The Pitch
Graphing Calculator Activity

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

Advanced Activity

Calculator skill -- basic graphing. You wouldn't take your calculator to the ballpark ... but you can bring the ballpark to your calculator!

Directions are based on the use of a TI83 graphing calculator.

The distance from the mound to the plate is measured in feet, the ball's speed is measured in miles per hour and most measurements in science classes are expressed in meters! We need to have one system of units to make sense of what we're doing, so let's decide to use meters for distance, meters per second (m/s) for speed, and meters per second per second (m/s²) for acceleration.

Let's graph the motion of the ball from the instant the pitcher releases it until it crosses the plate. The ball's drop is given by d = 1/2 at², where d is how far it drops from its starting height because of the acceleration due to gravity. (The initial velocity term often seen in this familiar equation is on vacation this time, because the vertical initial velocity is zero remember, we're supposing that the ball moves horizontally in the beginning!)

Since one-half of -9.8 equals -4.9, enter -4.9t² in your calculator's equation editor. The ball starts 2.0 m above the field, so add 2 at the end.

To set the graphing window, remember that x stands for time and y stands for distance above the field. Set x_min to zero, since we start watching just as the ball is released. The time to reach the plate is x_max, and since d = vt for the horizontal motion of the ball, t = d/v. From our assumptions above, that means you'd enter 18.4/42.5 for x_max. The calculator will instantly figure the value for you when you move to the next line. The x_scl tells how often to make tick marks along the x axis. You don't need to set the scale, but a value of .1 will make a mark every tenth of a second of the ball's flight. The level of the field is our reference, so y_min = 0 and y_max = 2.5 or so, so that we can see the ball's whole flight. For y_scl enter .25 so we can check the ball's position to the nearest quarter of a meter.

Here's the pitch! Press [GRAPH] and watch the pitch come in. Your view is from the third base side of the field; the pitcher is at the left edge and the batter at the right. Press [TRACE], hold down the right arrow and the cursor will follow the pitch. The bottom of the screen will show the time (x) and the height off the field (y).

Going Further

1. So far we have the basic arc of the ball, ignoring the Magnus force from spin the pitcher puts on the ball. The force can be as great as one quarter of the ball's weight and can act up, down or to either side.
   
   Want to throw a curve ball that drops more than the batter expects? How about a rising fast ball? Just add a second equation for the Magnus correction. The greatest acceleration that the Magnus force can create is one-fourth of 9.8 m/s², or 2.45 m/s². Since the motion equation requires you to take half of the acceleration, you could use any value from -1.225 for a sharply biting curve to 1.225 for a fastball with the maximum "pop." For an example, let's use -1 for a pretty good curve ball. Enter -x² for the second equation. Then for the third equation, add the first and second equations:
   
   Y3 = Y1 + Y2
   
   De-select Y2; move the cursor to the equals sign after Y2 and press [ENTER]. That way the calculator will graph Y1, the pitch with no spin, and Y3, the curve ball. Press [GRAPH] to see both pitches.
   
   How different are they? Doesn't look like much, but it's sure enough to matter! Press [TRACE] and hold down the right arrow to follow the pitch. At any time, press the up or down arrows to switch between the two graphs. Notice the numbers by the Y coordinate; the curve ball drops nearly 20 centimeters below where the batter would expect the ball! There's a strike, or at least a grounder!

2. If you'd prefer a fast ball, change the value in equation Y2. Let's go with the maximum upward Magnus force enter 1.225x² for Y2. (Remember to de-select it again by moving the cursor to the equals sign and pressing [ENTER].) You can see that since the Magnus force is no greater than one quarter of the ball's weight, the fast ball can never really rise above where it was released but it does come in higher than the batter expects.

3. Draw a strike zone! For a medium-sized batter, the strike zone starts about 0.50 m above the plate and reaches to about 1.35m. Enter these equations for Y4 and Y5 to graph that zone:
   
   Y4 = .5 (x>.4)
   
   Y5 = 1.35 (x>.4)
   
   To make the "greater than" sign, press [2nd][MATH]3. Set both of these graphs to dot mode; to do that, move the cursor all the way over to the left margin in front of Y4 and then press [ENTER] six times. Do the same for Y5. (What's the .4? Remember, for the calculator, x represents time, so the horizontal bars made by these graphs only appear after the ball has traveled for 0.4 second, when it's near the batter.)
   
   Your calculator is probably set to plot the graphs one after the other, so you'll see the first graph of the ball with no spin, then the ball with spin, and then the strike zone. It is more fun to plot all the graphs at once. To do that, press [MODE] then move the cursor down to Simul and press [ENTER], then [2^nd][MODE] to quit. Then press [GRAPH] for a different kind of show! The strike zone will appear just as the ball gets there.