Sailing downwind faster than the wind: resolved?

[mender]

For the sake of accuracy, how would you conduct the experiment? So far, a treadmill that has been leveled and is in a room of still air has been proposed. Is there more that you would add/specify?

It sounds like you've had experience testing devices that interact with the wind. What variables should I consider in my design? I know that the power generated from wind will be dependent on the amount of air influenced (propeller disc for example), efficiency of the interface (design of blades in terms of surface area, pitch, profile), and the energy available to be harnessed (wind speed). Anything else?

 

[mender]

This is unrealistic since when vehicle speed gets closer to airspeed, all movement stops ...

All movement of what stops? The rolling surface and the air are controlled to keep the relative speed between the two exactly the same. If the cart starts to move, the wheels have to roll and the prop being geared to the wheels has to turn. All the drag is present. No movement stops that I can see.

In a test of a cart that has a simple bluff body mounted on top, this method should allow the measurement of the speeds and forces. A treadmill test alone would not work for that scenario (directly downwind slower than the wind), nor would it work for a vehicle that went DDWFTTW.

Are you referring to the possible outcome of the test rather than the test itself? I still want to clarify the test conditions first.

 

[yoavraz]

All movement of what stops? The rolling surface and the air are controlled to keep the relative speed between the two exactly the same. If the cart starts to move, the wheels have to roll and the prop being geared to the wheels has to turn. All the drag is present. No movement stops that I can see.

I'm sorry, I misread your text and misunderstood your scenario. What I said was an answer to a different scenario and incorrect regarding yours.

If we continue your scenario, air speed drops until it cannot overcome the wheels' friction with the treadmill, the wheels stop rolling (with air speed > 0) and the treadmill starts to move the vehicle backwards (remember: the treadmill is going backwards to keep the air-pushed vehicle stationary!). Now then air flow relatively to vehicle is getting faster (and Treadmill changing to slower and slower), its force increases, and it starts pushing the vehicle again forwards. This repeats in a cycle, loop, (or steady state) and we never get to wind speed!

(This is equivalent to what happens outdoor without a treadmill.)

The only possibility to get to wind speed and pass it is with a side-wind component (meaning that wind direction is <180) that does not change during the experiment. The side component continues to generate forward force on the vehicle (sail, wing, wind-torbine, does not matter), overcome the friction, and accelerate until friction (drag in general) equals the forward force. Then the vehicle continues at constant speed, that can be larger than wind speed. If VMG is greater than wind speed, then the vehicle's velocity component in the wind direction at constant speed can also be larger than wind speed.

 

[mender]

So to extrapolate what you're saying, there would be a point where the air speed is zero and the treadmill is at 10 mph. At that point (or well before that point by your consideration), the cart would have no air speed to work with and quickly run off the back of the treadmill if the various speeds were not adjusted to compensate.

So what is happening in the videos with the treadmill? The vehicle should not be able to advance or even maintain position on the treadmill according to your explanation, but it clearly does. Are you saying that the treadmill test is not valid now?

It appears to me that you are stating that the treadmill test is invalid solely on your belief that the cart is not doing what it looks like it is doing. I'm not trying to discuss DDWFTTW at this point but I find it hard to follow your logic unless I conclude that you are.

In your opinion, what is it about the treadmill test that needs to be changed to represent what would happen in an outdoor test? The results or the procedure?

 

[schroder]

I'm sorry, I misread your text and misunderstood your scenario. What I said was an answer to a different scenario and incorrect regarding yours.

If we continue your scenario, air speed drops until it cannot overcome the wheels' friction with the treadmill, the wheels stop rolling (with air speed > 0) and the treadmill starts to move the vehicle backwards (remember: the treadmill is going backwards to keep the air-pushed vehicle stationary!). Now then air flow relatively to vehicle is getting faster (and Treadmill changing to slower and slower), its force increases, and it starts pushing the vehicle again forwards. This repeats in a cycle, loop, (or steady state) and we never get to wind speed!

(This is equivalent to what happens outdoor without a treadmill.)

The only possibility to get to wind speed and pass it is with a side-wind component (meaning that wind direction is <180) that does not change during the experiment. The side component continues to generate forward force on the vehicle (sail, wing, wind-torbine, does not matter), overcome the friction, and accelerate until friction (drag in general) equals the forward force. Then the vehicle continues at constant speed, that can be larger than wind speed. If VMG is greater than wind speed, then the vehicle's velocity component in the wind direction at constant speed can also be larger than wind speed.

Yes. Finally another voice of reason! The cart will advance until it reaches treadmill velocity, or close to it, then it cannot advance any more. It will fall back, recover, and advance again but never achieve treadmill (wind) velocity. What is wrong with the videos is the treadmill is too short to allow the cart to get up to the speed of the tread before it runs to the end. Also, the operator may be speeding up the tread to achieve acceleration, but I am not sure about that. If the treadmill is running at 10 m/sec the Max possible velocity the cart can achieve is also 10 m/sec with reference to the floor, or 20 m/sec with reference to the moving tread. This is exactly equivalent to moving directly downwind in a 10 m/sec wind at 10 m/sec. That is something that has never been achieved, although the iceboats do come close to that. Incidentally, Thin Air Design, I notive this thread has so far been conducted very well but the only personal slights so far have all been coming from you.

 

[mender]

What is wrong with the videos is the treadmill is too short to allow the cart to get up to the speed of the tread before it runs to the end. Also, the operator may be speeding up the tread to achieve acceleration, but I am not sure about that.

shroder, that seems to be an odd thing to say, that the cart is not up to the speed of the tread before it runs off the front end of the treadmill. By running off the front, it exceeds the speed of the tread. It has to. To address your next statement about a possible way to fake the video, if the treadmill were sped up, the cart should run off the back of the treadmill. Obviously, it didn't. Just a question, do you want to discuss this part in more depth before continuing?

Max possible velocity the cart can achieve is also 10 m/sec with reference to the floor, or 20 m/sec with reference to the moving tread.

I'm not sure what you mean by this statement.

This is exactly equivalent to moving directly downwind in a 10 m/sec wind at 10 m/sec.

Again, the arguments about the treadmill test seem to be linked to a particular conclusion, not the test procedure itself. Please clarify what needs to be changed about the procedure, not the results, in order for the treadmill test to be valid.

At this point, from what I can see the treadmill tests have been conducted according to "requests" from various people in order to eliminate possible fakery. Maybe the next test should show the treadmill starting at zero mph with a smooth increase in speed to 10 mph. Or with fans blowing at the cart from either in front of or behind the treadmill. I don't know.

Perhaps one of you could propose a test that would be considered valid. I'm not sure what your objections are to the test procedure on the video. Just to be clear, the treadmill is running in the normal direction in all the videos.

 

[yoavraz]

So to extrapolate what you're saying, there would be a point where the air speed is zero and the treadmill is at 10 mph. At that point (or well before that point by your consideration), the cart would have no air speed to work with and quickly run off the back of the treadmill if the various speeds were not adjusted to compensate.

So what is happening in the videos with the treadmill? The vehicle should not be able to advance or even maintain position on the treadmill according to your explanation, but it clearly does. Are you saying that the treadmill test is not valid now?

It appears to me that you are stating that the treadmill test is invalid solely on your belief that the cart is not doing what it looks like it is doing. I'm not trying to discuss DDWFTTW at this point but I find it hard to follow your logic unless I conclude that you are.

In your opinion, what is it about the treadmill test that needs to be changed to represent what would happen in an outdoor test? The results or the procedure?

No, no, no. The test is fine, and the vehicles are great. I'm saying only one thing again and again: The wind is not exactly 180 in the tests! It is < 180 (maybe by little; or > 180 turning little to the other side) ! Then well engineered vehicles with good lift and low drag at < 180 can make it.

In other words, if you manage to control exactly 180, then the vehicle speed will steady state or oscillate below wind speed, and will never get there.

Time to put this to rest?

 

[ThinAirDesign]

Yes. Finally another voice of reason! The cart will advance until it reaches treadmill velocity, or close to it, then it cannot advance any more.

And yet, in spite of the arrival of this "voice of reason", it advances with boring regularity. One hundred out of one hundred times it can start at the back of the belt, advance *faster* than belt speed and run of the FRONT of the belt.

It will fall back, recover, and advance again but never achieve treadmill (wind) velocity

.

Then propose how we're cheating the video and we'll address it, because we are showing it in the video doing *precisely* what you are saying it can never do.

What is wrong with the videos is the treadmill is too short to allow the cart to get up to the speed of the tread before it runs to the end.

Clearly this is *not* true since you can watch the device go *faster* than the belt over and over.

Also, the operator may be speeding up the tread to achieve acceleration, but I am not sure about that.

There a really good way you can be sure we are not speeding/slowing the treadmill:

Time the seam in the belt as it passes by ... people accused of varying the speed so in video #7 we used a bar of soap to make marks that could be timed.

Also in video #7 we run the device at less than treadmill full speed and increase the slope of the treadmill until the device will essentially "hover" (the same speed of the wind). We do this so I don't have to be pushing it back down the treadmill and people can see is achieve DDWTSSATW (directly downwind the same speed as the wind) for some reasonable time.

If the treadmill is running at 10 m/sec the Max possible velocity the cart can achieve is also 10 m/sec with reference to the floor, or 20 m/sec with reference to the moving tread.

Your math is off by a large margin. It may be a misunderstanding on your part, it may be a simple error -- I have no way of knowing. Above, in your claim you state that the cart can move at 2x the belt speed -- something even I haven't claimed.

This is exactly equivalent to moving directly downwind in a 10 m/sec wind at 10 m/sec.

As an example of your bad math, the only "exact equivalent to moving directly downwind in a 10 m/sec wind at 10 m/sec." Would be the cart moving at 10 m/sec with reference to the belt and 0 m/sec with reference to the air in the room -- and that's not a option you list.

That is something that has never been achieved,

And yet I can demonstrate it at any time.

Incidentally, Thin Air Design, I notive this thread has so far been conducted very well but the only personal slights so far have all been coming from you.

To state that I have inflicted "personal slights", required one to consider statements of fact "personal slights". If that is your definition, I plead guilty.

If you say something that is demonstrably wrong, I will state so. If you claim to have a devised a test which will return differing results in two different IFORs, I will call you on it -- for it is still impossible.

mender has asked a totally reasonable question. "Someone tell me why a treadmill isn't a valid test of a DDWFTTW device". The only answers he has received have been nothing more than inconsistent private assertion rather than statements based in the laws of physics.

Just the facts Ma'am.

JB

 

[ThinAirDesign]

No, no, no. The test is fine, and the vehicles are great. I'm saying only one thing again and again: The wind is not exactly 180 in the tests! It is < 180 (maybe by little; or > 180 turning little to the other side) ! Then well engineered vehicles with good lift and low drag at < 180 can make it.

In other words, if you manage to control exactly 180, then the vehicle speed will steady state or oscillate below wind speed, and will never get there.

Time to put this to rest?

Even the detractors don't agree -- what's to rest?

Mender, I know this thread is about the treadmill only so I will respect that.

But as I'm sure you're bright enough to know this, if someone believes that the sole reason the cart advances with authority up an inclined surface faster than the wind is that we are a half a degree or so off of the wind:

A: with a guide, that can be fixed and the results will not change.

B: the only reason we don't use a guide is so the device is free floating and people won't accuse us of cheating by using the guide to push, pull or power the device.

JB

 

[ThinAirDesign]

mender:

Maybe the next test should show the treadmill starting at zero mph with a smooth increase in speed to 10 mph.

Any test you need mender, I'll happily contribute to as far as possible.

No test will convince some on this thread because they are attempting to make the facts fit their conclusions.

JB

 

[Phrak]

To discuss this in unambiguous manner, we need to define some terms. Running in circles, is not running directly downwind, but is in fact, a form of tacking.

You think?

The question here is about a wind powered vehicle proceeding directly on a downwind course, outrunning the wind on a steady basis.

Step a little way outside the box. It's obvious and feasable, but I won't be jumping hoops for dog treats any time soon.

 

[mender]

No, no, no. The test is fine, and the vehicles are great. I'm saying only one thing again and again: The wind is not exactly 180 in the tests! It is < 180 (maybe by little; or > 180 turning little to the other side) ! Then well engineered vehicles with good lift and low drag at < 180 can make it.

In other words, if you manage to control exactly 180, then the vehicle speed will steady state or oscillate below wind speed, and will never get there.

Time to put this to rest?

No, unfortunately not time. In order to remove your contention which has also been voiced a few times before, the treadmill test is conducted in a room where there is no air movement. There is no way that the vehicle can form an angle to the air in the room since the air is not moving. There is no angle to the wind. Zero. Nada.

Even if there was an angle to the wind, there is a minimum angle that will be needed to provide enough side-wind component to be considered tacking. Any scenario that is in a room with still air and has a vehicle on a treadmill that is running continuously in a straight line will be well below that threshold.

Your assertion that the vehicle running steady state on the treadmill can only be tacking is just that - an assertion based on your belief that it is only possible that way. Despite not being able to describe a way to isolate the device from any wind angle, you claim that it is being powered exactly that way.

It is my intention to build a non-tacking cart that goes DDWFTTW using the treadmill test as my method of developing such. If you have an observation of how to control the test better than it presently is, please let me know. I fully intend on proving that my device will go DDWFTTW once it is done but I need to have a way of proving without a doubt under controlled conditions that it is doing exactly that.

 

[yoavraz]

Even the detractors don't agree -- what's to rest?

Mender, I know this thread is about the treadmill only so I will respect that.

But as I'm sure you're bright enough to know this, but if someone believes that the sole reason the cart advances with authority up an inclined surface faster than the wind is that we are a half a degree or so off of the wind:

A: with a guide, that can be fixed and the results will not change.

B: the only reason we don't use a guide is so the device is free floating and people won't accuse us of cheating by using the guide to push, pull or power the device.

JB

Putting guides, rails is an excellent idea, but even then I do not believe you can well control wind direction in an open treadmill at +-10 degrees (not even thinking of +-1/2)... I challenge you to put your good engineering to work, measure direction accurately (close-around and downwind the vehicle, not the far away fan orientation), and see what happens.

You may consider smoke stripes generated upwind by a net, and videoing from above to later measure angle of stripes relative to guides, rails (stripes should be completely parallel to rails in exact 180, with some bending around vehicle).

 

[schroder]

shroder, that seems to be an odd thing to say, that the cart is not up to the speed of the tread before it runs off the front end of the treadmill. By running off the front, it exceeds the speed of the tread. It has to. To address your next statement about a possible way to fake the video, if the treadmill were sped up, the cart should run off the back of the treadmill. Obviously, it didn't. Just a question, do you want to discuss this part in more depth before continuing?

Yes, let’s discuss this part in more detail, as this is the conceptual problem which is leading you astray in your thinking. The fact that the tread is coming from the front, makes you think that by simply advancing, the cart is outrunning the tread. That is simply not true. In order to outrun the tread, it must advance relative to the floor, at the tread velocity. Just think about the wind coming from the rear, no treadmill. The wind velocity pushes the cart with reference to the floor. The cart, in order to outrun the wind, must advance on the floor at greater than wind velocity, which it cannot. The fact that the tread is coming from the front is confusing many people, but it is essentially the same as a wind coming from the rear, and the cart must still advance against the floor at tread (wind) velocity! The cart has never done that! That is the entire point I have been trying to make. Do not be fooled by advancing on the tread. It is the floor which is the reference frame always.

 

[mender]

Putting guides, rails is an excellent idea, but even then I do not believe you can well control wind direction in an open treadmill at +-10 degrees (not even thinking of +-1/2)... I challenge you to put your good engineering to work, measure direction accurately (close-around and downwind the vehicle, not the far away fan orientation), and see what happens.

You may consider smoke stripes generated upwind by a net, and videoing from above to later measure angle of stripes relative to guides, rails (stripes should be completely parallel to rails in exact 180, with some bending around vehicle).

I guess I'm not understanding this. Is this in reference to the treadmill test? If so what wind are you trying to measure? The test is conducted in a room with still air.

 

[ThinAirDesign]

Putting guides, rails is an excellent idea, but even then I do not believe you can well control wind direction in an open treadmill at +-10 degrees (not even thinking of +-1/2)... I challenge you to put your good engineering to work, measure direction accurately (close-around and downwind the vehicle, not the far away fan orientation), and see what happens.

You may consider smoke stripes generated upwind by a net, and videoing from above to later measure angle of stripes relative to guides, rails (stripes should be completely parallel to rails in exact 180, with some bending around vehicle).

I'm confused by your statement yoavraz.

You do realize that the air in that room is still don't you. The doors are closed. The windows are closed. Even the curtains are closed and hanging still. The only moving air is being moved by the device itself.

There will be no "smoke stripes generated upwind" Any smoke released up ahead of the treadmill. will just form a billowing cloud.

JB

 

[ThinAirDesign]

Schroder:

It is the floor which is the reference frame always.

Sigh. Well at least we know where our problem lies.

Once the cart is moved onto the treadmill, the ground is *no longer* relevent.

With this device, it has an "air interface" (prop) and a "rolling surface interface"(wheels) It interfaces with no other mediums. Once on the treadmill, the ground is no longer involved.

On the treadmill, using the floor as a frame of reference makes as much sense as using a car passing by on a distant freeway.

JB

 

[schroder]

Schroder:


Sigh. Well at least we know where our problem lies.

Once the cart is moved onto the treadmill, the ground is *no longer* relevent.

With this device, it has an "air interface" (prop) and a "rolling surface interface"(wheels) It interfaces with no other mediums. Once on the treadmill, the ground is no longer involved.

On the treadmil, using the floor as a frame of reference makes as much sense as using a car passing by on a distant freeway.

JB

No. You are wrong. You are fond of quoting reference frames. When the cart is on the floor, with the wind at its back, it advances against the floor. When it is on the tread, the movement of the tread simulates a wind at its back, and it must still advance relative to the floor. That is a fact. Think about it and you must agree.

 

[mender]

No, shroder. The tread simulates the ground. The floor underneath the treadmill keeps the treadmill from falling into the basement.

If you are standing in the room before the test starts, do you feel a wind? When the treadmill is running, do you feel a wind? If you are right about the floor being the proper reference, you should feel a 10 mph wind.

The correct answer is of course that you can't feel the wind unless you are on the treadmill and moving backwards at 10 mph. If you are on the treadmill, do you have to run at 20 mph to keep up with the treadmill?

 

[ThinAirDesign]

No. You are wrong.

'Fraid not.

When the cart is on the floor, with the wind at its back, it advances against the floor.

That is correct.

When it is on the tread, the movement of the tread simulates a wind at its back, and it must still advance relative to the floor.

Nope, what's sitting under the treadmill might well be a carpeted belt moving at 100mph in the opposite direction as the treadmill and belt -- it is and will remain irrelevant.

The cart must only advance against the still air in the room to meet DDWFTTW.

That is a fact. Think about it and you must agree.

It's not a fact, and will never be a fact that the cart must advance against the floor in the room. The floor of the room is out of the picture once the cart no longer rolls on the floor.

Remember, in any scenario involving this device, there are only *two* mediums that matter -- the air and the surface upon which it rolls.

JB

 

[russ_watters]

schroder, ThinAirDesign is right. This is a problem of reference frames and due to the well established principle of relativity, the treadmill scenario is exactly identical to running downwind in a steady wind.

The first time I saw the video of the device on the treadmill, I got the scenario twisted around in my head. Then I drew myself a diagram and figured it out. That's what I suggest you do. Draw a diagram for each scenario and see if you can detect any difference in the motion vectors for wind and surface relative to the craft. The only surface that matters is the surface that is spinning the wheels and the only wind that matters is the one that is spinning the propeller or being pushed by the propeller.

In the previous thread, we used the following values:
Cart on treadmill:
-5mph, tredmill relative to ground
-1mph, cart relative to treadmill
-6mph, wheels spinning
-1mph, wind being generated by the propeller

Cart on ground:
-6mph, cart relative to ground
-6mph, wheels spinning
-5mph, wind speed
-1mph, wind being generated by the propeller

If the vectors all add up and you find no math errors, then the scenarios must be identical.

 

[russ_watters]

All experiments I have seen on videos do not prove the 180 possibility, to my opinion.
It is impossible in such conditions to keep the wind at 180 all the time. Even small fluctuations generate deviation from 180, which makes it equivalent to tacking at <180.

Certainly, the wind could not be constant, however a propeller isn't a sail and the cart has no control system to adjust for wind speed variation. So the only thing that variations in the wind speed or direction can do is make the cart not work. They can't make it work better.

 

[ThinAirDesign]

You understand it perfectly Russ.

Thanks

JB

 

[yoavraz]

OOOPS. I knew there was a disconnect, but until now I have not realized we are talking about different experiments. I understand you have previous context. I apparently was thinking about a different forum from the one that was mentioned (I see this subject is popular...). At least we agree about the outdoor experiment description, I hope. Though I can guess now the scenario, I do not want to guess anymore.

Mender, Could you kindly refer me to your treadmill experiment description, or describe it here, or send me a private message. Also pls point me to the related video. Thnx

 

[russ_watters]

Here's the other video: http://www.youtube.com/watch?v=aJpdWHFqHm0&feature=related

 

[mender]

yoavraz, it sounds like something went awry. The Jack Goodman video which was taken under actual wind conditions indeed will be subject to the vagaries of the shifting wind; however as Russ stated, for this device anything off directly downwind will serve to decrease the cart's performance, not enhance it. That's the one that Russ linked.

The treadmill test is as shown by TAD, and consists of a treadmill in an enclosed room. The speed of the treadmill is set at 10 mph, which simulates an outdoor test by providing a speed difference between the air in the room (still ) and the ground (10 mph).

I'm going to continue this tomorrow, I'm not quite up to snuff tonight. Please feel free to pose any questions or clarify what you think is relevant regarding a treadmill test regime. My intent is to build a non-prop cart device and I would like to make sure I'm not leaving anything out of my test program.

 

[schroder]

schroder, ThinAirDesign is right. This is a problem of reference frames and due to the well established principle of relativity, the treadmill scenario is exactly identical to running downwind in a steady wind.


If the vectors all add up and you find no math errors, then the scenarios must be identical.

Sorry to tell you this Russ, but it is you who are wrong about this. Just calm down and think about it rationally. Imagine two frames. We will keep the treadmill in both, but in the first one the tread is stationary with respect to the floor, and the air is moving from the back at 10 mph. You have to consider the floor and the tread together as the stationary reference which remain stationary throughout. All that is moving initially is the wind from the back. The wind now blows the cart forward, and we will assume 100% efficiency, so the cart now moves forward at 10 mph with respect to both the tread and the floor. I think we can all agree with what is happening here.
Scenario number two: The cart is sitting on the tread, the air and the floor are both stationary and will remain stationary throughout. The only thing that is moving initially is the tread trying to drag it backwards against the stationary air and the equally stationary floor. There is no difference between this scenario and scenario one. The cart must advance against the stationary reference of air and floor and it must do so at the tread velocity. We have all seen the cart advance against the tread and against the floor, but it has never reached the velocity of the tread (wind).I know that this is difficult at first to see, but once you do see it, it explains everything! Until or unless the cart achieves tread velocity, it is not going DDWFTTW or DDTFTTT where T is for Tread. You Must tie the floor to the air because it is otherwise impossible to gauge the velocity against the still air! They are both stationary so it is the most convenient reference. Once you break through and clarify this in your own mind, all will be clear and the mystery of these carts is solved. NOTHING goes DDWFTTW there are NO exceptions!

 

[swerdna]

Why I think traveling directly downwind faster than the wind is impossible . . .

The only energy sources involved are the wind and the kinetic energy of the vehicle.

The kinetic energy of a vehicle alone can’t be geared up to make the vehicle travel faster (that would be perpetual motion/free energy).

Whatever help the vehicle gets from a tailwind is completely lost when the vehicle reaches the speed of the wind.

At the speed of the wind the vehicle only has it’s own kinetic energy to accelerate faster than the wind (impossible)

When the vehicle travels faster than the speed of the wind it loses the advantage of a tailwind and gains the disadvantage of a headwind.

 

[Hurkyl]

Sorry to tell you this Russ, but it is you who are wrong about this. Just calm down and think about it rationally. Imagine two frames. We will keep the treadmill in both, but in the first one the tread is stationary with respect to the floor, and the air is moving from the back at 10 mph. You have to consider the floor and the tread together as the stationary reference which remain stationary throughout. All that is moving initially is the wind from the back. The wind now blows the cart forward, and we will assume 100% efficiency, so the cart now moves forward at 10 mph with respect to both the tread and the floor. I think we can all agree with what is happening here.

The equivalent scenario to this is the following:
The treadmill is running at 10mph, and the cart is moving backwards at 10 mph along with it, and there is no wind. Air resistance slows the cart, and because of our 100% efficiency assumption, the cart slows to a stop, and thus moving 10 mph forwards relative to the treadmill.

 

[schroder]

The equivalent scenario to this is the following:
The treadmill is running at 10mph, and the cart is moving backwards at 10 mph along with it, and there is no wind. Air resistance slows the cart, and because of our 100% efficiency assumption, the cart slows to a stop, and thus moving 10 mph forwards relative to the treadmill.

Relative to the treadmill yes, but Not relative to the stationary air and the floor. You are using the wrong reference for your measurement.