Re: free fall data

[brian whatcott <inet@intellisys.net>]

Ludwik's note ( quoted below ) raises several interesting associations.
I should first acknowledge the person who worked this problem rigorously.

He said, " ...we tried other distances, comparing the time for the whole
length with that for the half...;in such experiments REPEATED A FULL ONE
HUNDRED TIMES
[my caps] we always found that the spaces traversed were to each other as
the squares of the times..."
(Naturally Accelerated Motion, Theorem II Prop II Corol I, Third Day, 2 New
Sciences, G. Galilei) 

I dare to suppose that the instrumental dificulties faced by this first
experimentalist, where time was measured by heartbeats and by measuring the
efflux of water from a metering pipe, were answered satisfactorily by
statistical averaging measures which are still available to us.

I recall too, that an enlightened distance-learning institution (OU)
determined that it would serve their students' purpose to rerun these
celebrated Galilean experiments, using his inclined plane and a smooth ball.

   It was thereby exceptionally easy for them to detect which of the
students were teachers and other people of good background:
 their results tended to represent the square law rather well.
     It was later disclosed that they ( And Galileo ) had omitted the
correction for angular momentum which a ball gains when rolling down an
incline.

People who create dynamic simulations ( e.g. for flight simulators )
are warned to start the computational path at the highest difference
equation then summate ( for integrations ) in order to avoid the errors
involved with
subtracting successive differences - to whick Ludwik alludes. 

Sincerely
Brian Whatcott

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At 11:50 9/13/97 EDT, Ludwik wrote:
....
> 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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brian whatcott <inet@intellisys.net>
Altus OK