Advanced Activity
Calculator skills curve fitting and simultaneous equations
Directions are based on the use of a TI83 graphing calculator.
| Starting points: |
- Distance from home plate to first base = 90 ft
- 90 ft (1 m/3.28 ft) = 27.4 m
- Time for a ground ball to reach third base = 1.2 second
- Time for the third baseman to field and throw = 1.0 second
- Distance from third to first = 127 ft = 38.7 m
- Speed of an infielder's throw to first = 80 mph
- 80 mph (1 m/s / 2.24 mph) = 35.7 m/s
- Motion of a base runner to first base -- as shown
(From The Physics of Baseball, Robert Adair)
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Runner?s position vs. time
Starting from rest
A graphing calculator takes an equation and produces a graph. But here we have a graph from an authoritative book on baseball, and we can use the calculator to go backwards and make an equation from the graph. Then we can have some fun with it!
Any time you need to find an equation that fits some data, press the [STAT] key and [ENTER] to edit your data lists. In list L1, enter the time values, from zero through 5, by steps of .5. In list L2, enter your best guess of the distance the base runner has moved by that time. So, your first value in L2 will be zero. How far has the runner moved in the first half second? Looks like about 1 m, so enter that as the next item in list L2. After 1 second, it looks like the runner is now 3 m away. Enter that number and keep going, making your best estimate from the graph for the distance reached at each time. Be sure that you have 11 items in each list.
The calculator can use about 10 different kinds of equations to compare with your data for the best fit. To see how well any test curve fits, turn on Diagnostics. Go to CATALOG; that's [2nd][zero]. Cursor down to Diagnostics On and press [ENTER].
You can see that the curve is not a straight line, so a linear fit is probably not the best. Let's try a quadratic curve. Press [STAT] and move the cursor to CALC, then down to 5 and press [ENTER] two times. The screen will quickly display the values for the equation that best fits your data points. Using data from the graph above, you might have come up with this equation, rounding off to three significant figures:
y = .953x2 + 2.48x -.266
Your numbers might be a little different, depending on how you estimated the distance the runner traveled at each time.
Notice the value of R2, which tells how well the equation fits the points. A value of 1.00 would be a perfect fit, and anything close to that value means a good fit. In general, when you explore some data set, you could try many different kinds of curves and the one with the highest R value fits the data the best.
To plot the graph, first set the graphing mode of your calculator. Press [MODE] and move to Func, press [ENTER], and then Simul and [ENTER] again. That sets your calculator to plot ordinary functions and to plot them simultaneously. Press [2nd][MODE] to go on.
Since first base is the goal that both the batter and the thrown ball will try to reach, let's use the x axis, where y = 0, to represent the first base bag. Press [Y=] and enter the equation as shown above, but include one more term for the distance from home plate to first:
y1 = .953x2 + 2.48x - .266 - 27.4
You could, of course, combine those last two terms into one. Press [WINDOW] to set the viewing rectangle. Since x stands for time, graph from zero to 5 seconds with one-second intervals. Y represents the distance "behind" first base, so plot from about -30 to 5 meters or so with a scale of, say, 5 meters. Press [GRAPH] to see your work! It should, of course, look just like the one pictured above, but with the x axis near the top of the graph. You have digitized the graph and now can use it!
How much time will the runner need to reach first base safely? You can press [TRACE] and move along until the cursor is near the intersection. Or go to CALC that's [2nd][TRACE] and choose menu option 2 to find the zero or root of the function. Press [ENTER] with the cursor anywhere below the x axis, and then press the cursor-right key until the cursor pops anywhere above the axis and press [ENTER] twice more. The x value on the screen shows that the runner will arrive in about 4.25 seconds.
Going Further
What matters in the game is not how long it takes the runner to reach first, but if he or she gets there before or after the ball! Let's suppose the batter hits a grounder down the third base line. The third baseman fields the ball cleanly, wheels and throws to first. Does the runner beat the throw?
To find out, check the assumptions at the top of the page. Suppose the ball needs 1.2 seconds to reach third base and the third baseman needs 1.0 second to field the ball and make the throw to first at 80 mph or 35.7 m/s. Ignoring air friction for the throw, its motion is uniform and not accelerated, so the equation for the ball's motion is:
y2 = 35.7(x - 2.2) -38.7
The 35.7 is the ball's velocity in m/s, the 2.2 is the time after the batter makes contact until the third baseman releases the ball and the 38.7 m is how far the ball must travel from third to first. Again, it's negative because we are using the x axis to represent first base.
Imagine the crack of the bat when you press [GRAPH]. You'll see the runner hustling toward the safety of the axis before the graph of the ball appears. Then suddenly, there's the ball, too, moving so fast, and it reaches the axis well before the batter's graph hits the axis. It's an easy out! When does the ball arrive at the base? Again, use [2nd][TRACE], menu option 2, to see.
You'll need to press the cursor-down key once to switch to the second graph, the graph of the ball. Press [ENTER] with the cursor on the ball's graph but anywhere below the axis, then cursor-right until the ball moves above the axis, and press [ENTER] twice more. Only about 3.25 seconds! You can see why routine ground balls turn into easy outs. But if anything happens to cost the third baseman just one additional second -- chasing a slow roller, having to dive for the ball and standing back up, losing the ball in the glove or bobbling the ball before the throw -- then it's a close call.
Change equation 2 to graph that new situation, where the third baseman takes one extra second to field the ball. Your equation would read:
y2 = 35.7(x - 3.2) -38.7
When you graph the motions this time -- it looks too close to call! Find the runner's time to reach first once again and now press cursor-down one time to switch to the ball's graph. It still has over a meter to travel before it's in the first baseman's glove, and any umpire could see that difference! Baseball is a game of inches, and that's why you should run out every ground ball.
To get an ump's-eye view, change the WINDOW. Try graphing x from 4 to 4.5 seconds, and y from -5 to 1 meter or so. Now it's easy to see that runner reaches the safety of the x axis before the ball arrives. How much faster would the third baseman have to throw to still get the runner out? Use a larger speed in equation 2 to see how a faster throw could compensate for a slight bobble.
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