Differential Equations - Project #2
Due: December 13, 1995

Introduction:

In this project we will attempt to explain, mathematically, the reason why a curve ball really curves. There are many different theories and beliefs, but the one that has gained the most respect is based on the Kutta-Zhukovskii Theorem. This theorem says that if an object is spinning through a fluid, and there is a net circulation of the fluid around the object, there results a force mutually perpendicular to the direction of the motion of the object and the direction of axis of spin. There has long been the question of whether the amount of curve depends on velocity(V) or velocity squared (V2). Wind tunnel results conclude that it is dependent on V. We will set up a system of equations that describe the motion of the baseball, in three dimensions, as it travels from the pitchers mound to home plate; taking into account initial velocity of the ball, gravity, air resistance, and force due to spin. To further complicate these equations we could also consider the density of the air and its effect on the spin-force and also lift.

Section 1:

To write these equations of motion, we must take into account Newton's Second Law of Motion, which says that the sum of forces acting on an object equals the mass of the object multiplied by it's acceleration. The first thing we have to consider when writing equations of motion is the velocity of the ball. A typical major league curve ball travels at 80-85 MPH. For this project our values for two ideal pitches are 85 and 89 MPH. With these initial velocities the time it takes for the pitch to make it from the pitchers mound to home-plate is 7/10 of a second (.7s), traveling a distance of 60.5 ft. From the book, "Keep your Eye on the Ball" by Robert G Watts and A. Terry Bahill, we used 2.2 ft as the maximum y-span, which makes the Y velocity a maximum of 3.14 ft/sec. Next we are given that the air resistance is .006v where v the velocity. The lateral force on the ball is given as .0013v. This number is derived from the wind tunnel experiments' equation F(l)=6.4*10-7wv, where w is the rotational velocity in revolutions per minute, and v is the velocity in ft/sec. By substituting w=2000rev/min we get .0013v. This w assumes that the ball has been thrown with a vertical axis of rotation, but in reality this is not easily accomplished.

The equations of motion of the baseball are:

The variables are defined as follows: x is the distance from the pitchers mound to home plate, y is the horizontal displacement of the ball, and z is the height of the ball above the ground. One of these variables could be eliminated because of it's minimal force on the ball as compared to the other factors. The -.0013y' can be left out because if you substitute the maximum y velocity into -.0013y' we get .004 compared to the .72 lbs of force, this would only account for .5% of the force on the ball.

Section 2:

We used the spacecurve command in Maple to plot a good representation of this model. All of our units are in feet (ft) and seconds(s). To use this command we had to first, solve the system of equations for T. Then, by using these solutions we were able to get a plot of the curve. Building on this plot, we could modify the variables to get the best possible pitch. We assumed a right handed pitchers' throw and decided that the best possible throw would end up at the bottom-left of the strike-zone, which would be (0,-.708,2)* in our coordinate axes. Another possible curve ball that would result in a strike would be one thrown to the top-right hand side, viewed from the pitchers mound. This would be represented as (0,.708,5)* Both of these ideal curve-balls are represented below in our plots. For the first curve-ball our variables are as follows:

For the second pitch we used:

Comments:

http://www.kluge.net/~felicity/proj/deq2.html