free fall data

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

On 12 Sep 1997 Bob Carlson (Raacc@aol.com) refered to a simple PC program 
which also extracts coordinate data from digitized movies (see below).

This question has to do with propagation of errors when data from digitized 
movies are used to measure g. You know that free fall position data (x1, 
x2, x3, x4, ...) must be very accurate to calculate speeds (assuming errors 
are negligible for t). My data are never accurate enough. One approach is to 
do averaging. For example, we can eliminate x1, x2, x3 and to keep their 
average instead of x2, then do the same for the next group of data, etc. 
Averaging on the groups of 5 or more data points (as for example, in a 
program like MacMotion from Vernier) would help but this approach can not 
be used when only 5 to 10 usable data points are available. 

It turns out that accelerations calculated from the v2, v3, v4, v5 data 
fluctuate so widely that it is often embarassing to say "as you can see, 
the acceleration has a constant value close to 9.8". The mean value often 
differs from what is expected by about 30% while individual accelerations 
fluctuate between -100 and +100 m/s^2, or so. Do you agree?

1) How to deal with this? My approach, so far, was to avoid plotting
   accelerations. Only the velocity-versus-time data are plotted and 
   "the best" straight line is drawn through the data points. The slop 
   of this line is the acceleration. But the linear dependence is not 
   at all obvious and I am forced to say "as you know the relation must 
   be linear and we will use it to find g". I am not very happy with that 
   kind of "learning from experiments". A camcorder + computer setup 
   is more expensive than the sparking wire apparatus, it is not better.

2) A steel ball, dropped from an elevation of 2 meters, hits the floor. 
   Its motion is recorded at the rate of 30 pictures per second. How 
   accurate should the last six values of x be if the accelerations 
   computed from them (four individual differences of differences) are 
   not to fluctuate by more than 10%?

   One way to answer this is to use EXCEL. Assuming the last picture was
   taken at t=0.6000s I enter six values of t in the first column and
   the corrsponding distances (from the initial location) into the next
   column, as shown below. These were calculated with a=9.8. The third 
   column has formulas for v and the last one has the formulas for a.

      t(sec)    d (m)    v(m/s)    a              t=0 --> d=0
     .................................    last frame entered first, etc
     0.60000   1.764000          
     0.53333   1.393776   5.5534
     0.46666   1.067081   4.9000   9.80
     0.40000   0.784000   4.2462   9.80
     0.33333   0.544434   3.5935   9.80
     0.26666   0.348427   2.9401   9.80

  I change the distance 0.54443 to 0.5 and I see that the last two values
  in column 4 change to -0.2 and + 29.8 m/s^2. I restore 0.54443 and change 
  0.78 to 0.7. This changes of the last three accelerations to -9.1, +47.6 
  and -9.1 m/s^2. The effect of two same-direction-changes in col 2 is very 
  dramatic. To increase the last acceleration by 10% (9.8 --> 10.8) I must 
  change the last distance from 0.34847 to 0.353, that is by only about 1%.

  All this is not surprising; the percentage error on d=x1-x2 is much
  larger than the errors on x1 and x2, when d<<x1 and d<<x2.
        
                                                        Ludwik Kowalski
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> There is an alternative to VideoPoint, at least for pc users, that may
> provide an easier to use interface.  World-in-Motion Version 3.0 is soon
> to be released.  It has an html and conventional interface.  If you are
> interested, there is a new demo version that works with twelve avi files
> available at:
>                    http://members.aol.com/raacc/wim.html
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