Re: Air resistance

[LUDWIK KOWALSKI <KOWALSKIL@alpha.montclair.edu>]

Everythig so far may become irrelevant after the effect of air turbulances
on the motion detector is confirmed. See our next message on the topic. For
the time being let me post this:
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On 13 Dec 1997 John Mallinckrodt <ajmallinckro@CSUPomona.Edu> wrote:

>  (BTW, where *did* you get your 20% value anyway? I never understood that.)

As you know, I accepted the calculated accelerations at their face values
and used them to calculate R. Then I plotted the ten R=f(v) data points and 
traced a smooth curve through them. R changes by the factor of 3 while v 
changes by the factor close to 2. Thus, if one wants to parametrise the
data with the R=b*v^n then n must be about 2. The estimate of the "error 
in n" was also very crude. Using the R=b*v^n curve I calculated the R2/R1
ratios for several n and compared them with the experimental value of 3.
The acceptable range of R2/R1 was determined on the basis of distances
between the smooth curve and the individual data points. Contrary to what
one finds when averaging is 3 (or 7 which is the default value) those 
distances are relatively small. [In other words, the end points of an
R=b*v^n curve, when n is outside of the 1.6 and 2.4 range, do not fit 
into the region in which the data points are scattered.]

The validity of the apparent accelerations, calculated internally by Mac
Motion, was not questioned. I am glad John M. questioned the data and
thus demonstrated that my conclusions are wrong. Very resonable agreement
between my wrong results and the results produced by using the empirical
formula must be considered as coincidental. Apparently, the so-called
averaging (which is much more that averaging) leads to a serious destortion 
of data. Otherwise an analysis at the lavel of d would lead to the same
conclusion as the analysis at the level of a.

> I think your basketball data (and I mean the *data*, not the calculated 
> values of v and a) are *extraordinarily* good.  After all, they appear to 
> have an accuracy of less than 1 mm!  I have not used a sonic ranger myself, 
> but I can't imagine getting much better results with one. 

The accuracy of a single d measurement is about one wavelength (8mm).
The *extraordinary* goodness comes from the internal smoothing, as described 
by John G. This smoothing, and the second smoothing to get accelerations, 
are apparently responsinble for large systematic errors. 

> Just to reiterate, the data are not at *all* bad.  But the fact remains
> that they don't *begin* to be sufficient to determine "n"  with any
> precision. This is primarily because the data are taken over far too
> limited a range of velocities. 

According to the Mac Motion manual, the maximum d is  3.7 meters when
the sampling rate is 40 Hz. The minimum d is close to 0.4 m. The 
corresponding range of v (in a vacuum) is 2.8 to 8.5 m/s. This is better
than the range of 2 but not dramatically better. Working at 50 Hz (the 
highest possible sampling rate) the maximum d is even smaller. Too bad, 
one would prefer to use 50 Hz to maximize the number of data points and 
benefit from the correctly performed averaging.

Brian's concern about fitting the data with an inappropriate formula,
even in a limited region of v, is ligitimate. But the main issue is to 
collect good data, not how to fit them. A piecewise linear fit may be 
practically as useful as a fit to a very complicated formula. (Unless 
the goal is to determine some parameters of a theoretical formula, such 
as viscosity, etc.)
                                                      Ludwik Kowalski