Re: air resistance

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

   On 10 Dec 1997 19:32 John Gastineau <gastineau@mindspring.com> wrote:
> "15 point" averaging means that in finding the slope of the distance vs
> time graph, a linear least-squares fit is done to 15 points centered on the
> time value to be calculated. This means that the first 7 and last 7 points
> of the data series are not calculated in the same manner, but have
> truncated sets of fewer than 15 points. The word "averaging" is not used
> very appropriately in MacMotion.

This prompted me to look more carefully into the table produced by Mac
Motion, for example, the one posted earlier for the falling ball. And 
I now see why John M. had such a different range of n than I had for the 
same data. I just assumed, wrongly, that after averaging over the distances
the values of v are calculated as v(i)=(d(i)-d(i-1))/dt. But this is 
not at all true, as one can verify by calculating velocities by hand. 
There is some kind of smoothing going on at the v level. The same thing 
happens again when the values of a are calculated from the values
of v. The final a=f(t) curve generated by Mac Motion is not at all the
same the one generated by hand (or with a spread sheet). The values of 
R (air resistance) calculated from the spread sheet accelerations
fluctuate widely, the values of R calculated from Mac Motion displayed
accelerations fluctuate much less. 

I did not realize (or did not want to realize what was at once obvious
to John M) that in trying to improve the look of data (double smoothing)
Mac Motion falsified them. Here is an illustration:

      t(s)    a_Mac_Motion (m/s^2)     a_spr-Sheet (m/s^2)

    1.500        9.60                       11.2
    1.525        9.61                        3.2
    1.550        9.59                       16.0
    1.575        9.54                        6.4
    1.600        9.48                        9.6

               mean=9.56                  mean=9.28

The Mac Motion data are as bad as camcorder data. And I was under the
impression that they are much better. Mac Motion fulled me. But, as they 
say, it is never too late to learn. Sorry for creating so much noise. 
I hope I was not the only one to learn from this episode. 

We all face a challenge. How to collect good data on the R=f(v) relation.
Is it possible, with the equipment we have, or not?

> In Logger Pro (soon available on the Mac, and destined to replace
> MacMotion) you have separate choices of the number of points used to
> calculate derivatives (the fitting scheme mentioned above) and true
> averaging, in which case a moving average is applied to a data column 
> with the goal of data smoothing. Calculating a derivative and averaging 
> are thus two distinct operations. 

Do you have a beta version of this program, John G? Try it and share the
results. I do not think that the problem can be solved with better software. 
They must use higher ultrasonic frequency (d_error ~ lambda) and an analog-
to-digital convertor which does not truncate distances too early. 

                                             Ludwik Kowalski