Wind Power Vehicle Traveling Down Wind Faster Than The Wind

[spork]

Can you answer my question?

I'll certainly try.

Where can I find Nikos's comments on the experiment?

Do you mean professor Nikos Mourtos?

 

[OmCheeto]

I'll certainly try.



Do you mean professor Nikos Mourtos?

Is there more than one Nikos that worked on the project?

 

[spork]

Is there more than one Nikos that worked on the project?

You're talking about someone that worked on the project?
(let's see how long we can answer a question with another question)


I don't know that you can read his opinions or comments about the project anywhere. I certainly don't recall seeing them. I'm quite certain he doesn't know how it works, and I guess it'd be a coin toss as to whether he believes it works.

Why do you ask about his comments in particular?

 

[OmCheeto]

You're talking about someone that worked on the project?
(let's see how long we can answer a question with another question)


I don't know that you can read his opinions or comments about the project anywhere. I certainly don't recall seeing them. I'm quite certain he doesn't know how it works, and I guess it'd be a coin toss as to whether he believes it works.

Why do you ask about his comments in particular?

To quote Leonard; "Fascinating"

10-26-2009, 04:43 PM
ThinAirDesigns ThinAirDesigns is offline
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Join Date: Dec 2008
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San Jose State University Aero Professor and Stanford Phd Dr. Nikos Mourtos along with a team of students and advisors have taken on a project to construct and document DDWFTTW in a more thorough way than ever before. Their goal is to achieve a documented 2x windspeed DDW.

Follow their blog at www.fasterthanthewind.org

http://www.boatdesign.net/forums/pr...ctly-downwind-faster-than-wind-25527-10.html"

and now, fasterthanthewind.org, has no mention of his existence?

sorry. You can PM the mentors/admin, and have them delete this post, as I have, on many occasions stated, that (I am of course paraphrasing, as I'm old, and have not a clue what I originally stated); "This is the most incredibly mindboggling physics homework problem in the world"

 

[spork]

and now, fasterthanthewind.org, has no mention of his existence?

We set out to take this project on with a group of SJSU aero students. Dr. Mourtos was their professor - and agreed to the project. And that was about the end of it. One of the students took it upon himself to start working with us relatively late in the project. The others were effectively no-shows. Neither they nor the professor contributed in any way to the project.

Because we had accepted money from sponsors to accomplish a stated goal, JB and I saw to it that the project was completed.

But if it eases your concerns in any way - I strongly suspect he still exists.

 

[OmCheeto]

We set out to take this project on with a group of SJSU aero students. Dr. Mourtos was their professor - and agreed to the project. And that was about the end of it. One of the students took it upon himself to start working with us relatively late in the project. The others were effectively no-shows. Neither they nor the professor contributed in any way to the project.

Because we had accepted money from sponsors to accomplish a stated goal, JB and I saw to it that the project was completed.

But if it eases your concerns in any way - I strongly suspect he still exists.

Why of course he does. I checked him out on facebook. I didn't try an befriend him. I figured he had way too many dd'er fans chasing him.

But I did just find a paper written by one of http://www.engr.sjsu.edu/nikos/projectsMSAE.htm" students:

2010 http://www.engr.sjsu.edu/nikos/MSAE/pdf/Shethal.F10.pdf"
Shethal Thomas Kodiyattu

Unfortunately, it's a 125 page pdf, and I was supposed to be in bed about 3 hours ago, so I don't have time at the moment to read much.

zzzzzzz:zzz:zzz:zzz:

 

[spork]

But I did just find a paper written by one of http://www.engr.sjsu.edu/nikos/projectsMSAE.htm" students:

Unfortunately, it's a 125 page pdf

That paper is incidental to the project. We take no credit or blame for it.

Incidentally, I think you'll find there are fewer than 30 pages to read. But I couldn't force myself to read much more than the abstract. The 17.5 foot propeller seems to have become a 16 inch propeller according to this paper.

 

[Fun Value]

Will you build a cart and demonstrate it in class?

Depends on the time and their motivation. If they are highly motivated I might be able to demonstrate it and expect them to build their own on their own time while we move on. I like to cover the entire text in order and usually the all important high tech stuff needed now days comes near the end. I do plan to have them watch AT's U.tube presentations of ATPFTTP and UTRFTTR and at least build those things.

 

[Subductionzon]

I don't know if spork mentioned it, but he does have a series of videos that tell how to build the small cart they used on the treadmill. Look under the name of spork33 on YouTube. I believe the parts should run under $40.00.

 

[A.T.]

Depends on the time and their motivation. If they are highly motivated I might be able to demonstrate it and expect them to build their own on their own time while we move on. I like to cover the entire text in order and usually the all important high tech stuff needed now days comes near the end. I do plan to have them watch AT's U.tube presentations of ATPFTTP and UTRFTTR and at least build those things.

The paper and ruler videos are not mine, but from someone who posts as "Michael C". My youtube channel contains some schematic animations:
http://www.youtube.com/user/eyytee#g/u

Personally I think it would be nice if you would do Michael's experiments live with the students, so they can see that it is not a trick. But you should show the nicely made ruler video first, and stop it after the puppets present their answers. Then let the students make their predictions.

Building your own propeller model and testing it on a treadmill is of course a more advance project. But useful to teach Galilean Relativity. Michael has a nice video on that too:

https://www.youtube.com/watch?v=9Yt4zxYuPzI

 

[chingel]

But then what is the correct way of thinking about the cart moving faster than the wind, and also the Yo-Yo and tumbleweed? What is the most intuitive and logical explanation?

 

[spork]

But then what is the correct way of thinking about the cart moving faster than the wind, and also the Yo-Yo and tumbleweed? What is the most intuitive and logical explanation?

We have never found that any given explanation is better than another overall. Each person that gets it seems to find one more intuitive while others claim that one makes no sense, and think another is more intuitive. Some are certain that none are intuitive, and that DDWFTTW is impossible.

 

[Fun Value]

Thank you Subductionzon and A.T. I am taking your advice. I am going to try to build my own this summer and demonstrate it to my students next fall. I have yet not looked at Spork's videos, but plan to asap. Right now it seems to me that maybe the most difficult part is the gear-box - one that changes direction from the plane of the wheels to a perpendicular plane so as to run the propeller. I thought I might find one at the local hobby shop, but they didn't have one. I guess I might have to make my own. Wonder if I can make it highly efficient.

 

[spork]

Right now it seems to me that maybe the most difficult part is the gear-box - one that changes direction from the plane of the wheels to a perpendicular plane so as to run the propeller. I thought I might find one at the local hobby shop, but they didn't have one. I guess I might have to make my own. Wonder if I can make it highly efficient.

For my first one, I purchased two bevel gears and made the gear box on a mill from Delrin. In the videos though, I use a tail-rotor gearbox from an R/C heli. I give the part numbers and source in the video as well.

 

[chingel]

We have never found that any given explanation is better than another overall. Each person that gets it seems to find one more intuitive while others claim that one makes no sense, and think another is more intuitive. Some are certain that none are intuitive, and that DDWFTTW is impossible.

But what are the explanations? One is that you can view the propeller as two sails moving at an angle and they also rotate. What are the others?

 

[rcgldr]

But what are the explanations?

One is that when you have two media moving at different speeds, and a device that extracts energy from the relative movement of those two media (by interacting with the media), then it is possible to construct such a device so it can continue to extract energy from the relative movment of the two media even when the devices speed is greater than the relative speed of the two media. The under the ruler cart, sailcraft like ice boats, and DDWFTTW carts, are examples of such devices. The limit of speed is related to how much the device slows the relative speed of the two media (if it slows the relative speed to zero, it can't extract energy), and when energy ouput to overcome the factors related to speed equals energy input.

 

[mender]

Just a nitpick; this:
if it slows the relative speed to zero, it can't extract energy

should read "once it slows the relative speed to zero, it can't more extract energy."

 

[chingel]

The extracting energy from two different moving media doesn't really mean anything. It isn't really an explantion. It is like asking how a car works and then you get an answer that it extracts energy from a chemical reaction. Ok, but how?

 

[spork]

The extracting energy from two different moving media doesn't really mean anything. It isn't really an explantion. It is like asking how a car works and then you get an answer that it extracts energy from a chemical reaction. Ok, but how?

You're kind of right. It doesn't give you the "how", but it does show you that it's not energy for free. You have wind that had energy, and you slow that wind down. That energy has to go somewhere. It goes into moving/accelerating the cart, and is ultimately lost to heat.

So again, that by itself doesn't give you the how, but it shows that we're not talking about a perpetual motion machine. That means it's just an engineering problem at that point - not a matter of re-writing physics books.

 

[A.T.]

The extracting energy from two different moving media doesn't really mean anything. It isn't really an explanation.

It's all there is to say about where the energy is coming from.

It is like asking how a car works and then you get an answer that it extracts energy from a chemical reaction. Ok, but how?

If you are not happy with such answers then you should not insist on explanations in terms of energy.

 

[rcgldr]

The extracting energy from two different moving media doesn't really mean anything.

Two points being covered in my previous post:

1 - Say there's only one moving media the cart can interact with, for example wind but frictionless ground. The cart will eventually move at wind speed but no faster.

2 - The two media's relative speed with respect to each other provides an energy source that isn't directly related to the speed of the DDWFTTW cart. The cart can be designed to extract energy from the relative movement of the two media, even when the cart's speed is faster than the relative speed between the two media.

The BB cart achieved 2.8x wind velocity. If you switched the frame of reference to be the air, then the BB cart acheived -1.8x relative ground velocity, so from either frame of reference, the cart speed is faster than the relative speed between the two media.

Ok, but how?

I think that the yo-yo case is somewhat intuitive since the outcome is what most people would expect. If you pull a yo-yo on the ground by pulling it's string horizontally, the yo-yo moves faster than the string. Then you could point out that if the ground didn't have any friction, then the yo-yo could would only move as fast as the string, so it's clear that the the yo-yo needs to interact with both the ground and string, which are the two media moving at different speeds in this example, in order for the yo-yo to move faster than the string. The energy input is the force times distance the string moves. The energy output is the gain in linear and angular kinetic energy of the yo-yo as it accelerates (plus losses due to rolling resistance, and friction, which end up as heat).

 

[chingel]

It's all there is to say about where the energy is coming from.


If you are not happy with such answers then you should not insist on explanations in terms of energy.

Yes that's all nice and the only place it could possibly extract energy, but it doesn't explain anything, what, how? It is very unsatisfying. I cannot just go and start extracting energy from two different medias, I have to have some sort of a way or a mechanism of doing it.

I also have the same problem with the Yo-Yo. If the middle part is freewheeling, the Yo-Yo goes at the speed of the string, but when you connect it rigidly to the outer parts, it uses their energy to spin against the string and work itself up. Something must be wrong with my idea and it is bothering me.

Maybe a better way would be thinking about it like some sort of a fulcrum? If the Yo-Yo is hexagonal for example, the ground is the fixed point of the fulcrum, the force is applied somewhere in between the ground and center of the Yo-Yo and since the center of the Yo-Yo is further along the lever it would move faster. But this would mean that the closer the point where the string applies it's force is to the ground the faster it would move, but that is not the case with the Yo-Yo. Or maybe I'm thinking about it wrong?

 

[spork]

Yes that's all nice and the only place it could possibly extract energy, but it doesn't explain anything, what, how? It is very unsatisfying. I cannot just go and start extracting energy from two different medias, I have to have some sort of a way or a mechanism of doing it.

This is exactly what we've been telling you. If you insist on asking where does the energy come from, we'll tell you where the energy comes from (and you'll tell us that's not satisfying). If you want to know how it works - ask how it works - and then think about the answers.

I also have the same problem with the Yo-Yo. If the middle part is freewheeling, the Yo-Yo goes at the speed of the string, but when you connect it rigidly to the outer parts, it uses their energy to spin against the string and work itself up.

This is strictly a kinematics problem. It does not rely on the yo-yo's mass, momentum, or energy. If you pull the yo-yo by it's spindle it will roll itself up the string. The problem is that our intuition tells us that wheels rotate about their axles. In the ground frame, the wheel is rotating about the point of contact with the ground. So you're pulling on the yo-yo well above it's pivot point (the ground) and it's doing the natural thing - rotating about that pivot point toward the thing that's pulling it.

Maybe a better way would be thinking about it like some sort of a fulcrum? If the Yo-Yo is hexagonal for example, the ground is the fixed point of the fulcrum, the force is applied somewhere in between the ground and center of the Yo-Yo and since the center of the Yo-Yo is further along the lever it would move faster. But this would mean that the closer the point where the string applies it's force is to the ground the faster it would move

I should have read ahead. That is 100% correct.

...but that is not the case with the Yo-Yo.

It is the case with a yo-yo. Try it.

 

[chingel]

Yes in the end the energy can only come from the wind, and since there are videos of it going faster than the wind, it does extract it somehow. What I would like to know is how does it do it, where does the energy come from when the cart is up to wind speed?

Isn't it that the smaller the diameter of the center part, the faster the yo-yo goes up the string? But in the case of a fulcrum, the closer you are to the pivot point, assuming constant speed of the string, the faster the top of the lever moves, no?

 

[cjl]

Yes in the end the energy can only come from the wind, and since there are videos of it going faster than the wind, it does extract it somehow. What I would like to know is how does it do it, where does the energy come from when the cart is up to wind speed?

That depends on which frame you are looking at the problem from. From the ground frame, the energy comes from the fact that as the cart goes by, the air is slowed down. From the cart frame, the energy comes from the fact that the ground is applying a force to the wheels at a nonzero velocity.

Isn't it that the smaller the diameter of the center part, the faster the yo-yo goes up the string? But in the case of a fulcrum, the closer you are to the pivot point, assuming constant speed of the string, the faster the top of the lever moves, no?

For a yo-yo, the larger the diameter of the center part, the faster it will move (seriously). Look at this in the limiting case - if the center part had zero diameter, such that no string wound up, the yo-yo would simply roll at the same speed the string was pulled. That is clearly the slowest it will go. Now, if the center part has half the diameter of the outer part of the yo-yo, the yo-yo has to move faster, since as the string is pulled and the yo-yo moves, the string is being shortened as it is wound up. In the case of the center having half of the diameter of the outside, the yo-yo will travel at twice the speed at which the string is pulled. If the center part is 90% of the diameter of the outside, the yo-yo will travel ten times faster than the string is pulled (assuming no slipping).

 

[rcgldr]

Yes in the end the energy can only come from the wind, and since there are videos of it going faster than the wind, it does extract it somehow. What I would like to know is how does it do it, where does the energy come from when the cart is up to wind speed?

Although the cart is moving at or faster than wind speed, the air accelerated by the propeller is moving slower than wind speed, and that air is slowing down part of the wind, and that is the source of the energy. All of this is from a ground based frame of reference.

From the carts frame of reference, the forward force (from the wheels) on the surface of the Earth is slowing down the Earth (relative to the cart) a very tiny bit, and that is the source of energy that allows it to accelerate the relative headwind.

From the air's frame of reference, again the forward force (from the wheels) on the surface of the Earth is slowing down the Earth (relative to the air) a very tiny bit, and that is the source of energy that allows the propeller to accelerate the air.

The DDWFTTW cart is like a hybrid car, except that the braking energy is being used to drive the propeller as opposed to charging a battery. This only works if there's a tailwind to supply more energy than is consumed by the braking energy.

Isn't it that the smaller the diameter of the center part, the faster the yo-yo goes up the string?

If the center part (and string) had zero diameter there would be no winding of the string, and the yo-yo would move at the same speed as the string.

To cover all the cases from a yo-yo like device, imaging that there are two hubs on the outside of the yo-yo and that the hubs rest on a pair of rails, with a rail on each side of the yo-yo, with the yo-yo otherwise suspended in air, only resting on the hub rails. If the string is wound above the center axle and the string is pulled forwards, the yo-yo moves at less than string speed, and this is the equivalent of a negative advance ratio. If the string slides on the center axle, then it's the equivalent of a zero diameter axle, and a zero advance ratio, and the yo-yo moves at the same speed as the string. If the axle diameter is smaller than the hub diameter, the yo-yo moves faster than the string, the equivalent of advance ratio greater than zero and less than one. If the axle and hub diameters are the same, the yo-yo doesn't move unless it slides, this is an advance ratio of 1. If the axle diameter is greater than the hub diameter, but less than double the hub diameter, the yo-yo moves away as the string is pulled, but faster than the string, an advance ratio greater than 1 but less than 2. If the axle diameter is double the hub diameter, the yo-yo moves away at the same speed as the string, advance ratio of 2. If the axle diameter is greater than double the hub diameter, the yo-you moves away, but at less than string speed, an advance ratio greater than 2.

The closer ratio between the axle and hub diameters is to 1, the faster the yo-yo will move relative to the string, until the ratio gets too close to 1 and then the yo-yo "stalls" and won't move without sliding.

 

[chingel]

I would tend to think the smaller the diameter of the center part, the faster the yo-yo moves, because when the surface of the center part moves at 1 cm/s, if the outer diameter is twice of it it moves at 2 cm/s, 3 cm/s if triple etc. Since they are connected they have to make the same amount of turns and a bigger wheel means the surface has to move faster to make the turns at the same rate. But in this case the fulcrum analogy doesn't apply, because it produces different results, it goes faster the further away the point where you apply the force is from the pivot point, or not? And why doesn't it apply, because if the yo-yo is hexagonal, why can't I view it as a lever?

The places where it ultimately must get the energy are undeniably correct since the cart is actually moving faster than the wind and has to get it from somewhere. But what happens at windspeed, how does it do it?

 

[spork]

The places where it ultimately must get the energy are undeniably correct since the cart is actually moving faster than the wind and has to get it from somewhere. But what happens at windspeed, how does it do it?

You must stop asking "but where does the energy come from when at wind speed" if you want to know how it happens, and start asking "HOW?". I see you have asked "how?" now a couple of times. I have to run, but I will gladly answer that if someone hasn't beaten me to it by the time I return.

 

[rcgldr]

I would tend to think the smaller the diameter of the center part, the faster the yo-yo moves, because when the surface of the center part moves at 1 cm/s, if the outer diameter is twice of it it moves at 2 cm/s, 3 cm/s if triple etc.

Take the triple case. Assume the string moves at +1 cm/s, then the yo-yo moves in the same direction as the string at +1.5 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at -1.5 cm/s, the inner diameter moves at -.5 cm/s second (1/3rd the outer diameter surface speed), and string moves at -1 cm/s.

In the double case, the yo-yo moves at twice the string speed (relative to the ground) at +2 cm/s (relative to the ground). From the yo-yo's perpective, the ground moves at -2 cm/s, the inner diameter moves at -1 cm/s second (1/2 the outer diameter surface speed), and string moves at -1 cm/s.

The places where it ultimately must get the energy are undeniably correct since the cart is actually moving faster than the wind and has to get it from somewhere. But what happens at windspeed, how does it do it?

As mentioned above, from a ground based frame of reference, the propeller's thrust slows down the wind, even when the cart is moving at faster than wind speed. Using the BB's stated advance ratio of .8, and max speed of 2.8x wind speed, then if wind speed is 10 mph, cart speed is 28 mph. From the cart frame of reference, ground speed is -28 mph, prop thrust speed is (-28 x .8 =) -22.4 mph, and wind speed is (-28 + 10 = ) -18 mph, and the air affected by the propeller is accelerated (-22.4 - -18 =) -4.4 mph. From a ground frame of reference, the wind speed is +10 mph, the cart speed is +28 mph, the propeller thrust speed is (28 - 22.4 = ) +5.6 mph, slower than the +10 mph wind speed. So the air affected by the propeller is slowed down by 4.4 mph.

 

[spork]

...how does it do it?

There are a bunch of ways to describe this, but let's take a simple kinematic approach first...

First let's imagine our cart was submersed in Jello and the prop has a very fine pitch (in other words it only tries to screw it's way through the jello by about 1 foot per rotation). So now we move this entire block of jello that the cart is in - but it's wheels are still rolling on the stationary ground. By pushing the jello forward, we end up pushing the cart forward too - and that makes the propeller rotate. So let's imagine that we gear the wheels pretty low so we have to push the cart forward 100' to make the prop do one full turn.

See what happens there? We push the jello forward 100', the prop makes one full turn, so the cart moves forward 100' with the jello *plus* the one foot its prop pulled it through the jello.

So if we do this over a period of one minute, the jello goes 100 feet per minute. But the cart goes 101 feet per minute (because it pulled itself forward in the jello). We're now going down jello faster than the jello - right?

 

[cjl]

There are a bunch of ways to describe this, but let's take a simple kinematic approach first...

First let's imagine our cart was submersed in Jello and the prop has a very fine pitch (in other words it only tries to screw it's way through the jello by about 1 foot per rotation). So now we move this entire block of jello that the cart is in - but it's wheels are still rolling on the stationary ground. By pushing the jello forward, we end up pushing the cart forward too - and that makes the propeller rotate. So let's imagine that we gear the wheels pretty low so we have to push the cart forward 100' to make the prop do one full turn.

See what happens there? We push the jello forward 100', the prop makes one full turn, so the cart moves forward 100' with the jello *plus* the one foot its prop pulled it through the jello.

So if we do this over a period of one minute, the jello goes 100 feet per minute. But the cart goes 101 feet per minute (because it pulled itself forward in the jello). We're now going down jello faster than the jello - right?

That's an awesome way to visualize it

I think that's my favorite explanation yet. It makes it really obvious how it works.

 

[chingel]

Take the triple case. Assume the string moves at +1 cm/s, then the yo-yo moves in the same direction as the string at +1.5 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at -1.5 cm/s, the inner diameter moves at -.5 cm/s second (1/3rd the outer diameter surface speed), and string moves at -1 cm/s.

In the double case, the yo-yo moves at twice the string speed (relative to the ground) at +2 cm/s (relative to the ground). From the yo-yo's perpective, the ground moves at -2 cm/s, the inner diameter moves at -1 cm/s second (1/2 the outer diameter surface speed), and string moves at -1 cm/s.

I don't understand. Are you saying that the smaller the diameter of the middle part the faster it moves or the slower it moves? If the yo-yo moves 1,5 cm/s, and the middle part moves -0,5 cm, then it isn't moving up the string, it should be slipping on it? The middle part has to move faster than the string to climb it up. In the under the ruler faster than the ruler example, the smaller the diameter of the middle parts of the wheels where the big wheel applies the force, the faster it moves, or wasn't that the case?

If it moves faster when the middle part is smaller, then it would mean that the effect is different from the lever example, but why can't I view it as a lever, especially when making the yo-yo hexagonal?

The cart would indeed move along the jelly faster than the jelly. I am having difficulty imagining that when you get the cart up to jelly speed and then engage the propeller, how does the force apply, how does it not use it's own energy to propel itself? I guess it's the same principle as the yo-yo, where the wheels are pivoting around the contact point with the ground, which I am having trouble grasping.

 

[cjl]

It moves faster when the middle part is larger (for the yo-yo example).

 

[rcgldr]

Yo-yo ... take the triple case. ... (wrong string speed numbers from yo-yo frame of reference)

I don't understand.

My fault, the inner diameter speed and the string speed are the same, regardless of the frame of reference. The rest of the numbers were OK.

Are you saying that the smaller the diameter of the middle part the faster it moves or the slower it moves?

The slower it moves. If the diameter is zero, the yo-yo moves at string speed. The larger the diameter, the faster the yo-yo moves as long as it doesn't slide. If the axle diameter is the same as the yo-yo diameter, then it can only move by sliding. If the axle diameter was greater than the yo-yo diameter (assume the yo-yo is rolling on rails to allow this), the yo-yo will roll to the left when the string is pulled to the right. In both cases, the closer the ratio of inner_diameter to outer_diameter is to 1, the faster the yo-yo moves (except if they're equal to 1, the yo-yo can only move by sliding).

The formula with respect to the ground:

yo_yo_speed - string_speed = (inner_diameter / outer_diameter) x yo_yo_speed
yo_yo_speed = string_speed / (1 - (inner_diameter / outer_diameter))

Listing the case by inner_diameter to outer_diameter ratio:

1 to 3 case: yo-yo moves in the same direction as the string at +1.5 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at -1.5 cm/s, the inner diameter and the string move at -.5 cm/s second (1/3 the ground speed).

1 to 2 case: yo-yo moves in the same direction as the string at +2.0 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at -2.0 cm/s, the inner diameter and the string move at -1.0 cm/s second (1/2 the ground speed).

2 to 3 case: yo-yo moves in the same direction as the string at +3.0 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at -3.0 cm/s, the inner diameter and the string move at -2.0 cm/s second (2/3 the ground speed).

3 to 4 case: yo-yo moves in the same direction as the string at +4.0 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at -4.0 cm/s, the inner diameter and the string move at -3.0 cm/s second (3/4 the ground speed).

- - - -

3 to 2 case: yo-yo moves in the opposite direction as the string at -2.0 cm/s second (relative to the ground). From the yo-yo's perpective, the ground moves at +2.0 cm/s, the inner diameter and the string move at +3.0 cm/s second (3/2 the ground speed).

 

[A.T.]

In the under the ruler faster than the ruler example, the smaller the diameter of the middle parts of the wheels where the big wheel applies the force, the faster it moves,

No. It moves faster when the inner diameter tends towards the outer diameter.

then it would mean that the effect is different from the lever example,

No. It is the same. The blue wheel just allows to apply the force on top.

but why can't I view it as a lever

Sure you can. See the red dashed line indicating the leverage:

[PLAIN]http://img375.imageshack.us/img375/4229/dwfttwwheel09dashedleve.gif

I am having difficulty imagining that when you get the cart up to jelly speed and then engage the propeller, how does the force apply, how does it not use it's own energy to propel itself? I guess it's the same principle as the yo-yo, where the wheels are pivoting around the contact point with the ground, which I am having trouble grasping.

Here the same thing at windspeed. It obvious that the air will push the paddles forward: