Which Team Batting Statistic Predicts Run Production Best?

Objective

The objective of this experiment is to use correlation analysis to determine which team batting statistic is the best predictor of a baseball team's run-scoring ability.

Introduction

Baseball is an interesting combination of individual and team effort. For example, there is the one-on-one duel of pitcher against batter. But once the batter reaches base, he needs his teammates to follow with hits (or "productive outs") in order to move him up the bases so that he can score. From the scientific side, an interesting aspect of baseball is the rich trove of statistics on nearly every aspect of the game.

In this project, you will learn about correlation analysis, a statistical method for quantifying the relationship between two variables. As an example, consider as our two variables the age and height of male students in an elementary school. In general, individuals in this age range grow taller every year. If we made a scatterplot with height as our y-axis and age as our x-axis, we would expect the data points to show a consistent upward trend, with height increasing steadily along with age. The graph below shows simulated data (based on average growth charts).

In this case, the two variables are strongly correlated. As one increases, so does the other.

As a second example, suppose that we graph height as a function of birth month instead of age. Would you expect to find a correlation? Here is the same simulated height data, graphed now as a function of birth month (randomly assigned).

Our scatterplot is now a random arrangement of dots, with no apparent relationship. In this case, the two variables are not correlated.

To convince you that it is the same data, here is the same graph, with the different age groups (shown by grade level, K–6) each assigned a different symbol. You can clearly see the difference in average height of the different grade levels.

The statistic that describes this relationship between two variables is the correlation coefficient, r (or, more formally, the "Pearson product-moment correlation coefficient"). It is a scale-independent measure of how two measures co-vary (change together). The correlation coefficient ranges between −1 and +1.

What do the values of the correlation coefficient mean? Well, the closer the correlation coefficient is to either +1 or −1, the more strongly the two variables are correlated. If the correlation coefficient is negative, the variables are inversely correlated (when one variable increases, the other decreases). If the correlation coefficient is positive, the variables are positively correlated (when one variable increases, the other increases also). How close to +1 or −1 does the correlation coefficient need to be in order for us to consider the correlation to be "strong"? A good method for deciding this is to calculate the square of the correlation coefficient (r ²) and then multiply by 100. This gives you the percent variance in common between the two variables (Rummel, 1976). Let's see what this means by calculating r ² over the range from 0 to +1. (Note: for the corresponding values of r between 0 and −1, r ² will be the same, since squaring a negative number results in a positive number.)

Interpreting the Correlation Coefficienct Using r ²

r	r 2	% variance in common
1.00	1.00	100
0.90	0.81	81
0.80	0.64	64
0.70	0.49	49
0.60	0.36	36
0.50	0.25	25
0.40	0.16	16
0.30	0.09	9
0.20	0.04	4
0.10	0.01	1
0.00	0.00	0

As you can see from the table, r ² decreases much more rapidly than r. When r = 0.9, r ² = 0.81, and the variables have 81% of their variance in common. When r = 0.7, that might seem like a fairly strong correlation, but r ² has fallen to 0.49. The variables now have just less than half of their variance in common. By the time r ² has fallen to 0.5, r ² = 0.25, so the variables have only one-fourth of their variance in common.

For our simulated height data, the correlation coefficient for height vs. age was 0.88, indicating that age and height share 77% of their variance in common. In other words, 77% of the "spread" (variance) of the height data is shared with the "spread" of the age data. For height vs. birth month, the correlation coefficient was 0.03, so, to two decimal places, r ² = 0.00. There is no correlation between the variables (as we suspected).

It is important to remember that correlation does not imply that one variable causes the other to vary. Correlation between two variables is a way of measuring the relationship between the variables, but correlation is silent about the cause of the relationship.

If the correlation coefficient is exactly ±1, then the two variables are perfectly correlated. This means that their relationship can be described by a linear equation, of the form:

y = mx + b .

You've probably seen this equation before, and you may remember that m is the slope of the line, and b is the y-intercept of the line (where the line crosses the y-axis). If two variables are strongly correlated, it is sometimes valuable to use the linear equation as a method for predicting the value of the independent variable when we know the value of the dependent variable. This method is called linear regression.

Let's look again at the scatterplot of simulated height vs. age for elementary school students. If we draw a "best fit" line through the points, our scatterplot looks like this:

A "best fit" line means the line that minimizes the distance between the line and all of the data points in the scatterplot. If you wanted to predict a boy's height, and all you knew was his age, using this line to make a prediction would be your best guess. A spreadsheet program (like Excel) can do this "best fit" calculation for you, and help you get started with making a graph of the data and the regression line. You can also make a graph of the "residuals," which shows the distance of each data point from the regression line. Here is an example of a residuals graph, again using our simulated height vs. age data:

The residuals plot essentially rotates the linear regression plot by 45°, making it easier to compare how the data points are distributed around the regression line. It is easier to make the comparisons when the regression line has a slope of zero. The vertical scale can also be expanded, since the data is now centered within the area of the graph. If you see patterns in the residuals plot, these are features of the data that are not explained by correlation between the two variables.

This project will use correlation analysis to determine which team batting statistic is the best predictor of a baseball team's run-scoring ability (Albert, 20003). In addition to standard batting statistics, you'll also use batter's runs average (BRA), total average (TA), and runs created (RC). Each of these is defined in the Experimental Procedure section, where you can learn how to program them in to a spreadsheet with a formula.

There are many possible variations to this project that could apply similar methods, or extend them further for a more advanced project. See the Variations section below for some ideas. No doubt you can also come up with your own. You can also check out the book on which this project is based, Teaching Statistics Using Baseball, by Jim Albert.

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• baseball batting statistics:
  • hits (H),
  • doubles (2B),
  • triples (3B),
  • walks (BB),
  • strikeouts (SO),
  • batting average (BA),
  • on-base percentage (OBP),
  • slugging percentage (SLG),
  • batter's runs average (BRA),
  • total average (TA),
  • runs created (RC).
• correlation coefficient (or Pearson product-moment correlation coefficient),
• linear regression.

Questions

• If you find a correlation between two variables in your data set, can you conclude that one of the variables causes the other to change in a predictable way?

Bibliography

• Batting statistics are defined here:
  Forman, S.L., 2006. "Batting Glossary," Baseball-Reference.com - Major League Statistics and Information [accessed March 3, 2006] http://www.baseball-reference.com/about/bat_glossary.shtml.
• Here are two starting points for your background research on statistics:
  
  • Wikipedia contributors, 2006. "Correlation," Wikipedia, The Free Encyclopedia [accessed March 3, 2006] http://en.wikipedia.org/w/index.php?title=Correlation&oldid=42516704.
  • Rummel, R.J., 1976. "Understanding Correlation, Chapter 4.3, Interpreting the Correlation: Correlation Squared" Department of Political Science, University of Hawaii [accessed March 6, 2006] http://www.mega.nu:8080/ampp/rummel/uc.htm#S4.3.
• The following sites are good sources for baseball statistics.
  • This project uses annual team batting statistics from baseball-reference.com:
    Forman, S.L., 2006. "League Index," Baseball-Reference.com - Major League Statistics and Information [accessed March 3, 2006] http://www.baseball-reference.com/leagues/.
  • Here is another site where you can download historical baseball statistics:
    Lahman, S., 2006. "The Baseball Archive," [accessed March 3, 2006] http://www.baseball1.com/.
• Here is an Excel tutorial to get you started using a spreadsheet program:
  James, B., date unknown. "Excel 101," University of South Dakota, [accessed March 3, 2006] http://www.usd.edu/trio/tut/excel/.
• If you'd like more ideas for exploring baseball statistics, check out the book this project is based on:
  Albert, Jim, 2003. Teaching Statistics Using Baseball. Washington, D.C.: The Mathematical Association of America.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• a computer with Internet access and a spreadsheet program (the example below uses Microsoft Excel (Office 2003 version), similar functionality is probably available in other spreadsheet programs),
• a printer.

Experimental Procedure

1. Do your background research so that you are knowledgeable about the terms, concepts and questions above.
2. If you are not familiar with using a spreadsheet program, be sure to take the time to go through the Excel tutorial listed in the Bibliography.
3. Here is a short version of the data analysis steps you'll be following in order to find which batting statistic correlates best with run production. Use the links to jump down to the detailed sections, below. Use your browser's "back" button to return to this brief list:
   1. Download the historical data and import into Excel. (Downloading and Importing Data)
   2. Format the data for statistical analysis. (Formatting the Data)
   3. Add derived statistical measures (RC, TA, BRA or any others of your choosing) for testing. (Adding Derived Statistical Measures)
   4. Use the spreadsheet's correlation and linear regression analyses to see how the various batting statistics correlate with runs scored. (Running Correlation and Linear Regression Analysis)
   5. Compare the results. (Comparing the Results)
4. If spreadsheets are something new for you, then the detailed explanations that follow should help. If you are already comfortable with using spreadsheets, then you should be in good shape on your own.

Downloading and Importing Data

1. Download annual team battings statistics from Baseball-Reference.com: http://www.baseball-reference.com/leagues/.
   1. The data is arranged by year, and by league—National (NL) or American (AL).
   2. Choose the year and click on the league link (AL or NL) to get the team statistics for that league and year.
   3. Click and drag your mouse to highlight the batting statistics table, then copy and paste the table into Notepad.
   4. Repeat steps b and c for the other league. You can put both tables into the same text file, just include a blank line in between them.
   5. Save the tables as a text file (use the extension ".txt").
   6. The "Glossary" link (visible just above the table of team batting stats) has explanations of the batting statistics and their abbreviations.
2. Import the saved batting data into your spreadsheet program. Here's how to do it in Excel:
   1. From the menu, select File/Open.... You'll see a dialog like the one below.

   2. At the bottom of the File Open dialog, under "Files of type:" use the drop-down list to select "Text Files (*.prn, *.txt, *.csv)".
   3. Navigate to the directory where you saved your batting data file, select it, and click "Open."
   4. Excel now takes you through the "Text Import Wizard," a series of three dialogs. The first dialog looks like this (team batting data for 2005 shown):

   5. Make sure "Fixed width" file type is selected (as above), then click "Next."
   6. The second Text Import Wizard dialog is used to set the field widths. It looks like this:

   7. The lines with arrows show where Excel will be breaking the data into columns. Check to make sure that each of the data columns has been recognized (use the horizontal and vertical scroll bars to view all of the columns).
   8. For the team batting data, you'll probably find that you need to add a column break for the third-from-last data column (the cursor points to the spot in the image below). The dialog box has instructions for adding, moving and deleting column breaks.

   9. When all of the data columns are set to your satisfaction, click "Next".
   10. The final step of the Text Import Wizard is to select data formats for each of the columns, as shown below.

   11. The default selection, "General", is what you want for any column with numerical data. For columns containing only text (like the team names), you can select "Text", but this is optional. Again you can use the horizontal and vertical scroll bars to examine the data columns and make sure that all the data types are set properly.
   12. When you are satisfied with your selections, click "Finish" to import the data.

Formatting the Data

Here are some tips for getting your data organized before analyzing it. You'll learn how to remove unwanted rows (or columns), how to change the order of data columns, and how to freeze the column labels, so that they remain visible even when vertically scrolling the data table.

1. Removing unwanted rows (or columns). Sometimes an imported data file contains extra rows or columns that you don't need for your analysis. For example, with the team batting statistics, the second row is a dividing line made with characters. Since some of the analysis features we'll be using later require the data to be in contiguous blocks, we'll want to remove these extra rows. This is really simple. Here's how to do delete rows (or columns) in Excel:
   1. Right-click on the number(s) of the row(s) (or letters of the columns) you want to delete.
   2. The entire row (or column) will be highlighted, indicating that it is selected, and you will see a popup menu, as shown below.

   3. Select "Delete" from the popup menu (as shown above), and you're done.
   4. You'll want to delete any row that does not contain team batting data (except for row 1, which contains the column labels).
2. Rearranging columns. For the correlation and linear regression analysis we'll be running later, Excel requires the data of interest to be in contiguous blocks. We are interested in measuring the correlation between runs scored and batting statistics such as hits, doubles, triples, batting average, etc. The columns are arranged so that two data columns—games (G) and at-bats (AB)—separate runs per game (R/G) and runs (R) from the hitting stats of interest. If you want, you can simply delete the games and at-bat columns (as described above, 1a–d). Alternatively, you can move the columns. Here's how to rearrange data columns in Excel:
   1. First, you need to create two new columns to move the data into. We'll add them to the left of the runs/game column.
   2. Right-click on the column letters to select columns B and C.
   3. The entire columns will be highlighted, indicating that they are selected, and you will see a popup menu, as shown below.

   4. Select "Insert" from the popup menu (as shown above), and two empty columns are inserted to the left of your selection. (If you want to insert more columns, select more to start with.)
   5. Now select the games and at-bats columns, then cut-and-paste them to the new location.
   6. Finally, delete the now-empty columns and you're done.
3. Freezing the header row (data labels). If your data table has too many rows to fit on your screen, it can be nice to have the column labels stay put when you scroll vertically. That way, you can still see what data is in each column. Here's how to freeze your column labels in Excel:
   1. Select the row below your column labels.
   2. From the menu, select "Window/Freeze Panes." That's it!

Adding Derived Statistical Measures

In this section you will be adding derived statistics—those that are calculated from other statistics in your table. In addition to the derived statistics below, you can include other measures that you found in your background research, or you can try to create your own derived statistic.

1. First, insert at least three additional columns between the OPS+ and hmR/G columns. You'll be adding the three derived statistics mentioned in the Introduction: BRA, TA and RC.
2. Batter's runs average (BRA), is the product of OBP and SLG. Here's how to enter it in Excel:
   1. All formulas in Excel start with the equals sign: "=".
   2. The arithmetic operators for formulas are: "+", "-", "*", and "/", for addition, subtraction, multiplication and division, respectively.
   3. On our example spreadsheet, OBP is in column "M" and SLG is in column "N".
   4. So to enter the formula for BRA for row 2, you would type: "=M2*N2", as shown below:

   5. Hit "Enter", and Excel calculates the value for you, as shown below.

   6. Next, copy and paste the formula into the rest of the column.
   7. Finally, you will want to change the formatting of the column so that only 3 decimal places are displayed (the same number of significant figures as in the operands, OBP and SLG). Right-click to select the entire BRA column, and select "Format cells..." from the popup menu.
   8. In the Format Cells dialog, Choose the "Number" tab, then select "Number" from the Category list, and change the "Decimal places" value to 3, as shown below:

   9. Click "OK", and Excel displays BRA to 3 decimal places.
3. Total Average (TA), the ratio of the number of bases to the number of outs:
   TA = (TB + BB)/(AB - H). There is no column for TB (total bases) in the tables from Baseball-Reference.com. You can easily get TB from SLG, because:
   SLG = TB/AB ,
   
   so if we multiply both sides by AB, we can get TB:
   TB = SLG*AB .
   
4. Substituting SLG*AB for TB, our formula becomes:
   TA = (SLG*AB + BB)/(AB - H). Find the corresponding data columns on your spreadsheet and enter the formula. As you did for BRA, above, copy and paste the TA formula to the rest of the column, and change the number display to 3 decimal places.
5. Runs Created, devised by Bill James. RC = (H + BB)TB/(AB + BB). After the first two examples, you should be able to do this one on your own.

Running Correlation and Linear Regression Analysis

1. To run the correlation analysis, use the menu to select "Tools/Data Analysis...". [Note, if this choice is not available, select "Tools/Add-Ins...". Check the "Analysis Toolpak" box, and click "OK". The "Data Analysis..." choice should now be available on the "Tools" menu.]
2. In the Data Analysis dialog, select "Correlation" from the list of Analysis Tools (as shown below) then press "OK".

3. In the Correlation dialog, there are several pieces of information to fill in:
   1. You want to measure the correlations between runs scored (R) and each of the batting statistics (all of the columns from hits (H) to runs created (RC). You should include the first row, which contains the column labels. Excel will use the labels to identify the correlation data. Enter this range of columns in the "Input Range" field.
   2. Make sure that "Columns" and "Labels in First Row" are both checked.
   3. In the "Output Options" section, it's a good idea to put the output on a new worksheet, which you can also name.
   4. When you are satisfied with your selections (see the image below as an example) hit "OK" and Excel will add a new worksheet with the correlation analysis results.

4. Excel calculates the correlation coefficients for each pair of data columns in the range you supplied, and displays the results in a matrix on the new sheet you selected. Here is an example, using data from the 2005 baseball season:

5. The correlation matrix works like a mileage chart in a road atlas. To look up the correlation of batting average (BA) with runs (R), you look down the "R" column until you come to the value in the "BA" row (boxed coefficient in the image above).
6. Next you'll do the linear regression analysis. This time you will need to run the analysis separately for each pair of variables (e.g., runs vs. OBP, runs vs. SLG, etc.) you want to test. You can use the results of the correlation analysis, above, to decide which pairs to explore further with linear regression.
7. To run the linear regression analysis, use the menu to select "Tools/Data Analysis..." and then scroll down to choose "Regression" in the Data Analysis dialog, as shown below:

8. In the Regression dialog, there are several pieces of information to fill in:
   1. For the "Input Y Range," enter the range of cells for R (runs). Be sure to include the first row, with the column label.
   2. For the "Input X Range," enter the range of cells for the variable of interest (here, we're going to plot the regression line for R vs. OBP).
   3. Make sure that the "Labels" box is checked.
   4. Under "Output options," select "New Worksheet Ply" for the results, and give it a name (here, we're calling the new worksheet "2005_RvsOBP_Regr").
   5. Under "Residuals," make sure that boxes are checked as shown, so that Excel will automatically create regression and residuals plots for you.
   6. When you are satisfied with your selections (see the image below as an example) hit "OK" and Excel will add a new worksheet with the linear regression analysis results.

9. Excel does a lot for you automatically, but you will still need to tweak the formatting of the results. Here are some suggestions:
   1. Column widths. When Excel completes the Regression analysis, it will display the new worksheet containing the results, and all of the numerical results will be highlighted, indicating that they are selected. The first thing to do is to adjust the column widths so that you can read all of the headings. From the menu, select Format/Column/AutoFit Selection," and the widths will be set so that you can read everything.
   2. Next, you will need to reformat both the regression plot and the residuals plot. Think of Excel's automatic graphs as just a starting point (see below).
      To really see what's going on, you will want to expand the size of the graphs, and adjust both the x- and y-axis scales (see below). You may want to make other tweaks as well.
10. Here are some suggestions for formatting the graphs.
    To change the size of a graph window, click on the "Chart Area" (between the graph ("Plot Area") and the border around it. (If you hold the mouse still over the graph, a tooltip will pop up and tell you where you are.) The border of the Chart Area will be highlighted, and there will be "handles" to click and drag so that you can size the graph window to your liking.
    1. You can change the size of the "Plot Area" (the graph itself) in a similar manner.
    2. To make changes to the x- or y-axis, double-click on the axis labels in the graph. You should see a dialog like this one:

    3. On the "Patterns" tab (see above), change the "Major tick mark type" and "Minor tick mark type" to "Outside," so that the tick marks don't obscure your data points. (Note: when you are formatting the Residuals graph, under "Tick mark labels" select "Low.")
    4. On the "Scale" tab (see below), adjust the limits so that your data just fits within the graph area. Choose a value for the "Minimum" that is just below your lowest data point, and a value for the "Maximum" that is just above your highest data point. The "Major unit" is the step size for the major tick marks. This value determines the interval for labeling the axis with numbers. The "Minor unit" is the step size for the tick marks falling between numbered values.

    5. On the "Number" tab (see below), first select "Number" in the "Category" list. Then you can set the number of decimal places to display on the axis labels. Here the axis is OBP (on-base percentage), so we've chosen 3 decimal places, which is the customary way of displaying OBP.

    6. Push "OK" and you can see the updates on your graph. If you need to make more changes, just double-click on the axis label again. Make these changes for both the x- and y-axis.
11. Finally, here are a few more tweaks you can make to the graph: refining the graph title and axis labels, removing the legend, and changing the regression line from a series of symbols to a solid line.
    1. Right-click on the graph and select "Chart Options...". You'll see a dialog like the one below:

    2. On the "Titles" tab (see above), you can name the graph, and change the labels for the x- and y-axis.
    3. On the "Legend" tab (see below), you can de-select "Show legend." Then push "OK."

    4. Next, point your mouse at one of the data points for the regression line. (If you hold the mouse still over a data point, a tooltip will pop up to tell you which data series the point is from.) Double-click on the point, and the "Format Data Series" dialog will pop up (see below).

    5. It's best to have the regression line appear as a simple line, not as data symbols. In the "Marker" section, select "None". In the "Line" section, select "Custom", choose a solid line style in black, and then push "OK."
    6. Here's the finished result (below). Now the data points are clearly visible, as is their relationship to the regression line.

Comparing the Results

1. Use your correlation analysis results to decide which batting statistics are more highly correlated with scoring runs.
2. Make a table or graph of your results, using the r² statistic to interpret the significance of the correlation.
3. For each of the highly-correlated statistics, work through the linear regression analysis, and make a regression plot and a residuals plot. Compare the graphs and see which measure is best at predicting the number of runs scored.

Variations

Many variations of this project are possible. We're sure that you can think of more yourself, but here are a few ideas to get you started.

• Do you get the same results if you run this analysis for a different year? For a different baseball era? Can you think of reasons to explain any differences you find?
• Are there other derived statistics (besides RC, TA, and BRA) that might do a better job at predicting runs scored?
• You have to score at least one run to win a baseball game, so we expect teams that score more runs to win more games. However, you also have to keep the other team from scoring more runs than you do. So how well does a team's run-scoring ability correlate with winning percentage?
• Investigate correlations between team pitching statistics and winning percentage. Which pitching statistic is the best predictor of success?

  Baseball Economics

• How well do player salaries correlate with offensive performance? In baseball it is generally expected that the three outfielders and the first and third basemen will produce runs for the team by being skilled with the bat. Assemble the individual batting and salary statistics for this group of players for a single season. How well does salary correlate with the various batting statistics used above? You can take this further by expanding your sample to multiple seasons.
• How well does team payroll correlate with winning percentage?

  More Advanced Project Ideas

• Baseball and Athletic Longevity. History tells us that, over a human lifetime, the trajectory for most individual accomplishments is an arc. We all start off pretty much helpless as infants, grow in physical and mental skill through childhood, teenage years and young adulthood. If we are fortunate enough to live into old age, we also, inevitably, start to notice a decline in those same skills as the body and mind age. Baseball statistics provide a way to measure the trajectory of athletic ability for large numbers of individuals. There are many, many questions you could explore along these lines. What is the "average" age for peak performance? How much variance is there in this age? Does it differ for pitchers and batters? Which position has the greatest longevity? The shortest? Has peak performance age changed over time? Use year-by-year career statistics for individual players to identify their peak years by some measure that you devise. Compile and analyze tables of peak performance data for groups of players to answer one of the questions above, or a similar question that interests you.
• For more ideas, see Teaching Baseball Using Statistics, by Jim Albert (listed in the Bibliography).

Credits

Andrew Olson, Ph.D., Science Buddies

Sources

• Albert, Jim, 2003. Teaching Statistics Using Baseball. Washington, D.C.: The Mathematical Association of America.
• Rummel, R.J., 1976. "Understanding Correlation, Chapter 4.3, Interpreting the Correlation: Correlation Squared" Department of Political Science, University of Hawaii [accessed March 6, 2006] http://www.mega.nu:8080/ampp/rummel/uc.htm#S4.3.

Golf Clubs, Loft Angle, and Distance

Objective

The goal of this project is to measure how the initial launch angle of a golf ball affects how far the ball travels.

Introduction

As everyone knows, the goal in golf is to hit the small white ball into the cup on the green, using the smallest number of strokes possible. Golfers can carry up to 14 different clubs, each intended for a specialized purpose. There are drivers to get the maximum distance when teeing off, fairway woods for hitting long distances when the ball is no longer on the tee, irons for various distance ranges, pitching wedges and sand wedges for chipping shorter shots onto the green, and a putter for the final shot(s) on the green.

This project will focus on the irons, the mid-range clubs often used for shots approaching the green. The irons in a typical set of clubs are numbered 3–9. As the number of the iron increases, so does its loft angle—the angle of the club face with respect to the shaft of the club (see Figure 1). A three iron has a loft angle of 15°. The nine iron has a loft angle of 45°. In between, the loft angle increases by 5° with each iron.

Figure 1. The angle between the club face and the shaft is the loft angle of a golf club. The diagram shows the club head as it would look if you were facing the golfer and your eyes were at the level of the ball.

As the loft angle of the club increases, the ball will be launched at a higher initial angle. The distance that the ball travels will be determined by how fast the club is moving when it hits the ball, the angle of the club face to the ball at impact, and the amount of spin imparted to the ball. Just after impact with the club, the ball will have an initial launch angle, an initial velocity, and an initial spin rate. These parameters will largely determine the flight of the ball (other factors such as wind or collision with an object can also influence the flight).

The goal of this project is to determine the relationship between the loft angle of the club and the initial launch angle of the ball, and between the loft angle of the club and the distance that the ball travels.

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• Loft angle
• Launch angle
• Spin
• Initial velocity

More advanced students should also study:

• Momentum
• Elastic collisions
• Inelastic collisions
• Projectile motion

Questions

• The irons in a set of clubs differ slightly in shaft length. What effect do you think these differences would have on:
  • speed of the club head at impact with the ball?
  • distance that the ball is hit?
• Are there differences in mass of the club head between the irons in your set of clubs? If so, what effect would these differences have on:
  • speed of the club head at impact with the ball?
  • distance that the ball is hit?

Bibliography

• For basic information on golf clubs (purposes of the different types, typical distances for clubs), see:
  • Pinemeadow Golf Products, Inc., 2007. "Golf Lesson #1: The Basics of Golf Clubs," [accessed April 12, 2007] http://www.pinemeadowgolf.com/clubs101_1.html.
  • Kelley, B., 2007. "Golf Club Distances," About: Golf, Golf.About.com [accessed April 12, 2007] http://golf.about.com/od/beginners/l/blclubdistance.htm.
• For more information on projectile motion and momentum, see:
  
  • Henderson, T., 2004a. "Vectors and Motion in Two Dimensions," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed April 12, 2007] http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/vectoc.html.
  • Henderson, T., 2004b. "Momentum and Its Conservation," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed April 12, 2007] http://www.glenbrook.k12.il.us/gbssci/Phys/Class/momentum/momtoc.html.
• This project idea is from:
  Goodstein, M., 1999. Sports Science Projects: The Physics of Balls in Motion. Berkeley Heights, NJ: Enslow Publishers, pp. 83–85.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• Three different golf clubs (irons)
• Driving range with yard markers
• Helper
• Notebook
• Pencil
• For measuring the loft angle of your shots you will need:
  • Video camera
  • Tripod
  • TV or monitor for video playback
  • Plastic wrap
  • Marker
  • Protractor

Experimental Procedure

1. Do your background research so that you are knowledgeable about the terms, concepts, and questions above.
2. It's a good idea to call ahead to the driving range and ask for a good time to conduct your experiment (sometime when the range is less busy). You want to make sure you'll have enough room to set up your video camera so that you can measure the initial angle of the shot.
3. Select three (or more) different irons to test. Measure the loft angle for each iron.
4. Set up the video camera so that you can record a good view of the ball as it is launched by the club.
   1. You want to be able to measure the initial launch angle by playing back the video. The ball travels fast, so you'll need a wide field of view in order to catch the ball.
   2. It's a good idea to take a practice swing or two, and then go back and review the video to make sure that you can see the ball as it takes off. That way you'll be confident that your video data will be useful.
   3. If you should happen to have access to a high-speed video camera, that would be even better.
5. For each iron that you are testing, hit at least 20 balls. For each ball you should:
   1. Try your best to use a consistent swing. The speed of the club and the angle of the club face with respect to the ball should be the same (as near as possible) for each shot.
   2. Record video so that you can measure the launch angle.
   3. Measure the distance that the ball travels. Most driving ranges have yard markers that you can use for this measurement.
   4. It's also a good idea to note how much the ball deviated from a straight line. This will tell you how squarely you hit the ball.
6. Here is how to measure the initial launch angle for each shot:
   1. Set up the video camera to play back on a TV monitor.
   2. Cover the screen with a piece of plastic wrap.
   3. Advance the video to the shot you wish to analyze.
   4. Use a marker to mark the starting position of the ball on the plastic wrap.
   5. Advance the video one frame at a time through the swing until the ball is hit and starts to fly. Mark the new position of the ball.
   6. Continue advancing one frame at a time, marking the position of the ball each time, until the ball leaves the frame.
   7. The initial positions of the ball should fall on a straight line. Draw this line.
   8. Measure the angle that this line makes with the horizontal. This is the launch angle for the shot.
7. For each club, calculate the average launch angle.
   1. How does the launch angle compare the loft angle of the club?
   2. More advanced students should also calculate the standard deviation of the launch angle. Did you have a similar "spread" of launch angles for each club?
8. Make a graph of the distance (y-axis) vs. launch angle (x-axis) for each shot. Make a separate graph for each club.
   1. How does the distance vary with the measured launch angle?
   2. Which club gives you the greatest distance?
   3. More advanced students should also calculate the standard deviation of the distance. Did you have a similar "spread" of distances for each club?

Variations

• Another important launch variable is the initial speed of the ball. This is determined by how fast the club is moving when it hits the ball. You need very high speed photography to measure this. You can find this type of equipment at some golf pro shops. Maybe you can think of ways to enhance your experiment using this "Ball Launch Monitor" technology to measure your swing with the various irons in your club set.
• If you'd like to explore the aerodynamics of golf balls, you might be interested in the Science Buddies project A Cure for Hooks and Slices? Asymmetric Dimple Patterns and Golf Ball Flight.

Sources

This project idea is from:

• Goodstein, M., 1999. Sports Science Projects: The Physics of Balls in Motion. Berkeley Heights, NJ: Enslow Publishers, pp. 83–85.

Tee Time: How Does Tee Height Affect Driving Distance?

Objective

The goal of this project is to investigate how the height of the tee affects the distance of a drive.

Introduction

To be a successful golfer, you need to combine distance and accuracy to get the ball from the tee to the cup with the fewest strokes possible. Drives on the fairway need to be long and straight. As you approach the green, you need to be a good judge of distance in order to select the right club to put the ball where you want. Once on the green, you need to be able to read its contours so that you can predict the ball's path in order to sink your putt. This project will focus on one aspect of getting the most distance and accuracy from your tee shot: the importance of the tee height.

The height of the tee can be an important factor in determining the point in your swing when the clubface makes contact with the club. If your club is further along in the upswing, you are more likely to get a higher launch angle. Many golf pros recommend a higher launch angle for increasing drive distance (Stachura, 2003; Zander, 1999).

Figures 1–3 (from Bastable, 2006) illustrate three different tee heights (low, medium and high) that you could try for your experiment. For the low tee setting, the top edge of the ball is just about even with the top of the clubface (Figure 1).

Figure 1. Golf club and tee at low height. The top surface of the ball is about even with the top of the clubface. (Fred Vuich, Golf Magazine, 2003. Used with permission.)

For the medium tee setting, the center of the ball is even with the top of the clubface (Figure 2).

Figure 2. Golf club and tee at medium height. The center of the ball is about even with the top of the clubface. (Fred Vuich, Golf Magazine, 2003. Used with permission.)

For the high tee setting, the bottom of the ball is just above the top of the clubface (Figure 3).

Figure 3. Golf club and tee at high height. The bottom surface of the ball is just above the top of the clubface. (Fred Vuich, Golf Magazine, 2003. Used with permission.)

Do you think that altering the tee height will make any difference in the distance of your drives? Which tee height will give you the longest drives? Do an experiment to find out!

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• Projectile motion
• Initial conditions when the ball is hit:
  • Initial launch angle
  • Club speed
  • Spin rate

Questions

• How does tee height affect the initial launch angle for the ball?

Bibliography

• For a good article on the effects of tee height on drive distance and accuracy, see:
  Bastable, A., 2006. "Tee It High or Low?" Golf.com [accessed April 18, 2007] http://www.golf.com/golf/instruction/article/0,28136,1565244,00.html.
• Here are two articles on the importance of correct launch angle for achieving the maximum distance on drives:
  
  • Stachura, M., 2003. "Why Everybody Needs to Try More Loft -- and That Means You!" Golf Digest, November, 2003 [accessed April 17, 2007] http://www.golfdigest.com/instruction/index.ssf?/instruction/gd200311loft.html.
  • Zander, J., 1999. "Max Out Your Ball," Zander Golf (related article appeared in Golf Digest, February, 1999) [accessed April 17, 2007] http://www.zandergolf.com/articles/max2%20_dec98.html.
• For more information on projectile motion and momentum, see:
  
  • Henderson, T., 2004a. "Vectors and Motion in Two Dimensions," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed April 12, 2007] http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/vectoc.html.
  • Henderson, T., 2004b. "Momentum and Its Conservation," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed April 12, 2007] http://www.glenbrook.k12.il.us/gbssci/Phys/Class/momentum/momtoc.html.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• One golf club (a driver)
• Driving range with yard markers
• Golf tees
• Ruler for measuring tee height
• Two helpers
• Notebook
• Pencil
• For measuring the launch angle of your shots you will need:
  • Video camera
  • Tripod
  • TV or monitor for video playback
  • Plastic wrap
  • Marker
  • Protractor

Experimental Procedure

1. Do your background research so that you are knowledgeable about the terms, concepts, and questions above.
2. It's a good idea to call ahead to the driving range and ask for a good time to conduct your experiment (sometime when the range is less busy). You want to make sure you'll have enough room to set up your video camera so that you can measure the initial angle of the shot.
3. Select three (or more) different tee heights to test. Here is one idea for setting the three heights (from Bastable, 2006; see photos in the Introduction section, above):
   1. For the high tee setting, adjust the tee so that the bottom edge of the ball is just above the clubface when the driver is resting on the ground.
   2. For the medium tee setting, adjust the tee so that the center of the ball is about even with the top of the clubface.
   3. For the low tee setting, adjust the tee so that the top edge of the ball is about even with the top of the clubface.
   4. Measure each of these tee heights in centimeters, after setting the tee with reference to the driver. Then use the measured height to set the tees for each shot.
4. Have the first helper use a ruler to set the tee height for each shot while you look away. You may still notice the different heights as you set up to take your swing, but if you don't know the exact height for each shot, your swing is less likely to be affected. Don't focus on the tee height, focus on making a consistent swing.
5. You first helper should mix up the tee heights so that the order of the shots is not a factor. For example, if you get tired your later drives might be shorter than your earlier ones. If you mix up the tee heights, factors like fatigue should affect shots taken from all of the different tee heights equally.
6. Have your first helper record the following data for each shot:
   1. Tee height (low, medium, or high)
   2. Distance
   3. Amount the ball deviated from a straight line, in degrees
   4. Later you'll add the launch angle data, from the video recording.
7. Set up the video camera so that your second helper can record a good view of the ball as it is launched by the club.
   1. You want to be able to measure the initial launch angle by playing back the video. The ball travels fast, so you'll need a wide field of view in order to catch the ball.
   2. It's a good idea to take a practice swing or two, and then go back and review the video to make sure that you can see the ball as it takes off. That way you'll be confident that your video data will be useful.
   3. If you should happen to have access to a high-speed video camera, that would be even better.
8. For each tee height that you are testing, hit at least 20 balls. For each ball you should:
   1. Stand at the same distance from the tee for each shot.
   2. Try your best to use a consistent swing. The speed of the club head and the angle of the clubface with respect to the ball should be the same (as near as possible) for each shot.
   3. Video record so that you can measure the launch angle.
   4. Measure the distance that the ball travels. Most driving ranges have yard markers that you can use for this measurement.
   5. It's also a good idea to note how much the ball deviated from a straight line. This will tell you how squarely you hit the ball.
9. Here is how to measure the initial launch angle for each shot:
   1. Set up the video camera to play back on a TV monitor.
   2. Cover the screen with a piece of plastic wrap.
   3. Advance the video to the shot you wish to analyze.
   4. Use a marker to mark the starting position of the ball on the plastic wrap.
   5. Advance the video one frame at a time through the swing until the ball is hit and starts to fly. Mark the new position of the ball.
   6. Continue advancing one frame at a time, marking the position of the ball each time, until the ball leaves the frame.
   7. The initial positions of the ball should fall on a straight line. Draw this line.
   8. Measure the angle that this line makes with the horizontal. This is the launch angle for the shot.
10. For each tee height, calculate the average launch angle and drive distance.
    1. Is there a relationship between tee height and launch angle?
    2. Make a graph of the launch angle (y-axis) vs. tee height (x-axis) for each shot.
    3. Make a graph of the drive distance (y-axis) vs. tee height (x-axis) for each shot.
    4. How does the distance vary with the measured launch angle?
    5. Compare the average straightness of shots at the different tee heights.
    6. Did one tee height have a consistently better distance than the others?
    7. More advanced students should also calculate the standard deviation of the launch angle and of the drive distance. Did you have a similar "spread" of launch angles for each tee height? Did you have a similar "spread" of drive distances for each tee height?

Variations

• Test various "performance" tee designs vs. wooden tees for distance, shot dispersion, or both. Examples of performance tees are:
  • Brush t
  • Epoch-3
  • Zero Friction golf tees
• Consider paying for a club fitting session using ball launch monitor technology at a local golf pro shop. These use high speed photography to measure club head speed and the initial ball speed, launch angle, and spin. This data can be used to calculate the distance that the ball would travel. Combine the club fitting session with before and after driving range sessions to see if the ball launch monitor data helps you improve your drives.
• Expand your study to include more golfers. Do you get the same results?
• For a more basic golf-related experiment focusing on club selection and distance, see the Science Buddies project Golf Clubs, Loft Angle, and Distance.
• For another golf-related experiment that focuses more on the aerodynamics of the golf ball, see the Science Buddies project A Cure for Hooks and Slices? Asymmetric Dimple Patterns and Golf Ball Flight.

Sources

This project was based on:

• Bastable, A., 2006. "Tee It High or Low?" Golf.com [accessed April 18, 2007] http://www.golf.com/golf/instruction/article/0,28136,1565244,00.html.

Nothing But Net: The Science of Shooting Hoops

Objective

The goal of this project is to determine if the ball's starting position for shooting a basketball affects a player's shooting percentage. In this project, you'll measure shooting percentage when players shoot baskets from chest height, chin height, and over the head.

Introduction

Click here to watch a video of this investigation, produced by DragonflyTV and presented by pbskidsgo.org

If you're an avid basketball player, then few things are more satisfying than sinking a perfect jump shot, hitting nothing but net. What goes into making that perfect shot? What can you do to make your shots more accurate?

Jay and Jonathan both like basketball, and they're also pretty observant. They noticed that some kids liked to shoot from chest height, others from chin height, and others from over their heads. They wondered what effect the different starting positions had on shooting success. They set up an experiment to find out. Check out the video clip at right to see what they did.

Jay and Jonathan took a great approach to their project. Starting with a subject they knew and liked, they used observations about shooting styles to come up with an interesting question: does the height where the player holds the ball affect shooting percentage? They designed an experiment to test their hypothesis, keeping all experimental conditions constant, except for the variable of interest. Jay and Jonathan also did a great job with their data analysis. When their data disproved their initial hypothesis, they didn't stop. They dug deeper to see if their data might hold some other clue for shooting success.

If you've read this far, it's a good bet you're thinking about doing a basketball-related project, too. You could repeat Jay and Jonathan's experiment yourself. We've suggested a couple of improvements in the Experimental Procedure section below, and you can probably come up with some of your own. Or, you could use your powers of observation to come up with your own basketball question to test scientifically. The Variations section also has some ideas that might help you get started.

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• arc,
• trajectory,
• backspin,
• basketball shooting percentage.

More advanced students should also study:

• physics of projectile motion.

Questions

• Which trajectory do you think improves your chance of making a basket: higher or lower? Explain why.
• Do you think that there will be a relationship between the ball's starting position (chest height, chin height, or over the head) and the ball's trajectory? Why or why not?

Bibliography

• TPT, 2006. "Basketball by Jay and Jonathan," DragonflyTV, Twin Cities Public Television http://pbskids.org/dragonflytv/show/basketball.html.
• Cull, T., 1999. "Why is putting rotation on a basketball when you shoot good?" MadSci Network: Physics [accessed June 15, 2006] http://www.madsci.org/posts/archives/jun99/929552614.Ph.r.html.
• Henderson, T., 2004. "Lesson 2: Projectile Motion," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed June 15, 2006] http://gbs.glenbrook.k12.il.us/Academics/gbssci/phys/Class/vectors/u3l2a.html.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• basketball,
• basketball court, hoop, and backboard,
• volunteer players to shoot baskets (the more the better),
• masking tape.

The following items are optional, but can give you a better understanding of what is going on with the different hand positions.

video camera and tripod to record the shots,

small TV or video monitor for camera playback,

transparencies,

marker or stickers to mark ball position,

protractor for measuring launch angle.

Experimental Procedure

1. Pick a single location on the court for all players to shoot from, and mark it with masking tape. Choose the location so that your volunteers can make a good percentage of their shots. A straight-on shot would be a good choice, e.g., from the middle of the free-throw line. If your volunteers are younger players, you may want to move the shooting location in closer.
2. If you have a video camera, set it up on a tripod to record the shots. Make sure that the camera can capture the ball all the way from the player's hands to the basket. The camera should be perpendicular to the path of the ball. The camera should remain in the same position for the entire experiment.
3. Have each player shoot the same number of baskets from each hand position (chest height, chin height, and over the head). Things to think about:
   1. How many shots? You want to collect a large enough sample so that your results are reliable. Ten shots per hand position per player is a bare minimum.
   2. You might want to let each player take a few "warm-up" shots that don't count, so that they can home in on the right distance.
   3. Along that same line, since the players are getting practice as they go along, it would be a good idea to change the order of the hand positions from player to player. Have the first player shoot chest high, then chin high, then over the head. Have the second player shoot chin high, then over the head, then chest high. Have the third player shoot over the head, then chest high, then chin high. Keep rotating through the hand positions like this, so that each hand position is first, second, or third equally often.
   4. To help keep everyone's mind on doing their best, you might want to make a contest out of it. For example, you could offer prizes for the top three shooters, or for all the players who make a certain percentage of their shots.
4. Keep track of how many shots each player made from each hand position. Keep track of the order you assigned them for making the shots.

   Analyze Your Data

5. Calculate the shooting percentage for each individual for each hand position. Make a bar graph of the results. Is any position clearly better for all players? Is any position clearly better for individual players?
6. Did the players' shooting percentage improve or worsen during the experiment? Make symbol-and-line graph of shooting percentage vs. 1st, 2nd, and 3rd hand position to find out.
7. If you were able to videotape the experiments, you can analyze the trajectory of the ball, as Jay and Jonathan did in the video clip in the Introduction.
   1. Connect the video camera to a small TV or video monitor for playback.
   2. For each shot you analyze, tape a transparency over the monitor.
   3. Label the transparency with the player, shot number, hand position, and whether the shot was made (1) or not (0).
   4. Use your video camera's frame-by-frame playback mode to "freeze" the ball on the screen so that you can mark the position of the ball on the transparency. Trace the ball's position with a marker.
   5. You will want to advance the video by something like 3–6 frames between each position. Use the same number of frames each time. (Each video frame represents 1/30 of a second. If you mark the ball's position at 10–12 points on its way to the basket, you'll have a good picture of its trajectory.)
   6. You can use a protractor to measure the launch angle. Fit a line through the first few points, and measure the angle between the floor and the line.
   7. Calculate the average launch angle for each hand position for each player and graph the results. Is there a clear relationship between between launch angle and hand position?

Variations

• There are other aspects of hand position that are important for good shooting technique. A basketball coach will have you work on shooting with one hand (the other hand supports the ball, but the shooting hand launches the ball). Design an experiment to compare shooting percentage for one-handed vs. two-handed shooting. Which works better and why?
• You could focus your study on the trajectory of the ball. Is a higher trajectory really more successful as Jay and Jonathan hypothesized? Design an experiment to see whether a high, medium, or low arc results in a better shooting percentage.
• What are the effects of backspin on shooting percentage? Have players shoot from the same position both with and without backspin. Calculate shooting percentage for each method and compare. Does video analysis show any difference in average trajectory for the two cases? Does the ball bounce differently for the two cases? Does this have any effect on shooting success? (TPT, 2006)
• What's the scientific difference between a "soft" shot, and a "brick?" Is it the speed of the ball, the spin, or the arc? Design an investigation to figure out the difference! (TPT, 2006)
• More advanced students can do a quantitative analysis of the video data. Make sure that you have an accurate distance reference in the frame to use for calibration (built-in if you're shooting from a free-throw line—just measure the distance to the basket). Using the equations for projectile motion, you should be able to reproduce the trajectory of the ball. This will give you the launch angle and initial velocity of the ball. What range of launch angles yields the best shooting percentage?

Sources

The idea for this project came from this DragonflyTV Podcast:

• TPT, 2006. "Basketball by Jay and Jonathan," DragonflyTV, Twin Cities Public Television http://pbskids.org/dragonflytv/show/basketball.html.

Under Pressure: Ball Bouncing Dynamics

Objective

In this experiment you will test if air pressure will affect the dynamics of ball bouncing.

Introduction

Bouncing is what makes playing with a ball so much fun. But what makes a ball bounce? If you were to see a ball drop to the ground in slow motion, you would see its shape change dramatically. When it hits the ground, the bottom of the ball becomes flat against the floor. Then when it bounces up, it returns to its original shape. The changes in shape are due to a balance of forces and energy:

"When you drop a ball, gravity pulls it toward the floor. The ball gains energy of motion, known as kinetic energy. When the ball hits the floor and stops, that energy has to go somewhere. The energy goes into deforming the ball-- from its original round shape to a squashed shape. When the ball deforms, its molecules are stretched apart in some places and squeezed together in others. As they are pushed about, the molecules in the ball collide with and rub across each other." (Doherty, 1991)

In this experiment you will use a ball filled with air. When it is dropped, the air molecules will squish together and then spring apart. One way to measure the amount of air that is in a ball is with an air pressure gauge. You will use an air pump, a basketball, and an air pressure gauge to test if changing the air pressure will change the bounciness of the ball.

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the internet, or take you to your local library to find out more!

• air pressure
• forces
• energy
• motion
• gravity

Questions

• How high will a ball bounce?
• Will the air pressure change the bounciness of a ball?
• How can you measure the height a ball bounces?

Bibliography

• Doherty, P., 1991. "That's The Way The Ball Bounces," The Exploratorium Museum, San Francisco, CA. [accessed August 4, 2006]
  http://www.exploratorium.edu/sports/ball_bounces/index.html
• Here is another activity that tests temperature and bounce using golf balls and baseballs:
  Exploratorium Staff, 1998. "Science Activity: Bouncing Balls," The Exploratorium Museum, San Francisco, CA. [accessed August 4, 2006]
  http://www.exploratorium.edu/baseball/bouncing_balls.html
• There are many other great ball projects in this fun book:
  Goodstein, M., 1999. Sports Science Projects: The Physics of Balls in Motion. Berkeley Heights, NJ: Enslow Publishers.

Materials and Equipment

• ball air pump
• ball pressure gauge (available at a sporting goods store)
• basketball or soccer ball
• stepladder
• video camera with a tripod
• paint
• tape measure (preferably metric)
• large roll of paper
• volunteer

Experimental Procedure

1. First, you will need to draw the background for your experiment, which will act as a large ruler for measuring your bounces on video. Use the paint and a tape measure to make a series of lines on your roll of paper. It should be tall enough to hang up behind your ladder, about 5 feet tall. Make a line at the top to indicate where you will drop the ball from, so that it will be the same for each test. Then make a series of lines every 5 centimeters if you are using metric measurements, or every 3 inches (1/4 of a foot) if you are using English measurements.
2. Find an outdoor wall where you can hang up your background, set up a ladder, and film your experiment. It should be on a smooth cement surface for optimal ball bouncing. Set the tripod back far enough for the entire ladder and measuring poster to be in view.
3. Fully pump up the basketball and check the pressure with the pressure gauge. Write the pressure in the data table:

   Air Pressure in PSI (Lbs)	Height of Bounce (cm)

4. Have your volunteer get the video camera ready. Carefully climb the ladder with the ball. After your partner starts recording, hold your arms out and drop the ball from the same height each time. It is important to let the ball fall out of your hands, and not to push it down with your hands, wrists, or arms.
5. Let the ball bounce until it stops, and then stop recording.
6. Use the pressure gauge to release some of the air pressure from the ball. Record the new pressure reading in your data table and repeat steps 4 and 5.
7. Keep releasing pressure and repeating the experiment until the ball no longer bounces.
8. Now, take your video and watch it to gather your data. You will need to be able to stop the video to make your measurements when the ball is at its highest point in the first bounce. Write each measurement in the data table.

   A ball bounces in an arc, so measure the height at the top of the arc of the first bounce. (Image by Harold E. "Doc" Edgerton)

9. Make a line graph of your data. Set up the graph by putting a scale of the air pressure on the left side of the graph (Y-axis) and the height of the bounce on the bottom of the graph (X-axis). Now use your data table to put a dot where your data intersect. Then draw a line to connect the dots. Is it a straight line? Does it slope up or down? What do you think this means about your results?

Variations

• Does height make a difference? Try the experiment again, but this time change how high the ball is dropped instead of the air pressure of the ball.
• Do different kinds of balls bounce differently? Try bouncing different kinds of balls to test this question. Some balls to try are: soccer, basketball, volleyball, golf, tennis, or handball.
• Check out these Additional Sports Project Ideas that explore the physics of bouncing balls.
• Does the weather make a difference? Try your bounce test in different weather conditions. Try comparing a hot day to a cold day. Try comparing two days with different atmospheric pressure. You can check the weather stats at http://www.weatherunderground.com
• There are many other great ball projects in this fun book by Madeline P. Goodstein, Sports Science Projects: The Physics of Balls in Motion published by Enslow Publishers. Check it out from your local library!

Credits

Sara Agee, Ph.D., Science Buddies