dr_dave said:
Jim,
I'm not exactly sure what you mean by your you last (
bolded) statement, but maybe your whole paragraph has something to do with the slight trend changes we saw with the squirt vs. speed data from our robot tests (see Diagram 2 in my
February '08 article).
Regardless, I agree that any variation in squirt with speed is negligible (practically speaking). Now, "effective squirt" or "squerve" (the combined effects of squirt and swerve) is another story.
Regards,
Dave
Dr. Dave,
Glad to see you're apparently feeling better. What I meant was that the effective offset b' is given by:
b' = Rsin[arcsin(b/R) - A + k(Wav)T]
where b is the initial contact point offset, A the squirt angle, and the last term is the amount of ball rotation during impact that contributes to the final effective offset. For an approximately symmetric force function over time, Wav (the time-averaged angular velocity) is about half the final angular velocity. The factor k, as best I can figure, is something like 1/5 or 1/4, which comes from considerations of the time-averaged torque divided by the time averaged force. It's somewhat dependent on the shape of the force-time curve, but doesn't vary much with different curves.
So if you compare two shots struck at (2/5)R at two different speeds, with total ball rotations of 2 and 4 degrees (WavT), respectively, using k=.25, and assuming squirt angles of 2.5 degrees with a high squirt stick, and 1.8 degrees with a low squirt stick, the ratios of effective tip offset are:
lower speed (2 degrees of rotation):
(b'/R)lowSq/(b'/R)highSq = sin[23.578 -
1.8 + .25(
2)]/sin[23.578 -
2.5 + .25(
2)] = 1.031
higher speed (4 degrees of rotation):
(b'/R)lowSq/(b'/R)highSq = sin[23.578 -
1.8 + .25(
4)]/sin[23.578 -
2.5 + .25(
4)] = 1.030
The calculation is a "first order" type since everything affects everything else. But the "perturbations" are very small. Nevertheless, it seems I grossly overstated the effect of ball rotation at higher speeds smoothing out the differences between cues! Intuition suggested it would be more than this. Do you agree, more or less, with the logic and conclusion?
One thing that it indicates is that the difference in resultant spin/speed ratios between the hypothetical high and low-squirt cues is about 3%. This is pretty small, but larger than a fraction of a percent. Does this sound reasonable?
I re-read your article and think it goes a long way to dispelling some myths. As always, thanks for the work and for making it available to us. Have you an explanation for the dip in the curve for squirt angle versus speed graph yet?
Jim