One of your mistakes was to use a point of great inherent interest (your choice of T0) as the left endpoint for your "region of interest" instead of choosing a training set that would place T0 within its interior.
My purpose was to reveal trend detail during the descent, and as you can see...
...the Poly(10) curve is a pretty good approximation.
Can I reveal more detail around T
0 with an additional curve, sure, and as I said I may do so. (Not sure if it's worth the bother given I already have the S-G curve though)
When compared to the NIST curve and the S-G curve, it is clear that the Poly(10) curve does indeed reveal a lot more detail than the NIST curve, which was the intent.
Another of your mistakes was to select a class of approximating functions whose derivatives tend to be least accurate near the endpoints of your so-called region of interest.
Whilst the italicised section is technically true, bearing in mind intent, your suggestion of *mistake* is specious. Could behaviour around T
0 be made
more accurate with an alternate curve, sure, though at the expense of detail at other times (it's end-points, and fitting from a near-linear section to a curve.)
Another of your mistakes was to ignore boundary conditions.
Not ignored, a compromise. End-point is the end of available data, as the roofline becomes obscured.
Another of your mistakes was to make unsupportable claims about your model, such as no "
abrupt" or "
instantaneous" change in acceleration.
The trend, as you know, matches the S-G curve very well, and so such claims are shown to be supported using alternate method. You're talking about the first ~0.25s after T
0, perhaps a little stretch to be made directly from the Poly(10) curve alone, but true nonetheless.
Would you rather rely upon the NIST curve ? Is that a more accurate representation of trend in your opinion ?
By denying the applicability of your model to
neighborhoods of T
0, you cannot say whether there is an abrupt change of acceleration at T
0. Indeed, a mere glance at your acceleration graph for Poly(10) would suggest that the acceleration either changes abruptly or is rapidly diminishing from a large upward acceleration at your T
0.
Getting silly now. You are performing the same kind of literal interpretation that has caused folk to latch onto such phrases as "2.25s of freefall" verbatim. The S-G curve shows that the transition is pretty sharp, but still takes a while to reach gravitational acceleration. (~0.75s)
In your measurements, however, that large upward acceleration isn't really there.
Of course it isn't
(The acceleration curve you obtained by Savitzky-Golay smoothing
maxes out at about 4 ft/s^2 instead of the 12 ft/s^2 of your Poly(10) polynomial at T
0 and 138 ft/s^2 at 11s.)
Indeed.
I think you mean well. I also think you're out of your depth.
I think you have come around nicely to accepting that the Poly(10) curve does indeed reveal a lot more information about the actual velocity profile during descent. It's all good
As I said, I may well perform an additional fit with an earlier ROI, with which we'll be able to argue some more, and find that it matches the S-G curve behaviour around T
0 pretty well.
)
)
