Rick,
RM: Perhaps I’ve never said it explicitly but equation 5:
fits the velocity profiles of ALL curved movements such as the ones you mention as well as those of the movements of the planets in their orbits, the path of a fielder running to catch a fly ball, and randomly generated movements, such as those produced by Maoz et al (2016).
That’s an extremely important fact about equation 5 and quite counter intuitive, I think. I glad you pointed it out. I was blown away by the fact that it accounts the velocity profiles of planetary orbits.
Perhaps that fact is counter-intuitive to you, but I think it has been self-evident to all others. What is more important is that it does not only fit to every movement as a trajectory but to every single point of every possible trajectory. Let’s think about an arbitrary point p. The studied object can pass p with different velocities even if the curvature remains the same and conversely the object can pass p with the same velocity even if the curvature in p were different.
So (5) does NOT show that there is a mathematical relationship between V and R. Quite the contrary it shows that the there is NO mathematical relationship between V and R and they can vary fully independently.
(5) is equivalent for example to D = V3 x 1/R which becomes a bit clearer if change the radius R to curvature C (C = 1/R) and thus we get:
(6) D = V3 x C
Now, let’s again compare this to an analogical but perhaps intuitively easier case of the size of a rectangular quadrangle. Area is Width times Height, or:
(7) A = W x H
Note that the forms of (6) and (7) are just similar so that there are two measurements (both based on different values of x and y variables) and from these two is calculated a third combined variable which describes something about the current combination of those two measures. As well as (5) or (6) fits to every single point of any trajectory as well (7) fits to every single rectangular quadrangle. And as well as (5) or (6) tells nothing about the relationship between the velocities and curvatures of these points as well (7) tells nothing about the relationship between the heights and width of these rectangular quadrangles.
Adam has stated principally this same argument to you in http://discourse.iapct.org/t/behavioral-illusions-the-basis-of-a-scientific-revolution/15614/118 and you did not answer it. Would you answer it now?