# The quickest 6-3 play in baseball: to bounce or not to bounce?

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.

• I assume the quickest way is to maximize the horizontal component of the ball's velocity, i.e. release the ball horizontally, if you have the arm to make it reach without bouncing. Assuming you don't have the arm muscles, the question is: is it faster to throw with a little vertical component, and hence less horizontal, or to throw it horizontal anyway and let it hit the ground (at least once) and be slowed down by the bounce(s)... I'd guess more than one bounce was suboptimal, but one bounce might be close... – innisfree May 25 '13 at 20:21
• Another factor is that the first-baseman can shorten the time of flight by stretching toward the fielder. This requires the ball to reach first closer to the ground, which, if recognized by the fielder, can further reduce the time... – DJohnM May 25 '13 at 20:52
• One more complication: the fielder could get a slower, arcing throw off on the run rather than stopping and planting for a faster throw. – DJohnM May 25 '13 at 20:56
• I think that one of the reasons that players bounce the ball is that they can throw faster if they throw down. I don't have any evidence of that, but it makes sense: the longer the ball is in your hand, the faster you can accelerate it. So it would be worth investigating if a throw can be faster slightly downward instead of straight ahead. – krs013 May 25 '13 at 21:46
• The pure ballistic and collision physics are probably not the guiding factors here. Human bio-mechanics and the danger of the ball hitting rough ground and taking a bad bounce seem like more important factors. – Olin Lathrop May 25 '13 at 22:07

## 1 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).

• Hitting the dirt, even if it is nicely leveled, is far from ideal unless the ball is spinning just right. Otherwise, some of the kinetic energy of the ball moving linearly will be converted to spinning, slowing down the linear movement. Given that bouncing is bad because it will almost certainly slow down the horizontal speed, a slight up angled throw is a no-brainer. You still get Cos of the up angle horizontal component, which is basically 1 for small angles. Put another way, a small up-angle is basically free compared to pure horizontal, so well worth it if it avoids a bounce. – Olin Lathrop May 25 '13 at 22:01