All the math I’ve done is on this post. It is very rudimentary; a “back of the envelope” calculation. Honestly, I don’t think it’s sufficient to prove much; actuator disk theory (assuming the rotating propeller is a circular plane that provides thrust) has a lot of limitations: for example, if an airplane is stopped at a runway, actuator disk theory says the energy available to the propeller (from the engine) provides the power to move the airplane (thrust times velocity) and rotate the propeller (propeller drag times RPM, more or less). Since the airplane is not moving (velocity = 0), then the RPM would shoot up, because all you’d have left is engine power = prop drag times RPM. This is obviously not the case, because in reality, the propeller is sucking air past it, and that’s consuming most of the engine’s power (hopefully; otherwise it’s a pretty crappy prop).

The two phases of the cart traveling, that is, phase 1, where it’s accelerating to wind speed, and phase 2, where it’s accelerating past wind speed, must be considered carefully.

In phase 1, the cart has beneficial drag in that the drag from the wind on the cart is helping it go forward. There’s a negative force in the “windmill torque” of the propeller, that is, the force of the wind trying to turn the propeller the wrong way. This is overcome by simple gearing; it is much easier for the wheels to turn the propeller than it is for the propeller to turn the wheels (and move the cart backward). It is from this that the discussions about pulling a spool of wire one way or the other arose, I believe. I also think that having a relatively flat propeller (low angle of attack on the blades) makes for a crappy windmill, which also helps: that is, the windmill drag of the propeller, which helps push the cart forward, is larger than the windmill torque, which would make the propeller turn the wrong way and pull the cart into the wind. Note that Jack (the guy who built the cart that started all this) did test his cart in the “windmill” configuration, and it does go into the wind, though obviously not faster than it.

When you get to phase 2—let’s assume when the cart is going right at wind speed—then there is zero drag (on the body of the cart), because the relative velocity is zero. But there is thrust, since the propeller has been turning this whole time. So obviously the cart will accelerate. As it goes beyond wind speed, drag starts pushing against it, so the limit is when thrust = drag + friction—it’s not a perpetual motion machine. So far, several tests have proven this; the treadmill tests prove this, and so do the rotating platter tests.

The real question remains whether or not the wind and the wind alone could get the cart to wind speed in the first place. If the cart had no prop (let’s say it’s a shopping cart in a parking lot), then the maximum speed in a constant wind will be when the drag on the cart equals the friction in it, and this velocity must be less than that of the wind speed. How do you rationalize this? Well, if the cart were going at wind speed, then there is no drag on it, because the relative velocity is zero. But there is friction pushing it back, so it would decelerate. A balloon floating in the wind, on the other hand, has no friction, thus it will go *at* wind speed, because if it went faster, it would have negative drag, and if it went slower, it would have positive drag.

So from one point of view, you can say the propeller drag can be dumped into overall friction, so you have a shopping cart scenario, but you have a propeller providing thrust, too, so if thrust + drag = friction, the cart will stop accelerating. From this point of view, and without going into more detail, you can’t say whether the drag that satisfies this will be positive (cart going slower than the wind) or negative (cart going faster than the wind).

The problem many people have is with the energy point of view. If the difference between the “speed of the ground” (zero) and the wind speed is providing all the energy, and the propeller’s source of energy is the motion of the cart over the ground, how could the cart possibly end up going faster than the wind, which is providing this energy anyway? I think there are some chunks missing in such an argument. I don’t know what they are. Part of it is that the energy standpoint forces you to put relationships between propeller drag and thrust, and I haven’t done this with too much care.

If you wanted to factor in the windmill torque of the propeller, I’d say the key is to just put a constant in front of that force in your equations, and work through until you are forced to make some sort of conclusion, and see if it comes out realistic or not. You’ll get something in the style of what I did, that is, I arrived at the conclusion that the required lift (thrust) to drag ratio of the propeller is realistic, which was enough to convince myself.

I’d also like to get my hands on the PDFs. I haven’t had any luck yet.

is there any chance you have worked out equations to describe the motion before reaching wind speed and at wind speed. I am trying to figure out what to include (and being more specific than your combined resistance for my own understanding.)

here are the sorts of questions I have:
when the cart is going slower than the wind, can the force of the wind on its area be considered a postive drag. Is there any negative sort of drag. How might you think of this in terms of relative velocity?

How do you describe the force of the wind on the propeller that is trying to make it rotate opposite to the direction that the wheels are forcing it to move?

thanks for any help you can provide!

Can I have these bauer.pdfs, or can you refer me to JB please?

Thanks!

I want Platt to flat out tell me what he wants to see to be convinced. Hopefully others from Make have noticed the ruckus this cart has caused on the Internet, and give us all a chance to chime in.

BTW, I have the original Bauer papers in .PDF If you wish to have them let me know and I’ll email them to you.

I’m JB and my partner and I are the ones that entered the Mythbusters video into their challenge.

Platt just doesn’t get the inertial reference frame thing with the treadmill, and he certainly doesn’t get the whole “the belt won’t reverse at the moment we go through 0 apparent wind speed”.

This little brainteaser twists so many people up it’s hilarious.

Again, nice work.

JB