What's the quickest way the get the ball (say from shortstop) into the first-baseman's glove, given some fixed initial (throwing) speed? Directly or with one bounce?
I'm fairly sure that the answer depends on the initial throwing speed, but where's the cut-off point (roughly)?
PS 1: Numbers, if needed:
Under the current rules, a major league baseball weighs between $5$ and $5\frac1 4$ ounces ($142$ and $149\,\mathrm g$), and is $9$ to $9\frac1 4$ inches ($229–235\,\mathrm{mm}$) in circumference ($2\frac7 8–3$ in or $73–76\,\mathrm{mm}$ in diameter).
The horizontal distance to be travelled is $36\,\mathrm m$.
PS 2: The surface is "clay" (around first base). The ball is covered in leather.
PS 3: Let's assume that throwing accuracy and catching abilities are not issues.
Best Answer
The time it takes to get to the first baseman only depend on the horizontal component of the velocity, so you want to maximize that. If you put no conditions on how far off the ground the first baseman catches the ball, and assume that the total speed is great enough, then the ideal throw is with a horizontal initial velocity. The first baseman catches the ball at whatever height the ball has fallen to during its trajectory.
If the initial speed is such that a horizontal throw can't make it to first base, then the problem is more complicated. Upon hitting the dirt, the ball will lose some horizontal speed (thanks Olin Lathrop for the clarification). The question is then to look at how far the throw goes before the hop and how much horizontal speed the ball loses at the hop. That gives you the total time for a perfectly horizontal throw. You can compare that to the time for a throw with just enough vertical component to avoid bouncing. You probably don't need much vertical component, so your throw is still almost perfectly horizontal, and you don't lose much time that way. So you probably want to avoid the bounce.
I tried solving the problem to write the ideal angle as a function of the initial speed and the distance parameters, but the bookkeeping gets complicated very quickly.
All that, of course, is assuming that ballistics is your limiting factor. In a real baseball play, the shortstop will probably always do better by making a faster throw (as long as it's sufficiently on-target).