Track and field records continue to be broken in modern times due to technological advances and enhanced understanding of nutrition, physiology and the underlying physics principles involved in these sports. Students will use the Internet to research track and field records and learn how physics principles are applied to these sporting events.

This project can be integrated into a unit on velocity and acceleration in a physics, advanced algebra or calculus class.

Students should be familiar with using a search engine on the Internet. They should have basic trigonometry knowledge and have studied the equations of motion for projectiles.

Materials Needed.

Internet Access. Spreadsheet, graphing calculator or graph paper.

To use the Internet to research statistical data related to world or local athletic records.

To fit a linear equation to Olympic discus throw records and interpret the slope.

To find the best "angle of elevation" to throw the discus to maximize the distance thrown.

To learn physics principles that apply to various track and field events.

Completion of the Discus Throw Activity Sheet and a 1-2 page paper discussing another track and field event.

Students may wish to research how physics principles apply in preventing sports injuries.

!!!! Warning !!!! The Web sites given in this lesson may have changed! Before using this lesson with your students, be sure to check if the sites are still working or if you must find another site. Sometimes the sites still have the relevant data but you may need to change the directions to access the data.

If this project is used in a mathematics class, the questions about describing physics principles which help explain coaching strategies should be omitted (#3 and #9 III).

If students have enough mathematics/physics background, the equation in #6 in the Activity Sheet can be derived as follows. If a projectile is thrown from the ground with velocity v, making an angle f with the horizontal, then its initial vertical velocity (at time t = 0) is v sin(f).. If we let the upward direction be positive and let ground level be zero, then vertical acceleration = -g, vertical velocity = -gt + vsin(f), vertical distance = -gt²/2 + vsin(f); then the vertical distance will be zero either for t = 0 or t = 2vsin(f)/g. In a calculus class, these velocity and distance equations can be obtained by integration.

The trigonometric identity needed to answer Question #7 is 2sin(f)cos(f) = sin(2 f). (This identity can be derived from the more general identity sin(a+b) = sin(a)cos(b) + sin(b)cos(a).)

Some helpful Internet sites to find more information about athletic topics are

http://www.olympics.org,

http://www.iaaf.org,

http://espn.go.com/editors/atlanta96/almanac,

http://ms.mathscience.k12.va.us/lessons/physics/intphys1.html,

http://acs5.bu.edu:8001/~rnyhan/introduction.html,

http://curiculum.qed.qld.gov.au/kla/eda/oly_gold.htm

Press STAT, choose EDIT and enter the year in, say, list L₁ and the distance data in list L₂. Press STAT and choose CALC. Cursor to 4:LinReg(ax+b) and press ENTER. Then type L₂,L₁,Y₁. The list name with the y-variable always is first. The regression equation will be stored as Y₁. The variable name Y₁ can be accessed from the VARS menu. The slope and y-intercept will be displayed.

To graph the data and the best-fit line use STAT PLOT. Turn ON one of the Plots, choose the first icon (showing a scatterplot), enter the year list name L₁ as the Xlist and the distance list name L₂ as the Ylist. Then press ZOOM and choose ZoomStat.

Type in the year and distance data in two columns (say, columns C and D). To find the line of best fit, highlight two adjacent cells in the same row. Then type

where D3:D24 contains the distance values and C3:C24 the year values. (Your columns may be different, but the column with the y-variable range always is first.) Then press Ctrl + Shift + Enter. The slope and y-intercept of the best-fit line will be displayed in that order.

To show the graph of this data, position the cursor in an empty cell and click the graph icon. Choose XY(Scatter), then Next. Type C3:D24 for the Data Range and click on Columns and then on Next. Type "Year" for the Value(X) Axis and "Distance" for the Value(Y) Axis. Click Next. Click on As Object In for Chart Location and then click Finish.

You may want to change the limits on the Year and Distance scales. To do so double click on the x-values in the graph to display the Format Axis window. Click on Scale and type in the Year limits. Similarly change the Distance limits if needed.

Finally click on Chart on the top bar and choose Add Trendline to display the graph of the best-fit line.

If you wish more background about the statistical concepts involved in the lesson, some good sites to check are:

http://davidmlane.com/hyperstat/index.html

http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html

http://www.math.unb.ca/~maureen/SSCEdCom/basicstats/basicstats.html

http://www.math.unb.ca/~knight/BasicStat/$content.htm

http://www.bbns.org/us/math/ap_stats

http://www.grad.cgs.edu/wise/linksf.shtml

http://www.cvgs.k12.va.us/DIGSTATS

http://www.statsoft.com/textbook/stathome.html

http://www.stats.gla.ac.uk/steps/glossary/index.html

http://www.crpc.rice.edu/CRPC/GT/sboone/Lessons/lptitle.html

http://forum.swarthmore.edu/library/topics/statistics

http://www.psychstat.smsu.edu/introbook/skb00.htm

TI-83 instructions:

http://www.ti.com/calc/docs/act/koehler001.htm

http://www.wku.edu/~neal/manual/ti83.html

The Calculator website at the Mathematics Department of the University of Wisconsin-La Crosse will perform basic statistical calculations. If you do not have access to a simple statistical computer package or calculators with statistics options, your students may access http://www.compute.uwlax.edu/stats_htdocs/newmenu.html to perform statistical computations on-line.

In order to print out just a copy of the student worksheet, highlight this section, then copy and paste it into your word processor. You may then revise the worksheet if you wish.

1. (10 points) Access the site http://Infoplease.lycos.com/ipsa/A0115041.html to find the Olympic discus results for all the past Olympic games. Click on Event by Event, Track and Field, Men or Women, and then Discus. You may use a spreadsheet or a graphing calculator to enter this data or, if not, plot the data on graph paper.

2. (5 points each) a. Use your spreadsheet or graphing calculator to find the best-fitting line to your data. If you do not have access to this technology, draw a line on your graph paper that seems to fit the data.

b. What is the slope of your best-fitting line?

c. Interpret the slope.

d. What distance do you predict for the gold medal discus winner in the next Olympic games? How did you calculate this distance?

3. (10 points) Use a search engine on the Internet to try to determine some coaching techniques for discus throwing. Describe how some of these techniques may be based on principles of physics.

4. (3 points) At what angle do you think the discus should be thrown in order to maximize its distance?

In this section, we will determine, by physics principles, the best angle to throw the discus (or any projectile) to achieve maximum distance.

5. (3 points each) If a projectile is thrown with velocity v making an angle f with the horizontal, what is its initial:

a. vertical velocity? ____________________

b. horizontal velocity? ____________________

6. (3 points) Assuming negligible air resistance and that the projectile is thrown from ground level, then the time it is in the air until it hits the ground is t = 2vsin(f)/g, where g is the acceleration of gravity (9.8 m/sec² or 32 ft/sec²). What is the horizontal distance the projectile has traveled during this time?

7. (3 points) Use a trigonometric identity to rewrite sin(f)cos(f) in your expression in #6.

8. (3 points) What value of f will maximize your expression in #7? Why?

9. (20 points) Write a 1-2 page paper about another track and field event.

Find information similar to the discus gold medal records. Try to predict the winning record for the next competition by graphing your data and fitting a line or other curve. Describe in words the mathematical relationship between time and your data records. But if a clear pattern is not apparent, point this out too.

Search the Internet to find coaching tips for your track and field event and try to explain any underlying physics principles for this sport. If algebraic equations are an important part of the physics for this event show the equations and explain units and the mathematical relationship that they convey. You may be able to find short video clips which helps explain correct techniques and strategies. If so, bookmark the video clip and demonstrate it to the class for 5 extra points.

Your paper will be evaluated using the following criteria.

I. Neatness and clarity.

II. Organized table of data, graph of a best-fitting line or curve, description of any mathematical relationships, and prediction for the next competition.

III. Description of coaching strategies and how physics principles can be applied.

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