The Pitch
Pitch Trajectory Activity

Drag Simulation Activity
The Magnus Force Activity
Pitch Trajectory Activity
Graphing Calculator Activity

Overview of Lesson

In this exercise students will construct a graph of the trajectory of a pitched ball using the equations of kinematics. The graph will have multiple plots representing a ball thrown with different initial conditions.

Goal

Students will apply their understanding of two-dimensional kinematics, gravity, air drag and the Magnus force to make a meaningful graphical representation of a real-world event.

Objectives

• The students will apply the equations of kinematics to a two-dimensional motion situation.

• The students will apply Newton's laws of motion to an object subjected to multiple forces.

• The students will calculate and plot a baseball's trajectory for several different sets of starting conditions.

• The students will predict the effect of changing variables.

• The students will discuss the limitations of their graphical representation.

Ohio Academic Content Standards

Materials

• Graph paper: use at least 11-by-17-inch or chart-sized paper if it can be found

• Assorted colored pens or pencils

• Calculators

• Rulers

• French curves (optional)

Procedure

1. Give each student or small group of students a sheet of graph paper, and tell them that you want the graph to represent the side view of a pitched baseball in a big league park. See Appendix A for graph paper.

2. Give them as much or as little help as they need setting up the axes of the graph. The x-axis should represent about 70 feet to accommodate the distance from the pitcher's mound to home plate (60 feet 6 inches) and the y-axis should represent about 8 feet to accommodate the release point of the pitch to be about 6 feet above the ground. These distances may be given to the students or they can do research on their own to find them. In addition they might want to draw in the dimensions of the strike zone (bottom edge at 19 inches and the top edge at 45 inches above the ground for the average 6-foot-tall batter), and sketch in the silhouettes of the pitcher and catcher.
   
   Note: It is best to stay in the English unit system for this entire exercise because all of the measurements common in baseball are in feet and inches.

3. Graph plot #1: (no forces acting on the ball)
   Have the students plot a straight horizontal line at the 6-foot mark on the y-axis. This represents a ball thrown with an initial velocity in the horizontal direction without the influence of any forces. It will be used as a reference line as the rest of the graph is plotted.

4. Graph plot #2: (gravity only)
   Given that the ball is released with a velocity of 90 mph in the horizontal direction from an initial height of 6 feet, have students calculate the vertical position of the ball every 5 feet along the way from the mound to home plate if gravity is the only force acting on the ball. This is done using the equations of kinematics for uniformly accelerated motion.
   
   Method: Use the horizontal velocity (constant 90 mph) and horizontal distance to determine the elapsed time at each of the 5-foot horizontal intervals.
   
   Using the zero gravity line as a reference and the elapsed times, determine the vertical distance the ball would have fallen from that line at each of the 5-foot intervals along the way. With a pen of a different color, draw a smooth curve through the points you have plotted.

5. Graph plot #3: (gravity and air drag)
   In this plot we introduce the force of air drag. Air drag acts in the horizontal direction, slowing the ball down as it moves toward the plate. Have students recall the equation that is used to calculate the force of air drag. Explain that because of its complexity we will not use this formula to calculate the air drag. Instead, we will use a ballpark value for the deceleration of the ball that holds true to this relationship when it is applied using the aerodynamics of a baseball and average atmospheric conditions. The ballpark value for the deceleration that we will use is .5 g. That is, it will act as a horizontal deceleration with a magnitude of 16 ft/s².
   
   Method: The method to make the plot is similar to the previous one but the calculations are a bit more tricky as the motion now includes an acceleration in both dimensions. Begin by using the equations of kinematics to determine the elapsed time when the ball has moved 5 feet horizontally toward the plate; repeat this for each additional 5-foot interval. Using the elapsed times and the zero gravity reference line, determine how far the ball would have fallen from the reference line at each of the intervals. With a pen of yet another color, draw a smooth curve through the points you have plotted.

6. Graph plot #4: (the curveball: gravity, air drag and top spin)
   In this plot we introduce an additional acceleration in the downward direction caused by the Magnus force. Have students recall the equation used to calculate the Magnus force. Again explain that because of its complexity we will not use this formula to calculate the Magnus force. Instead, we will use a ballpark value for the acceleration of the ball that holds true to this relationship when it is applied using the mass and aerodynamics of a baseball, average atmospheric conditions and a spin velocity comparable to a curve thrown by a major leaguer. The value we will use will again be .5 g, so the downward acceleration becomes 1.5g or 48 ft/s².
   
   Method: The elapsed times calculated for plot #3 are used again for this plot as the horizontal motion of the object is not affected by the vertically acting Magnus force. Using the elapsed times from plot #3 and the zero gravity reference line, determine how far the ball would have fallen from the reference line at each 5-foot interval. With a pen of yet another color, draw a smooth curve through the points you have plotted.

7. Graph plot #5: (the rising fastball: gravity, air drag and back spin)
   This plot is nearly identical to that of the curveball (plot #4) except that the Magnus force acts in the upward direction. Therefore, the downward acceleration of the ball is reduced by a factor of .5 g, making it 16ft/s².

   Method: Again the horizontal motion is not affected by the Magnus force so the elapsed-time values from plot #3 may be used. Using the elapsed times from plot #3 and the zero gravity reference line, determine how far the ball would have fallen from the reference line at each 5-foot interval. With a pen of yet another color, draw a smooth curve through the points you have plotted.

8. Follow Up: After completion of all five plots on the graph, have students add a legend with a color code and any other labels needed. The graph can then be a reference when discussing age-old baseball questions like:

   Does a curveball really curve?

   Does a rising fastball really rise?

   Does a curveball break sharply in the last few feet before it reaches the plate? (Players describe the ball as "falling off the table.")

   Another part of the follow-up discussion should be about the simplifications and generalizations that were used in making the graph. Important things to point out are:

   The starting velocity of a real pitch is not likely to be purely horizontal. Most pitches are thrown with a downward angle.

   The starting velocity of a breaking pitch like a curve ball is more likely to be in the 75-80 mi/hr range.

   Air drag will act on the vertical motion of the object as well, although its magnitude is small in comparison to the horizontal drag.

The values for both the air drag and Magnus forces are approximate and derived using a drag coefficient that represents an average orientation of the seams on the ball and average atmospheric conditions.

Evaluation

The evaluation for this activity is the completion, accuracy and quality of the graph. See Appendix B to evaluate graph.

Note

An old-fashioned paper-and-pencil approach to making this graph seems to be superior to using graphing software or graphing calculators. There is something about the extra time invested in making it from scratch that gives students more time to ponder the concepts. Students seem to develop a sense of accomplishment when they can hang the final product on the wall.