(Or, how does a girl get a hit playing coed Slow Pitch softball?)

Hello!!!  My name is Nica Calderon, and I'm am a senior at California Lutheran University . I will be getting my BA in Mathematics in May, and for my Senior Mathematical Modeling Project, I decided to find out what it takes for a girl to get a hit in a slow pitch softball game.  I have been playing softball for 15 years, and the last five I have been playing slow pitch 'cause I'm too old and not good enough to play fast pitch anymore.  I still love to play, and I love it even more when my batting average is high! 

There are many things to take into consideration when dealing with this problem.  For starters, the rules of the game affect where a girl can hit.  Slow pitch softball allows for there to be 10 players on the field, meaning you have 4 outfielders, but you can play with as little as 8 players to avoid a forfeit.  There is a "line" in the outfield which is 175 feet from home. This line is not clearly marked, but it is there nonetheless.  When a girl is batting, one of the outfielders can play in front of this line (this fielder is called the rover), but the other three must play behind it until the ball is hit.  If there are less than 10 players on the field then all the outfielders must play behind the line.  The person playing rover must be a girl!!

The next thing to take a look at is the flight of the ball.  The basic equation that yields the range of an object (in this case, a ball) in projectile motion is
R = (v2/g)(sin2q)
where v is the velocity of the ball, g is the force of gravity which is -32 ft/sec, and q is the angle at which the ball is projected, measured against the horizontal (the ground).  But, THIS EQUATION DOES NOT INCLUDE THE FORCE OF AIR FRICTION!!! However, we'll take a look at the graphs of some of these flight patterns anyway.  Click here to see those graphs

Now we want to know what the flight of the ball will look like with friction........


Now, in order to hit the ball into the outfield, and about 160 feet from home we need a new equation that uses air friction.  This equation is
R = [(v2/g)(sin2q)] - [a(8/3)(sin2qcosq)]
where a is a value that can change, depending on the b in the air friction equation, and everything else is the same as the previous equation.  A convenient number to use is 1/10.  Before we look at the distances that this equation yields, we have to make some assumptions first:

1.  The ball is always hit on the "sweet spot"  of the bat.  For those of you who don't play softball, this is smack dab in the middle of the fat part of the bat.  Its not always easy to do, but without this assumption things get much more complicated.
2.  The velocity of the ball off the bat is three times that of the swing.  This is not magic, just momentum conservation.

Now we're ready to take a look at how far the ball will go!
Here's a chart with those ranges!

Now that we know exactly what it takes to get a hit, we can all start batting .1000, right?  Okay, maybe not.  If it was that easy then the game wouldn't be as much fun.  So what keeps us from doing it???
1.  Hitting the ball on the sweet spot of the bat is not always easy, even if you have great hand eye coordination.
2.  Not everyone plays with the intention of "just getting on base".  People are usually trying for home runs, which takes you mind off the fundamentals, and in turn reduces the likelihood that you'll get a hit, especially not a home run!!!
3.  We're not perfect.  Even Mark McGuire doesn't get a hit every time!