I don’t do any actual math. But I visualize the cueball path after contact with an object ball as the addition of two vectors. The first vector is the tangent line with a magnitude proportional to how hard I hit the shot. The second vector is the original cueball path with a magnitude relative to the amount of action on the cueball. My mind just sees both vectors and merges them into the predicted path. I see a lot of talk about things like the peace sign for predicting some rolling cueball trajectories. I was just wondering if others went off pure feel /familiarity or if they similarly had a mathematical visualization in mind.
Yes, there are easy, accurate ways to use vectors to describe where the cue ball is going, but you are leaving out some details.
First, the speed and direction of the cue ball off the object ball is given by a right triangle. The incoming speed and direction is the hypotenuse of the triangle and the two perpendicular legs are in the directions of the object ball and the cue ball. That triangle will give you the relative speed of the cue ball and object ball relative to the incoming speed.
Second, the follow/draw on the cue ball can be described as a vector. It's forward along the incoming path of the cue ball and back if you have draw. The length of that spin vector should be scaled so that if the cue ball is rolling smoothly, the length is the same as the hypotenuse above and in the same direction.
Finally, the final path of the cue ball is a mixture of the speed vector coming off the object ball (90-degree rule) and the spin vector where they are combined in the ratio of 5/7 of the speed vector to 2/7 of the spin vector.
The spin part may sound like magic, but in fact it was all worked out nearly 200 years ago. I'm sure Dr. Dave has an explanation of it somewhere on his site. I have a column or two I wrote about it somewhere in my online column archive.
The 2/7 and 5/7 mixture is the basis of the follow angle system that has been discussed extensively here recently.
The vector idea leads to several useful conclusions.