# Physics of Baseball: Batting

Summer is upon us, and so that means that we here at the Virtuosi have started talking about baseball. In fact, Corky and I did some simple calculations that illuminate just how impressive batting in baseball can be. We were interested in just how hard it is to hit a pitch with the bat. So we thought we’d model hitting the ball with a rather simple approximation of a robot swinging a cylindrical bat, horizontally with some rotational speed and at a random height. The question then becomes, if the robot chooses a random height and a random time to swing, what are the chances that it gets a hit?

### Spatial Resolution

So the first thing to consider is how much of the strike zone the bat
takes up. In order to be a strike, the ball needs to be over home plate,
which is 17 inches wide, and between the knees and logo on the batters
jersey. Estimating this height as 0.7 m or 28 inches or so, we have the
area of the strike zone

### Time Resolution

But getting a hit on a swing is different than getting a bunt. Not only
do you have to have your bat at the right height, but you need to time
the swing correctly. Lets first look at how much time we are dealing
with here. Most major league pitchers pitch the ball at about 90 mph or
so. The pitchers mound is 60.5 feet away from home base. This means that
the pitch is in the air for

#### Estimating the speed of a swing

I don’t know how fast you can swing a baseball bat, but I can estimate
it. I know that if you land your swing just right, you have a pretty
good shot at a home run. Fields are typically 300 feet long. So, I can
ask, if I launch a projectile at a 45 degree angle, how fast does it
need to be going in order to make it 300 feet. Well, we can solve this
projectile problem if we remember some of our introductory physics. We
decouple the horizontal and vertical motions of the ball, the ball
travels horizontally 300 feet, so we know

#### Coming back to timing

So, we have an estimate for how fast the bat is going. Knowing this and
estimating the length between the sweet spot and the pivot point of the
bat to be about 0.75 m or so, we can obtain the angular frequency of the
bat.

### Experiment

Saying that the robot swings at some random time during the duration of
the pitch is pretty bad. So I decided to do a little experiment to see
how good people are at judging times on half second scales. I had some
friends of mine start a stop watch and while looking try to stop it as
close as they could at the half second mark. Collecting their
deviations, I obtained a standard deviation of about 41 milliseconds,
which suggests a window of about 100 milliseconds over which people can
reliably judge half second intervals. Now, I have to admit, this wasn’t
done in any very rigorous sort of way, I had them do this while walking
to dinner, but it ought to give a rough estimate of the relevant time
scale for landing a hit. So instead of comparing our 15 millisecond ‘get
a hit’ window to the full half second pitch duration, lets compare it
instead to the 100 ‘humans trying to judge when to hit’ window. This
gives us a temporal resolution of about

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