Re: Air resistance

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

Jack Uretsky also wrote:

> I think that you have forgotten that the average of garbage is
> smooth garbage. More precisely, if the error on 1 measurement is sigma,
> and IF SIGMA IS THE SD OF A NORMAL DISTRIBUTION, then the error on
> N measurements is sigma/sqrt(N).

Yes, the absolute error is growing as sqrt(N) but it becomes less and
less significant in comparison with the true (~ mean) value.

> I have no reason to believe that the fluctuations in the first digit of 
> difference of differences of the position measurements bear any 
> resemblance to a normally distributed  random variable.

Neither am I. That is why the imaginary data on the "true y=x^2" relation 
offer a more useful illustration than real data from a "black box" where 
truncation may not be the only vilain. The last column below shows the
truncated values of y=x^2. Pretend this was done in a black box.

            x     y1      y2

          2.00  4.000    4.0
          2.02  4.080    4.0
          2.04  4.162    4.1
          2.06  4.244    4.2
          .....
          3.96 15.681   15.6
          3.98 15.840   15.8
          4.00 16.000   16.0

Note that all y2 are truncated (which is worse) rather than rounded 
representations of the true values. By reducing 100 data pairs to only 
four data pairs (x and y2_mean_of_25) I obtained the following data:

            x          y_mean   true y=x^2
         
          2.24         4.984      5.0176
          2.74         7.472      7.5056
          3.24        10.468     10.4979
          3.74        13.952     13.9876 

The least square fit to y=A*x^b yields A=0.987 and b=2.008. Do not
underestimate the power of averaging. That is why I am not totally
convinced that my original analysis of data (b=2 +/- 20%) was really 
wrong. Perhaps this stupid black box is not so stupid after all. 
There is no doubt that the data on v and a are of "lower quality that 
the data on d". That is why we have 20%. And I would yield easilly if 
somebody insisted that 20% is too optimistic and 30% is better. But 
1000% (b=0.2 --> 2) or 250% (from b=2 --> 5) is too much.

What else can I say? I do not think that the question, "how bad these 
data are ?", has been definitely answered. These "black boxes" were made 
to measure what we are measuring. The data are not as good as we would 
like them to be but how can they be so bad? I wish more people were dooing 
experiments and posted the data. Jack is right by saying that data from
differend labs must be compared to learn the truth. 

Yes, it is a wrong time of the year, we all have other priorities right 
now. But come back to this topic when you can. We have the same equipment
and a standard basketball ball is not hard to find. It does not have to
be exactly the same as ours. How good is the tool we are using? What are
its real limitations? And why not a topic for a student research project?

                                              Ludwik Kowalski