tfk, I have a question.
Your table contains raw video data, did I understand that correctly? If so, how did you differentiate it? Or are the derivatives from the NIST curve fit?
**** NERD ALERT **** NERD ALERT **** NERD ALERT ****
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H,
I don't have raw data. I started with the empirical curve that NIST presented.
And then simply differentiated this formula. Twice.
The most important questions to ask are not, "are they pretty?" Or "are they mathematically easy to manipulate?" (although this helps...)
The important question is "how well do these curves reflect reality?"
And the only data points that we have are the position vs. time data points in Fig 12-76. And these look damn good.
Now the interpolating function forcibly smooths this raw data, and you would therefore EXPECT the first & second derivatives to be well-behaved and essentially "noise reduced". That is a good thing.
Chandler did the opposite. He took good data and forced it (with his "central difference" velocity approximations) to be artificially GRANULAR.
The use of the "central difference approximation" to calculate instantaneous velocity amplifies slight data errors.
To see this, switch back & forth between the position vs time (Fig 12-76) and velocity vs. time (Fig 12-77) curves in the NIST report. (vol 2, pg 602 & 603). If you look carefully, you can see this "error amplification" effect.
If the successive points on the position curve are both on (@ t=2.6 & 2.9 sec), both below (@ t = 2.2 & 2.4 sec) or both above (@ t = 3.2 & 3.4 sec) the position interpolation curve, then the derived velocity point will be right on the velocity curve. Which is true for the velocity points at 2.7, 2.3 & 3.3 sec, respectively.
But if the first data point is below the interpolated position curve and the next one is above the curve , then the calculated velocity will be too high. (e.g., poisition points @ t = 5.0 & 5.2 sec => velocity point @ 5.1 sec.)
If the first position point is above the curve & the second is below, then the calculated velocity point will be too low. (e.g., position point at 1.6 & 1.8 sec => velocity point at 1.7 sec).
You can see that the operation of piecemeal slope calculation is AMPLIFYING small errors into medium errors. And taking the second derivative to calculate acceleration amplifies medium errors into large errors. (small, medium & large being relative terms, of course.) This is the opposite of what you want to do. You want to eliminate these errors as early as possible. NOT amplify them.
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There are two further checks that one can do (with the data available in the published reports) to see which method is better.
1. Integrate Chandler's velocity curves
Take Chandler's velocity graphs, numerically integrate them, and see how well they fit the raw position data. I haven't done this yet, but I'd predict that the answer is "not well".
2. Apply the central difference approximation to the velocity data points to see how they fit his "constant acceleration" conclusion.
This 2nd step will be MOST enlightening. On the velocity vs. time curve, estimate the slope of the straight line between successive velocity data points, and plot those values at the "timing midpoint" on a new acceleration graph.
Look at Chandler's velocity vs. time graph & compare the slope of the line connecting each successive data point. The successive slopes of his data shows a distinct "oscillation" above & below the average slope. Between 1.75 & 2 seconds, the slope is much higher than his interpolated (red line), the next slope below, the next above, the next below, etc. This oscillation is a classic "regression to the mean" that happens when your raw data is too lumpy.
So the answer to this question is that, when you apply Chandler's OWN technique to his velocity data, the conclusion is that "the acceleration ain't CLOSE to staying constant". Of course, these massive changes in acceleration are NOT real. They are simply artifacts of poor math technique.
The test of of how well the methodology described here fits WHAT REALLY HAPPENED lies buried in the raw data files. The answer would be found by using some of the "enhancement techniques" I mentioned in the last post. And by taking more data points. Perhaps I'll drop a note to NIST & see if they can provide raw video... Anyone know anyone over there?
But I am certain this techniques is far superior to Chandler's. You want to TAME data noise. Not AMPLIFY it.
Tom