MA2051 ODEs Project 2 B Term, 1995

Uncle Charlie

This project concerns why a curve ball curves. Along with the more obvious forces that act on a baseball in flight, there is an additional force due to the spinning of the ball. This force is given by the Kutta-Zhukovskii Theorem.

1. Please set up a model for the horizontal and vertical components for the motion of the ball. Take into account the initial velocity of the ball, gravity, air resistance (friction, drag), as well as the force due to spin. This means you will need to look up the Kutta-Zhukovskii Theorem. For convenience, you may assume that the axis of rotation for the ball is vertical and that the rotation rate is constant. As far as I know, no pitcher can actually throw a curve ball with a vertical axis of rotation, but this would be the ideal case for horizontal moment. You may consider other forces as well (for example, lift), but keep in mind that they will complicate the model, and they are not required for this project.

2. Please use Maple (or your favorite software package) to plot a few possible trajectories for a baseball. You may solve the model equations by hand and use the computer only for plotting, or you may do everything numerically. Be sure to label your plot(s), and give units for time and coefficients.

Here are a few of the facts you will need (you will need to find others on your own).

1. The distance from the pitching rubber to home plate is 60 ft, 6 in.
2. A baseball weighs about 5 ounces and has roughly a 2.8 in diameter.
3. The strike zone is above the plate and between the batter's arm pits and knees when he/she is standing in a normal stance. Alternatively, it is wherever the umpire says it is.
4. A good major league curve ball travels at about 80 -- 85 mph.
5. One of the most difficult things to deal with in this problem is the friction force (drag force). It depends on many parameters (temperature, humidity, etc.). For the purposes of this project, you may assume that it is given as where is the force of friction (in pounds) and v is the velocity (in ft/sec). This assumption is convenient but somewhat unrealistic; for a baseball, the force of friction is actually proportional to , not v. You may consider this more realistic assumption, but this is not required for the project.
6. Pitcher's mound is about 10 in above the plane of the field.
7. Home plate is 17 in wide.
8. There is an interesting reference on reserve in the library: Keep Your Eye on the Ball by Robert G. Watts and A. Terry Bahill (QC 26 W38 1990). But keep in mind that this book does much more than what you are asked to do for this project.
9. The New York Yankees should be forcibly removed from the control of their present owner, then relocated to Toledo.

As on the first project, you will be graded on how well you write (spelling and grammar), as well what you say (math content). Be sure to give an introduction to the problem, then explore the issues mentioned above in a short essay that shows understanding of the material. On the other hand, don't be long-winded; in general, short, accurate answers are best. Also don't submit many solution plots with little or no explanation; one, two or three well-chosen and carefully discussed plots will be clear. Do not attach copies of Maple worksheets.

First Draft Due: Thursday, 7 December 1995

Final Draft Due: Wednesday, 13 December 1995