This post intends to document the squirt, that is the deflection of the actual cue ball path from the aiming line. According to experimental results mentioned in [1] and [2] squirt seems to depend only on the amount of sidespin (that is the lateral distance from the center of the ball to the point of impact) and the effective endmass of the cue. The latter is essentially determined by the moment of inertia of the cue along the vertical axis (that is how easily it can be deflected sideways). For an explanation of the effect see [2] or, for a more technical approach, [1].
For the purposes of the experiment all cloth-related friction coefficients have been set to zero to eliminate the effects of swerve and therefore cue elevation. Also, the coefficient of friction between the cue and the ball was made infinite to ensure that the cue would always grip the ball thus eliminating the possibility of miscue. The goal was to see how well the simulation would match existing experimental data presented in [1] therefore the diameter of the ball was chosen to be 50.3mm and its weight 120g. The cue mass did not seem to make any difference (and it shouldn't as it is its inertia that matters) but it was nevertheless chosen to be 186g. The results are shown below:
d A B C ---------------------------------- 2.5 1.3 - 1.2 5.0 2.6 - 2.2 7.5 3.4 - 4.0 10.0 5.3 - 6.4 11.0 5.8 ~7 7.5 12.0 6.3 ~9.5 9.0 13.0 6.8 ~13 10.1
The first column above is the amount of english aplied in mm away from the centre of the ball. Column A is the squirt in degrees produced by the simulation. This was found to be pretty constant across several executions of the experiment and with varying impact velocities. The last column (C) is the experimental data presented in [1] (samples from the fitted curve actually). It can be seen that the results are very close as long as the amount of english is low enough. After a point though the measured data begins to deviate a lot from our results. One possible explanation is that, as the distance from the centre rises, the tip of the cue stick starts to slip on the ball resulting in 'mild miscues'. To see if this is the case the moment of inertia of the cue was fixed to the value that produced the results above and the coefficient of friction between the tip and the ball (hitherto infinite) was set to 0.8. This produced the results of column B which are only approximate as the actual numbers seemed to vary more accross various shots. Where there are no entries in the column the numbers are the same as those in column A which means that no slipping of the tip occured. The results now more closely match the experimental data although not perfectly, especially as the distance from the centre rises beyond the "miscue limit" (assumed to be half the ball radius). This is to be expected though as the simulation method, and especially the friction model employed by ODE is not meant to be absolutely accurate.
To carry out experimentation on your own you can start Billiards like this:
billiards -Oexperiment -Osquirt
This will setup the various parameters as described, including the amount of english. To fiddle with the parameters edit file scripts/experiment.lua and restart. To carry out the experiment simply shoot the ball (do not change the aim). A soon as the ball hits the far cushion the simulation will stop and the amount of squirt will be printed on the console in degrees. ------ References:
[1] Cue and ball deflection (or “squirt”) in billiards, Rod Cross, Physics Department, University of Sydney, Sydney NSW 2006, Australia
[2] The articles on squirt by David Alciatore, http://billiards.colostate.edu/bd_articles/index.html