle will be back to below wind speed.
Even in the Blackbird data for that 28mph record (I will not call that 2.8x times the wind speed as wind during the test varies between 8mph and 15mph) you can see the acceleration rate decreasing and it was not far from the peak speed maybe 32 to 35mph and if they waited maybe another 30 seconds they will have been there but I think the road was short so they could not have continued.

[electrodacus]

.... it seems you just need to run the experiment as you can not properly calculate the result.

Sage advice.  You should try following it.

Why will I do that ? I know what the result will be both by brain simulation and simple power equation.
It will be a waste of time for me to confirm what is super clear and there is no evidence proposed to the contrary (no experiment showing that or equation predicting anything other than vehicle moving from right to left).

[electrodacus]

And just in case this can help here is my first wheel only equivalent of blackbird
The large wheel is to scale as Blackbird propeller sweep area vs the wheels that are also to scale.



But this is more complicated so afer I see where people are most confused I reduced the model to just the one you see before with the G and M wheel of same size trying to advance on a treadmill.

[ejeffrey]

But the power doesn't have any directionality.  That is one of your fundamental misunderstandings.  The backwards force on the wheels is less than the forward force of the prop.  Net force: forward.  The power input at the wheels is less than the power consumed by the prop.  Net power: positive.  The fact that the force at the wheels is backwards is *irrelevant*

Power has directionality if that is what I chose.

Nope.

in order to know what direction the vehicle has moved you will want to have a sign (your choice of convention).

No, that is just not what energy means.  Maybe you are thinking of momentum?

The power input at the wheels is less than the power consumed by the prop.

Your sentence makes no sense. Wheels powers the propeller so in an ideal system the two can be at best equal and in a real system power at the propeller will be lower than power generated at the wheels as wheels provide power to propeller and nothing else.

Yes, I got that kind of confusing.

   

As mentioned direction can be added to both power and kinetic energy as you start with a stationary vehicle co zero kinetic energy and if you want to know the direction you can decide to use a sign to indicate the direction of travel.

Nope. Power has a sign, but that refers to time (i.e., is energy increasing or decreasing with time).  But neither has a spatial direction associated with it.  I mean, you can put a sign or a vector on it if you want, but it doesn't mean anything or follow any important conservation laws.

As I offered to others here if you can prove a vehicle as shown in my diagram moves from left to right I will pay for your expenses related to the experiment.

I don't understand your conditions which seem arbitrary and capricious since you think that it matters if the applied motion is from the top (as in the demo from veritasium) or is a board pushed by hand rather than an actual treadmill.  So I don't think you would accept any demo you saw.  As far as I am concerned the demo Derek did with the little cart was identical to your proposal, just with a board pushed from the top side instead instead of a treadmill and a horizontal linkage connecting the wheels.  So I really don't know how to do a demo you would accept.  Your only argument about the existing demo is "can't you see it is different"  I can't, so I can't reasonably do a demo.

Therefore I will make this super concrete so that nobody has to do a demo.

The treadmill is moving at 10 cm/second to the left (that would be negative in the sign convention I suggested).  The wheels both have a radius of 5 cm.  The gear ratio is such that for every one rotation of G, M rotates 0.5 turn in the same direction.  Please calculate the steady-state velocity of the cart as well as the rotation rate of each wheel (radians per second, or RPM if you prefer).  Numerical answers, and explicitly state the direction of motion for each (left/right for the cart, clockwise/anticlockwise for the wheel).

Can you solve that simple problem?  Feel free to use whatever formulas you like, but again: give the answer in the steady state when the cart is no longer accelerating.  I will check to make sure that your answers aren't slipping.

Can you or can you not solve this problem using your equations?  I will post mine here after you try.

[bdunham7]

Your explanation involves larger force at wheel M but that is irrelevant as long as power is lower than on wheel G.

I'm not explaining anything.  I'm asking you to clarify yours by explaining what happens in the simple, specific situation where M is held fixed so that the car cannot move and G is generating power.  Where does the power go?

[electrodacus]

Your explanation involves larger force at wheel M but that is irrelevant as long as power is lower than on wheel G.

I'm not explaining anything.  I'm asking you to clarify yours by explaining what happens in the simple, specific situation where M is held fixed so that the car cannot move and G is generating power.  Where does the power go?

How is M held fixed ? That is not the case in real world If you fix the wheel M to the vehicle then you will no longer call that a wheel.
To generate power from G the vehicle will need to move backwards (right to left) as you apply a breaking power.

[electrodacus]

I don't understand your conditions which seem arbitrary and capricious since you think that it matters if the applied motion is from the top (as in the demo from veritasium) or is a board pushed by hand rather than an actual treadmill.  So I don't think you would accept any demo you saw.  As far as I am concerned the demo Derek did with the little cart was identical to your proposal, just with a board pushed from the top side instead instead of a treadmill and a horizontal linkage connecting the wheels.  So I really don't know how to do a demo you would accept.  Your only argument about the existing demo is "can't you see it is different"  I can't, so I can't reasonably do a demo.

Therefore I will make this super concrete so that nobody has to do a demo.

The treadmill is moving at 10 cm/second to the left (that would be negative in the sign convention I suggested).  The wheels both have a radius of 5 cm.  The gear ratio is such that for every one rotation of G, M rotates 0.5 turn in the same direction.  Please calculate the steady-state velocity of the cart as well as the rotation rate of each wheel (radians per second, or RPM if you prefer).  Numerical answers, and explicitly state the direction of motion for each (left/right for the cart, clockwise/anticlockwise for the wheel).

Can you solve that simple problem?  Feel free to use whatever formulas you like, but again: give the answer in the steady state when the cart is no longer accelerating.  I will check to make sure that your answers aren't slipping.

Can you or can you not solve this problem using your equations?  I will post mine here after you try.

My conditions are not capricious.
What Derek from Veritasium shows is fairly different form what I have in my diagram and also different from what he claims it represents.
In his case the floor is the input source for the vehicle driving the small wheels (generator wheels) and those in turn will drive the large wheel (M wheel) that drives from left to right on the lumber.  The floor that you can consider the wind as it is what powers the vehicle moves in the opposite direction about 2.8x time faster than how fast the large wheel travels on the lumber.
So what he demonstrates there is a vehicle driving in the opposite direction to wind at 2.8x lower speed than the wind. Of course all of that is possible thus the reason it can be demonstrated.
Let me know what you disagree with from my above explanation.

The demo I will accept is super simple and exactly as in my diagram. So just have a treadmill set at any speed you want and at the back on the treadmill as shown in diagram put a box on the floor so that M wheel of the vehicle can stay at same level as the treadmill so that is level. No other conditions are necessary. You can use any wheels with any traction and any gear ratio between the G and M wheels.

What you suggest will act as a locked vehicle so it will be pushed back if the M wheel has less traction or if the G wheel has less traction it will just slip on the treadmill. But since treadmill surface is made of some sort of rubber most likely the back wheel M will slide back meaning from right to left.
If that red box is super slippery there may be some wheel rotation (clockwise) but net result will still be vehicle moving from right to left.
How fast the vehicle moves from right to left (only possibility) will depend on friction in multiple parts of the system.

Here is a vehicle on a treadmill that will be what Veritasium demonstrated and while you will think so it is not the same with the one in my diagram.

The fact that vehicle is flipped changes more than you think.

[Kleinstein]

One can look at the demonstration with the funny belt driven vehicle in different ways. It depends on which part ( belt or fixt structure) you use use to represent the ground or wind speed.

If you do it like the treadmill experiments with the propeller driven vehicles, the band represents the ground and the standing air / fixed structure are the moving air. With the band moving right to left, the wind direction would be from the left to the right. Because of the camera moving with the "wind", a stationary object would be at wind speed. An object moving to the left in the picture would be slower than the wind (e.g. more like the "ground"). An object moving to the right in the picture would be faster than the "wind".

So if you take the same indentification of reference frames for the wheeled vehicle in the short video, you get the vehicle going to the right an thus faster than the "wind".

[iMo]

A rotating propeller creates a forward lift independent of the surrounding wind direction and wind speed (under typical conditions).
At the beginning the vehicle is pushed forward by the force of the wind where the propeller acts as simple sail (no blade's wing effect, no additional propeller lift, the max achievable vehicle speed is equal the tail wind speed).
The moving mass of the vehicle incl. inertia of the propeller and mechanically coupled wheels creates a storage of the energy compensating fluctuations of the wind speed/direction and by the propeller induced lift (ie. like a filter capacitor in our electronics).
At a specific vehicle/blades speed the propeller blades start to act as the wings (as they are designed that way) and start to create the lift.
The by the propeller created lift/speed adds up to the speed of the wind.

With already rotating propeller (the blades create lift):
with the wind coming from front: v_vehicle = j*v_vehicle + k*prop_lift - v_wind
with the wind coming from back : v_vehicle = i*v_vehicle + k*prop_lift + v_wind
where j and i are some params related to vehicle losses, k is related to propeller lift efficiency..

With a variable-pitch propeller you would be able to regulate the lift at specific configuration - thus to maximize the vehicle speed.

The max speed of the vehicle would be determined by the losses (friction in gear, wheels, aerodynamics, etc.) and the lift efficiency of the propeller.
For example with 50% "propeller's lift creating efficiency" and 40% mechanical vehicle losses the speed of the vehicle will be higher than the speed of the tail wind.

[Brumby]

.... it seems you just need to run the experiment as you can not properly calculate the result.

Sage advice.  You should try following it.

Why will I do that ? I know what the result will be

But you don't.  You think you do - but you don't.

both by brain simulation

Very scientific ... NOT

and simple power equation.

Simply wrong equations ... or at the very least, incomplete.

It will be a waste of time for me to confirm what is super clear

Except, you are still wrong - and seem unable to acknowledge that possibility.  The mechanism is NOT "super clear" and the fact that you have made this statement condemns your claims as being made from a position of preconception - and not open to thinking beyond your expectation.

and there is no evidence proposed to the contrary (no experiment showing that or equation predicting anything other than vehicle moving from right to left).

There is an experiment - and it's freely available to view, review and consider.  The information is there - IF you care to expand your thinking past "the obvious" ... which you consistently refuse to do.

Those of us who have seen the demonstration and (sooner or later) understand how the mechanism works,  know full well that the blackbird would continue indefinitely - at higher than wind speed - as long as the wind was suitable.

The claim that the length of the test run was too small to demonstrate what you propose is something you need to PROVE.  Just waving your hand in the air and making such a statement is of zero value.  NONE of your hypotheses, "calculations" or claims fit the evidence which has been demonstrated.

... and please, don't decry the vehicle for it's structural issues.  It was good enough for the demonstration.

[ejeffrey]

My conditions are not capricious.
What Derek from Veritasium shows is fairly different form what I have in my diagram and also different from what he claims it represents.
In his case the floor is the input source for the vehicle driving the small wheels (generator wheels) and those in turn will drive the large wheel (M wheel) that drives from left to right on the lumber.  The floor that you can consider the wind as it is what powers the vehicle moves in the opposite direction about 2.8x time faster than how fast the large wheel travels on the lumber.

I say the board is the wind, the floor is the ground which is stationary, and the cart moves to the right faster than the board.  This is exactly as in the blackbird.  Why is that not ok?  To be clear you can label things differently and change your reference frame if you want, but why is my choice and Derek's unacceptable or different than the blackbird?

[bdunham7]

How is M held fixed ? That is not the case in real world If you fix the wheel M to the vehicle then you will no longer call that a wheel.

So if I set the parking brake on my car, the rear wheels are no longer called 'wheels'?  Do they become pumpkins?  And for the purpose of my question, what does it matter what you call them?

You can hold M fixed in any way you want, and I'm not proposing it is fixed forever, just long enough to start the experiment.  So, again, with M fixed, the belt moving and G loaded and generating power, where does the power go--or do I have excess power remaining?

[electrodacus]

A rotating propeller creates a forward lift independent of the surrounding wind direction and wind speed (under typical conditions).
At the beginning the vehicle is pushed forward by the force of the wind where the propeller acts as simple sail (no blade's wing effect, no additional propeller lift, the max achievable vehicle speed is equal the tail wind speed).
The moving mass of the vehicle incl. inertia of the propeller and mechanically coupled wheels creates a storage of the energy compensating fluctuations of the wind speed/direction and by the propeller induced lift (ie. like a filter capacitor in our electronics).
At a specific vehicle/blades speed the propeller blades start to act as the wings (as they are designed that way) and start to create the lift.
The by the propeller created lift/speed adds up to the speed of the wind.

With already rotating propeller (the blades create lift):
with the wind coming from front: v_vehicle = j*v_vehicle + k*prop_lift - v_wind
with the wind coming from back : v_vehicle = i*v_vehicle + k*prop_lift + v_wind
where j and i are some params related to vehicle losses, k is related to propeller lift efficiency..

With a variable-pitch propeller you would be able to regulate the lift at specific configuration - thus to maximize the vehicle speed.

The max speed of the vehicle would be determined by the losses (friction in gear, wheels, aerodynamics, etc.) and the lift efficiency of the propeller.
For example with 50% "propeller's lift creating efficiency" and 40% mechanical vehicle losses the speed of the vehicle will be higher than the speed of the tail wind.

As soon as the vehicle starts to move the propeller will create thrust as propeller is connected to wheel so propeller will start to rotate as soon as vehicle moves.
So vehicle body and even the propeller is used as a sail and that is most of the power acting on the vehicle but then as propeller starts to spin faster it will be like increasing the size of the sail and at the same time propeller will push air back against the wind so that pressure behind the propeller increases compared to a simple sail where pressure decreases as vehicle speeds up.
So large part of the wind energy during initial phase (below wind speed) will be stored both as two types of kinetic energy one in the vehicle body and one in the rotating propeller but the important type of energy storage and what allows vehicle to exceed wind speed is this pressure differential created by the propeller.

You can test what I'm saying or just imagine having a propeller where blades have no pitch so they will not create any thrust in that case they will be just like a sail and only kinetic energy is stored (flywheel) but to be called a propeller they will need to have some pitch so it will create thrust using energy to do so but also storing that energy by compressing the air.

You can imagine that a vehicle pushed by wind will accelerate faster if propeller had no pitch as energy will not need to be stored by compressing the air and of course that vehicle can not exceed wind speed unless you are allowed to drive the wheels from the flywheel energy by changing gears not the case with blackbird tho blackbird can change the propeller blade pitch and that will also allow for kinetic energy from flywheel to be used if that was desired.

So wind power available during below wind speed will be used as follow
- small part of it will accelerate the vehicle increasing vehicle kinetic energy
- most will be taken trough the wheel and transferred to propeller that will store most of it as kinetic energy (flywheel) and also as pressure differential by creating a pressure differential between the back and front of the propeller. You may think that this sort of energy storage is not useful as it will quickly dissipate if propeller was to stop same way the magnetic field around an inductor will disappear fast if current is removed but anyone accepts inductor as being an energy storage device yes propeller pressure differential that is almost a perfect analog is not considered by most energy storage.
The propeller is a combination of capacitor (flywheel) and inductor (pressure differential when used in air or other compressible materials so this type of storage is not available in water).
If the propeller will not have been connected to wheel the this type of compressed air storage will not have been possible.


Imagine there is no wind and you push the vehicle to some low speed say 1m/s and say we ignore the pressure differential stored energy (maybe propeller used in water) and there is no wind or water flow.
Say wheel is connected to propeller and say total efficiency of transferring power from wheel to propeller and including propeller efficiency is 70% (realistic number).
Then at a 10W of breaking power your propeller will put the equivalent of just 7W as thrust thus vehicle will slow down.
Now imagine vehicle is at 11m/s with a wind of 10m/s or water flow of 10m/s so again we ignore the pressure differential energy storage.
In this situation breaking at 10W will result in the same 7W of thrust as before.
So there will be no difference between 1m/s with no wind and 11m/s with a 10m/s wind (or water flow so that we can ignore the pressure differential energy storage that you do not want to accept anyway).

Sorry for the long replay but without a form of energy storage the vehicle will have no change getting to wind speed let alone exceed that.
And as mentioned the most convincing experiment for you (all) will be to push the vehicle to designed speed and see how vehicle accelerates faster than pushed speed while there is no wind (this only works in air not in water).

[electrodacus]

Simply wrong equations ... or at the very least, incomplete.

Please show how my equation is wrong or incomplete.
If you can demonstrate that then you can convince me that I'm wrong.
The equation is both correct and complete for what it is needed to know.

To remind you power at the generator will always be higher than power at the motor (equal only in ideal theoretical case) and since Motor is powered by generator only motor power will be lower resulting in vehicle moving from right to left.

If you can not disprove that simple (common sense to me) equation then you can not say using a different method of calculating will provide a different result since that will mean your method is wrong.

[IanB]

And as mentioned the most convincing experiment for you (all) will be to push the vehicle to designed speed and see how vehicle accelerates faster than pushed speed while there is no wind (this only works in air not in water).

The vehicle won't accelerate faster than pushed speed when there is no wind, because there is no wind! It won't accelerate at all, it will slow down and stop.

So we can do this experiment of pushing the vehicle when there is no wind, and the vehicle will coast to a standstill. And we will not be surprised, because this is the expected outcome. There is nothing for us to be convinced about, because we are already convinced about this.

[bdunham7]

Please show how my equation is wrong or incomplete.
If you can demonstrate that then you can convince me that I'm wrong.
The equation is both correct and complete for what it is needed to know.

To remind you power at the generator will always be higher than power at the motor (equal only in ideal theoretical case) and since Motor is powered by generator only motor power will be lower resulting in vehicle moving from right to left.

If you can not disprove that simple (common sense to me) equation then you can not say using a different method of calculating will provide a different result since that will mean your method is wrong.

This is the whole point we've been trying to hammer into your head for the past umpteen posts.  You are completely ignoring the basic principle of force multiplication.  Your so-called knowledge of physics doesn't even encompass the basic principles of Archimedes, let alone Newton.  A less-powerful motor can easily overcome a more powerful one with force multiplication via a gearset or some other mechanism.  This is how the propeller-powered vehicle can move upwind, which you your self seem to have acknowledged as a workable concept.  Similarly whatever force is applied to get your G wheel to generate power, that can be overcome with an arbitrarily lesser-powered motor given sufficient gear reduction.  The initial case I'm trying to get you to consider just starts with zero speed and thus zero power.  But you deflect this point with the objection that if we keep it from turning it then is not a wheel.  Really?  And so what?  And even if you persist with that, then instead let motor M be a very small, low-powered motor with a very high gear reduction ratio.  Each force will push the car in opposite directions, but the one with greater force will win, period.

[electrodacus]

I say the board is the wind, the floor is the ground which is stationary, and the cart moves to the right faster than the board.  This is exactly as in the blackbird.  Why is that not ok?  To be clear you can label things differently and change your reference frame if you want, but why is my choice and Derek's unacceptable or different than the blackbird?

When you change reference frames you need to be careful on how you interpret the data as to be correct all interpretations should provide the same result.
As for why I look in the way I do is because I know by looking at the system that small wheels drive the large wheel and not the other way around so the small wheels are the generator and the large wheel is the motor. If you do not believe that try to add a freewheel to the system to observe with wheel is the generator.
So since the small wheels are the generator that is where the energy will be cumming from thus if you want to say it is a Wind powered vehicle that generator wheel will be powered by wind.
So the wind is the floor witch as shown in video moves from right to left about 2.8x faster than how fast the large wheel moves from left to right on the lumber witch is the road. That will make this a vehicle moving opposite to wind direction at 2.8x lower speed than the wind speed.
There is no other correct interpretation of what that vehicle represent's.

The vehicle in my simple diagram has the wind considered at zero speed starting point and is represented by the fixed red box and then a vehicle moving on a road at the speed the treadmill runs at but instead of showing the vehicle in motion the road is moving (treadmill).
This is the same experiment as the propeller based vehicle ruining on the treadmill with the following differences
-Propeller is replaced by a wheel and so air is now represented by the fixed red box
-There no longer is the type of energy storage that will allow the vehicle to move from left to right so it is meant to prove just that.
Propeller will push against the air vs wheel push against a solid (red box) since air is compressible energy is stored there before releasing the vehicle from the hand while with the solid there is no elasticity where to be able to store energy unless as I shown you add a spiral spring to connect the motor to the wheel then you can simulate the same thing as with the propeller vehicle moving from left to right for some limited amount of time determined by the amount of stored energy.