The Flight
Graphing Calculator Activity

To Bounce or Not to Bounce
Projectile Range Activity
Graphing Calculator Activity

Advanced Activity

Calculator skills graphing parametric equations

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

In most graphs, the x axis (the domain) determines the value of the y axis (the range) through a function of some kind. Or we can say, x tells y what to be. For example, if y = 2x, then whatever x is, y is twice that much. If instead y = x +5, then whatever x is, y is that plus 5.

In this lesson you'll make a different kind of graph, where a third, hidden variable tells both x and y what to be. (Think of a puppeteer above the stage controlling both the arms and legs of a puppet. Or think of the wizard in The Wizard of Oz, running things from behind a curtain.) There's nothing sinister about these graphs, though -- time (t) is the variable that will be controlling x (the distance the ball moves across the field) and y (its distance above the field). When two equations depend on a third variable for their values, they are called parametric equations.

Let's start from the simple case where we ignore air friction and the spin of the ball. After the ball leaves the bat it has two separate motions: out across the field (the x direction) and up and down above the field (the y direction). We call those separate motions the components of the ball's motion. Our first step is to break down the ball's initial velocity (v_o) into its two components, v_x and v_y. Their magnitudes or sizes are given by

v_{x =} v_o cos 2

v_{y =} v_o sin 2

where 2 is the angle above the field at which the ball leaps off the bat.

The key idea is that gravity always pulls straight down, so the y component is accelerated motion but the x component is constant, steady, uniform motion. The distance away from the plate (x) and the distance above the field (y) are found by

x = v_o cos 2 t

y = v_o sin 2 t + _ at²

OK! So how do you make your calculator do that? First, be sure it's set to degrees, parametric mode and sequential graphing. Click the mode button next to the 2nd key.

Next, let's draw the outfield fence. Since the fence doesn't move, for all values of t it's 106.7 m away in our example. So enter 106.7 for equation X_1T. To make a vertical line for the fence, enter the height of the fence, 3.05 m, minus time.

Press [WINDOW] to set the graph viewing rectangle. Set T_min, x_min, and y_min to zero. T_max is the total time the ball is in the air. Try about 3 seconds. T_step determines the plotting speed and resolution of the calculator. The smaller the number, the more precise your results, but the longer it will take to plot. The calculator runs faster with a larger number but the graph will fail to represent the actual path of the ball. A good time step for this example is about 0.05 second, so set T_step = .05. x_max is the total length of the field. Make it a little bigger than the distance to the fence, or about 110 m. Finally, y_max is the distance above the field that we wish to view. Let's try 50 m for this example.

Press [GRAPH] and see if you have a little home run fence in the far corner of the screen.

Now let's draw the hit and see how it goes! Enter these equations as the second set in your equation editor:

X_2T = 49.1 cos (15)T

Y_2T = 49.1 sin (15)T - 4.9T² + 1

Look back to the original numbers for this example. Do you see where they fit in these equations?

Press [GRAPH], make a sound like the crack of a bat if you want to add some realism and watch the ball fly! It's outta there!!

Just how good a homer was that? Press [TRACE] and cursor down to choose equation set 2. Then cursor right to follow the ball's path. When x is close to the distance of the outfield fence (106.7 m), check the value of y and see how much bigger it is than the height of the fence, 3.05 m. You can also check the value of T to see how long the ball was in the air before it left the park.

Going Further

1. Try other values of initial speed and angle; just be sure to change the values in both the x and y equations. It's amazing how little you have to change either variable to make a homer into a catchable ball, or vice versa. Baseball is a game of inches!

2. Add the drag force, the retarding force caused by air resistance. At line drive speeds, drag approximately equals the weight of the ball, a force large enough to radically change the ball's motion. It's convenient for us that the forces are about equal; all we should have to do is add a -4.9T² term to the x equation. The trouble is, that would assume that the drag force is always horizontal, but it's not; it's always pointing opposite the motion of the ball. It would be way beyond the scope of these lessons to write equations that reflect the way air drag slows the ball in both the horizontal and vertical directions. You can get an approximate solution by changing your equations to look like these:

   X_2T = 49.1 cos (15)T - 2.6T²

   Y_2T = 49.1 sin (15)T - 7.5T² +1

Changes are based on curves of ball trajectories in The Physics of Baseball by Robert Adair. The x equation is fairly accurate, as air drag increasingly reduces the ball's horizontal velocity the longer the ball flies. The y equation is fairly accurate while the ball is rising because the air drag effectively reduces the ball's vertical velocity as if gravitation were stronger. However, after the ball reaches its peak, air drag tends to oppose gravity and should reduce the value of -7.5 to something smaller than -4.9.

Press [GRAPH] to run the play again. What a difference! The ball that was an easy home run, without air friction, is now just a routine fly ball. How far does it travel? When does it hit the ground, if no one happens to catch it? Use [TRACE] to find those answers.

Remember that the width of the window is 110 m, but the height is 50 m, so the graph is not exactly to scale. More importantly, remember that air drag is quite complicated and is impossible to represent accurately here.