Thanks for the diagram showing the "paddle" Humber. The really great thing is that you've shown other details also and so (as already noted by mender) you've exposed another error that I suspected you were making (and which was also the main motivation for my second question). Anyway, I had a long post mostly written but can throw some of that away.No, it was the number that jjcote used, so I adopted his version and use.
It is a nominal value, but of course it will vary within the flow.
You should note that I still very little idea what you're really trying to say here Humber. What distance are you talking about? (I'm not sure I really want to know actually.) Remember, I am not inside your head. I don't know precisely what you were looking at or thinking about when you wrote this. You seem to have very "different" ways of thinking about things and you've also stated that yourself. That probably means that when explaining something that you think is wrong or whatever, that you also need to spend a bit more time "filling in the gaps" as it were. Instead your comments often seem to add confusion rather than shedding light on the matter. To much like a stream of consciousness perhaps? No?, 6? Yes. I think so, no... m/s maybe. Or a distance. Do you think so? Who really knows?10 - 6 = 4m/s. Not distance, velocity.
Anyway, those diagrams of mine used the values that jjcote used in the original set-up. All the velocities I have shown out to the right are as "you" (Humber, standing still on Earth) would measure them. I thought that would be fairly obvious given that one part was always labelled as "still" but I guess I should have stayed with the "diagrams only" approach to minimise any possible new sources of "confusion".
So please just look at the diagrams again and see if you can at least follow those. They're just three rectangular "photographs" from a short movie really, "clipped" on the left and right. You can think of the asterisk and plus sign as puffs of different coloured smoke. There is really nothing tricky here at all. They don't prove the cart works or doesn't. They're about as simple as you could get short of actually filming the scenes (complete with smoke and a reference mark on the ground/belt) with a non-moving camera on a tripod and then posting three images each taken a second later than the previous one. In the first three we see air moving to the right over non-moving ground. In the second set of three we see the belt moving left while the air high above doesn't move. That is what the numbers and the diagrams both show and say. Please give me a cheap thrill by telling me you can understand them now!
Back to one of your main errors (and as already noted by mender). The "nominal" boundary layer's velocity must be some constant value relative to the surface in both the road and treadmill scenarios (because we're assuming those are essentially the same kind of surface in terms of drag with an air mass moving past at some particular speed). It's the result of friction with the surface below. The belt "drags" on the air above it. The road "drags" on the air above it. It's the interface between the surface below and the air mass above, and the speed (relative to the surface) builds up from 0 at the bottom at the same rate in both "real wind over ground" and "still air over moving belt" scenarios. You can't just take one "nominal value" in one frame of reference and then plug the same value into a different frame of reference.
Consider a layer that is just below the very top of the boundary region. That is where the "boundary layer" effect will be least. In other words that is where the boundary layer air is moving at very close to "wind speed". Relative to the road (with "real wind" above at 10 m/s) that might be moving at 9.9 m/s (for example). But relative to the "cart observer" (room) for the treadmill that same point in the boundary layer would be moving at 0.1 m/s. It's obviously not 9.9 m/s again! Also note that the two correct values add to wind speed and this must always be the case if they are being measured relative to ground and "cart observer" as you are doing.
So, instead of doing it correctly, you have often (always?) been using the same value (such as 6 m/s) in both cases. Mixed frames in, garbage out..
