<--
.. title: Physics of Baseball: Batting
.. date: 2010-05-19 18:21:00
.. tags: order of magnitude, fun, baseball
.. category: old
.. slug: physics-of-baseball-batting
.. author: Alemi
.. has_math: true
-->
[![image](http://4.bp.blogspot.com/_YOjDhtygcuA/S_RkXXs4DpI/AAAAAAAAAKo/jPSgwpl4qHA/s320/baseball.jpg)](http://4.bp.blogspot.com/_YOjDhtygcuA/S_RkXXs4DpI/AAAAAAAAAKo/jPSgwpl4qHA/s1600/baseball.jpg)
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 $$ A_S = (17") \times (0.7 m) = 0.3 \text{
m}^2 $$ when you swing, how much of this area does the bat take up?
Well, treat it as a cylinder, with a diameter of 10 cm, and assume it
runs the length of the strike zone, when the area that the bat takes up
is $$ A_B = (10\text{ cm}) \times (17" ) = 0.043 \text{ m}^2 $$ So
that the fractional area that the bat takes up during our idealized
swing is $$ \frac{A_B}{A_S} \approx 14\% $$ So already, if our
robot is guessing where inside the strike zone to place the bat, and
doing so randomly, assuming the pitch is a strike to begin with, it will
be able to bunt successfully about 14% of the time.
### 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 $$ t = \frac{ 60.5 \text{ ft} }{ 90
\text{ mph} } \approx \frac{1}{2} \text{ second} $$ i.e. from the
time the pitcher releases the ball to the time it crosses home plate is
only about half a second. Compare this with human reaction times. My
drivers ed course told me that human reaction times are typically a
third of a second or so. So, baseball happens quick! Alright, but we
were interested in how well you have to time your swing. Successfully
hitting the ball means that you've made contact with the ball such that
it lands somewhere in the field. I.e. you've got a 90 degree play in
when you make contact. How does this translate to time? We would need to
know how fast you swing.
#### 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 $$ v_x t = 300 \text{ ft} $$
where t is the time the ball is in the air, similarly we know that it is
gravity that makes the ball fall, and so as far the vertical motion is
concerned, in half the total flight time, we need the vertical velocity
to go from its initial value to zero, i.e. $$ g \frac{t}{2} = v_y $$
where g is the acceleration due to gravity. Furthermore, I'm assuming
that I am launching this projectile at a 45 degree angle, for which I
know from trig that $$ v_x = v_y = \frac{v}{\sqrt 2} $$ So I can
stick these equations into one another and solve for the velocity needed
to get the ball going 300 feet. $$ \frac{v^2}{ g} = 300 \text{ ft} =
\frac{ v^2}{ 9.8 \text{ m/s}^2 } $$ $$ v \approx 30 \text{ m/s}
\sim 67 \text{ mph}$$ So it looks like the ball needs to leave the bat
going about 70 mph in order to clear the park. ( This was of course
neglecting air resistance, which ought to be important for baseballs ).
Great that tells us how fast the ball needs to be going when it leaves
the bat, but how fast was the bat going in order to get the ball going
that fast? Well, lets work worst case and assume that the baseball - bat
interaction is inelastic. I.e. I reckon that if I throw a baseball at
about 100 mph towards a wooden wall, it doesn't bounce a whole lot. In
that case, the bat needs to take the ball from coming at it at 90 mph to
leaving at 70 mph or so, i.e. the place where the ball hits the bat
needs to be going at about 160 mph. That seems fast, but when you think
about it, if a pitcher can pitch a ball at 90 mph, that means their hand
is moving at 90 mph during the last bits of the pitch, so you expect
that a batter can move their hands about that fast, and we have the
added advantage of the bat being a lever.
#### 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. $$ v = \omega r $$ $$ \omega = \frac{ 160 \text{ mph} }{ 0.75
\text{ m} } \approx 100 \text{ Hz} $$ So, if we need to have a 90
degree resolution in our swing timing to hit the ball in the park, this
means that if our swing near the end is happening at 100 \text{ Hz}, we
need to get the timing down to within $$ t = \frac{ 90 \text{
degree}}{ 100 \text{ Hz} } \sim 15 \text{ ms} $$ So we need to get
the timing of our swing to within about 15 milliseconds to land the hit.
So if our robot randomly swung at some point during the duration of the
pitch, it would only hit with probability $$ \frac{\text{time to land
hit} }{ \text{time of pitch}} = \frac{ 15 \text{ ms}}{500 \text{
ms}} \sim 3\% $$ or only 3% of the time. If we take both the spatial
placement, and timing of the swing as independent, the probability that
our robot gets a hit would be something about $$ p = 0.03 \times 0.14 =
0.004 = 0.4 \% $$ or our robot would only get a hit 1 time out of 250
tries. Suddenly hitting looks pretty impressive.
### 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 $$ p = \frac{ 15}{100} = 15
\%. $$ So that now we obtain an overall hit probability of $$ p = 0.15
* 0.14 = 0.021 = 2 \% $$ So that it seems like a poor baseball player,
more or less randomly swinging should have a batting average of about
2\%. Compare this with typical baseball batting averages of 250 or so,
denoting 0.25 or 25% probability of a hit. I think this is a much better
estimate of how much better over random baseball players can do with
training. So it looks like practice can improve your ability to do a
task by about an order of magnitude or so. Either way, baseball is
pretty darn impressive when you think about it.