Patrick Johnson said:"Simply slapping or punching the ball" feels exactly the same to the ball as any other kind of hit. The key is where you hit the ball, and the key to that is how you stroke. The nature of the hit (slap, punch, "accelerating through", or what-have-you) is irrelevant except for its effect on tip/ball accuracy. I can build a robot (if you have the money) that will hit the cue ball with no followthrough and get precisely the same action you can with any kind of human stroke.
pj
chgo
For most of us mortals (my mechanics happen to suck), I agree that follow through seems to be critical to a straight stroke and hitting the cueball in the right spot. However, I would argue that if your mechanics are EXTREMELY good, the follow-through will not affect the contact point on the cue ball that much. What WILL be dramatically affected will be your ability to hit with the same speed time after time and this is almost as important.
You're right, a robot could be programmed to hit exactly the same without follow through (i.e. no deceleration at the point of contact). Humans cannot be that accurate, not even close! If a human tried to decelerate just after contact, he would end up slowing down before the hit at least half the time (trying to nail that 1/1000 sec). Only a real freak could pull that off consistently, but why bother? Follow-through is a relatively easy way to accomplish a consistent hit.
Now, speaking strictly from the world of the mathematics of physics, it doesn't matter whether you are actually accelerating, at constant speed or decelerating at the point of impact. The only parameter that affects the hit is the instantaneous speed of the cue at the instant of impact (assuming the same cue and cue ball). THe duration of contact is essentially neglible, which makes acceleration or deceleration irrelevant. However, if you are decelerating, it's likely that you're going to hit much softer that you intend. Following through will prevent this.
Back to the real world of pool, you're fine accelerating or at even speed as long as you don't slow down.
Well, I entertained myself with this post! I hope I didn't bore everyone!
Tom