Jsp, on the off chance you're still looking for some clarification, I hope this might help.
The reason the CB's post-impact tangent line speed is reduced less at larger CB-OB separations has to do entirely with the ghostball angles. To see this let's step back a moment. First, a reminder of a basic theorem from geometry: an exterior angle of a triangle is equal to the sum of the opposite interior angles.
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The angles are labeled to reflect shot geometry: C is the geometric cut angle (cut angle sans throw); P is the corresponding impact angle (Dr. Dave's term) between the line of centers of the CB/OB and the direction the OB would take without throw; and G is the ghostball angle.
Note that if you vary P while keeping the lengths 2R and d fixed, then G co-varies with P; increasing P increases G and vise versa. This is true up to where C (= P + G) reaches 90 degrees.The important point here is that C changes by a larger amount than P because of this covariance of G.
These angles are shown below for the two cases. Case 1 (C1, P1, G1) represents using no english, while Case 2 (C2, P2, G2) represents using outside english and throw to send the OB down the same path and at the same speed as in Case 1.
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Throw angles T1 (varies with cut angle) and T2 (always equal to 5 degrees for the graphs) are also indicated.
Only two assumptions are made regarding the CB's pre-impact speed. One is that for Case 1, it lies in the range of slow to moderate such that the balls end up gearing during the collision. This makes it both easier to calculate the throw angle T1, and, more importantly, T1 is independent of the specific speed of the CB at any particular cut angle. (The throw velocity component added to the OB and subtracted from the CB is always 1/7'th of the CB's velocity component in the tangent line direction.)
Second, for Case 2, the CB's speed is adjusted so that the OB's speed is the same as in Case 1. Adjustments are required because of the generally thicker hit with Case 2 (but actually thinner at very small cut angles around a couple of degrees and less) and the different throw velocities imparted to the OB between Cases 1 and 2 (T2 versus T1). However, these adjustments are quite small/negligible.
There are four things then that enter into the CB's post-impact speed down the tangent line in Case 2. Two are the just mentioned CB speed adjustments to equalize the OB's speed, one is the added tangent line velocity component from the throw T2 (equal but opposite forces), and the last is the change in the cut angle from C1 to C2.
In the last diagram above, the impact angle P is altered (reduced) from P1 to P2 to compensate for the throw angles T1 and T2. In the course of doing this, because of the covariance of the change in the ghostball angles (G2-G1), the cut angle changes by a larger amount than the impact angle. The closer the balls are together and the smaller the cut angle, the greater is this disparity between a change in P and a change in C. This, plus the fact that the CB's added tangent line velocity from T2 doesn't vary much with P, results in the general shape of the curves. (It would take a little math here to back those statements up.)
If the CB and OB are an infinite distance apart, the ghostball angle is always zero no matter where the CB strikes the OB, as is the
change in the ghostball angle from Case 1 to Case 2. Thus, the change in the cut angle C is exactly equal to the change in the impact angle P. Below is plot of the CB's post-impact speed reduction (sans the small speed adjustments to equalize the OB's speed):
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Without the ghostball angles coming into play, the reduction is nil.
Next, and finally (yay!), is a graph showing the separate factors mentioned above which contribute to the CB's post-impact velocity (the green line combines the cut angle reduction with the added throw velocity from T2).
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I hope all of this sheds a little light on the subject and doesn't totally lie in the realm of "more than I wanted to know." Before seeing the last graph above, I thought the CB's pre-impact speed adjustments for the varying throw (T2 versusT1 across the range of cut angles) would be much larger and a major contributor to the shape of a curve. Obviously, that's not the case.
As Dr. Dave indicated, swerve tends to vitiate all of this; the father the balls are apart, the worse it gets (even converting a reduction in CB speed to a boost!) So the graphs are only strictly valid where a cue's intrinsic pivot point is at the same height as the center of the cueball. In that case, vertical squirt keeps the applied force level and the vertical spin axis
truly vertical (i.e., no swerve).
Jim
).
) Where the curves drop down into negative territory, both the CB and OB would be traveling on the same side of their original line of centers (Bob Jewett's challenge).
