Jude Rosenstock said:
...After this collision whether the friction be collision-enduced-throw or spin-enduced-throw and assuming there is no draw or follow, the cue-ball will travel in a 90 degree path from the path of the object-ball. That is to say, if you could cut a ball 90 degrees, the cue-ball would not change course (see figure 4.2). To cut a ball sharper than 90 degrees would mean that the cue-ball would deflect on the same side of its original path as the object-ball it is colliding with.
The 90 degree rule is not an absolute. The cueball takes off at
approximately 90 degrees for most shots, but not exactly. Why? Because of throw (and some inelasticity of the collision, but we won't go there).
So suppose you hit the object ball at a geometric cut angle of 87 degrees, and throw added another 5 degrees. The cueball will take off down the original tangent line of the 87 degree cut, but will be sped up a little by the throw (the friction acts equally and oppositely on the balls). In the meantime, the object ball will have been propelled backward along the tangent line by the same friction (in addition to forward along line of centers by the compression force) for an effective cut angle of 87 + 5, or 92 degrees.
Okay, a slight modification to the above. The original tangent line will also shift because of the compression of the balls, probably about a degree for a hard hit shot. This should mean, but I'm not sure, that you'll get an effective cut angle of about 93 degrees.
Jude Rosenstock said:
I would be fascinated if someone took the time to plug-in our scenario into all of the equations that apply. However, I'm reserved to believe that the results will only convey my conviction that a cut sharper than 90 degrees, even with maximum side-spin is impossible. If anything, even if mathematics could prove the possibility existed, it's practicality would justify this debate moot.
If you were masochistic enough to read my last two posts, they indicate that the 90+ shot is possible, and with an amount of spin that is quite obtainable. For a cueball speed of 12 mph and a geometric cut angle of 87 degrees, throw is predicted to be over 5 degrees when using a spin/speed ratio (RW/V) of 1.034, where maximum throw occurs. However, at a ratio of 1.01, it drops down to less than 3 degrees (actually less than 2 degrees), yielding less than a 90 degree cut. At the other end, it drops below 3 degrees at a ratio of 1.17. That's what I meant by the tolerances being very tight! But the ratios should be obtainable in that you probably can get something closer to 1.25 with maximum tip offset.
These numbers depend somewhat on knowing the precise dependence of the coefficient of friction on surface speed, which to be honest, is not that well established.
Jim
