Physical Role of Batter in Baseball Physically, what is the role of a batter in baseball?
My question is inspired by How does the speed of an incoming pitch affect the speed of a baseball after it's hit?
The answer to that question, that a faster pitch results in a farther hit, surprised me. I had thought that a batter was solely applying impulse to the ball (mass * velocity of bat = Force * time on ball). Clearly the batter is doing more.
Specifically, I'm interested in what information we'd need to know and what math we'd use to calculate the final velocity of the ball.
 A: I'll try to answer in the context of cricket; baseball shall follow as a special case. (Cricket is richer in terms of the role of the batsman, as compared to baseball!)
The objective is to transfer momentum to the ball, which can be done in two broad ways (this isn't the standard terminology, this is how I see it)- 
1 Guiding - Here, you use the momentum of the ball to your advantage. Of course, you are also imparting momentum to the ball, but largely by deflecting the ball to the area where you want it go. A classic example is the ''uppercut'', another example is the first shot here. (Wait till the slow motion replay.)
From the point of view of physics, it can be seen very clearly from the first example that if you decompose the momentum vector of the ball (which was angled ''in'' towards the batsman), the component along the pitch was retained completely. The batsman only positioned the bat such that the ball's velocity would pick up a vertical component upon collision and hence sail over the keeper and the slips.
2 Smashing - This is where cricket and baseball are related. Here, the effort isn't really directed at using the inherent momentum, but rather, the aim is to impart sufficient amount of momentum to the ball by ''tonking'' it. One example is this. Flat pull shots always make a classic example see the first shot here. In fact, in straight hits, the component along the pitch is sometimes completely reversed. Naturally, to completely reverse the direction as in example 2(c), or even in the more agricultural examples 2(a) and 2(b), you would have to transfer a lot of momentum to the ball from your side (bat), which is why I call it ''smashing''. (In example 2(c) though, the guy makes it look effortless.)
Of course, there's some physics even in the innocuous Forward Defense, especially when it comes to playing these kind of nudges and stealing quick singles, but these can hardly be extrapolated to baseball!
Now, back to your question, the ''physical role'' of a batsman who is equipped to play each of these is to judge when would it be more appropriate to use case 1 and when to use case 2 - this (shot selection + placement + timing) is what the art of batting is about. If a guy is hurling the ball down to you at 140-150 km/h, the ball is already carrying a lot of momentum, and complete momentum reversal (''tonking'' it) would require need you to spend a lot of energy, plus if you mishit, there is a chance that it won't sail over the boundary and you get caught in the deep. That's why, whenever the ball is angled rightly and there is enough room to play such shots, people tend to resort to Example 1(a). In comparison, if the ball isn't turning/swinging viciously, slower bowlers tend to get smashed more often, because it is easier to reverse components/complete direction of the ball in that case. Occasionally people make errors of judgement, sometimes with heartbreaking results!. 
I think the most appropriate way for me to end the answer would be - yes, you are right that the batsman is only transferring momentum to the ball. But momentum is a vector, and that's why batting is an art! 
A: It helps to think of the extreme cases here. If the bat was not moving (like a bunt), the resulting bounce does depend on incoming speed. Now, make the incoming ball really slow and the batter hitting fast. The result does not depend on incoming speed. The true answer is the superposition of these two cases.
