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Catcher of the Fly

Catcher of the Fly. Introduction Statement of Problem Strong LOT Model 7 th Inning Stretch Unanswered Questions Competing Models Conclusion. Introduction. Janel Krenz. Favorite Baseball Team: Milwaukee Brewers. Ivan Lau. Favorite Baseball Team: New York Yankees. Lori Naiberg.

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Catcher of the Fly

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  1. Catcher of the Fly Introduction Statement of Problem Strong LOT Model 7th Inning Stretch Unanswered Questions Competing Models Conclusion

  2. Introduction

  3. Janel Krenz Favorite Baseball Team: Milwaukee Brewers

  4. Ivan Lau Favorite Baseball Team: New York Yankees

  5. Lori Naiberg Favorite Baseball Team: Chicago Cubs

  6. Ben Rahn Favorite Baseball Team: UW-Stout Blue Devils

  7. Chad Seichter Favorite Baseball Team: Oakland Athletics

  8. Our group has extensively studied the paper: “A Mathematician Catches a Baseball” by Edward Aboufadel. The paper discusses how an outfielder catches a fly ball. We will go on to discuss our findings on the mathematics of how an outfielder catches a fly ball.

  9. Statement of Problem

  10. Background • Old Theory • Complex calculations • Solved the problem in 3 dimensions • Fielders run straight path

  11. New Theory • Linear Optical ball Trajectory (LOT) • Outfielder uses a curved running path • Fielder keeps the ball on a straight line

  12. Aboufadel’s Mathematical Model H = Home Plate B = Position of Ball F = Position of Fielder B*= Projection of Ball onto Field I = Fielders Image of the Ball I* = Unique Perpendicular

  13. Aboufadel derived the Strong LOT Model, which is a special case of the LOT model. • The Strong LOT Model hypothesis: The strategy that the fielder uses to catch a fly ball is to follow a path that keeps both p and q constant. (p = yi/xi, q = zi/xi) • With F = (xf, yf, zf), B = (xb, yb, zb), and I = (xi, yi, zi)

  14. Strong LOT Model

  15. The Strong LOT Model Hypothesis • The strategy a fielder uses to catch a fly ball is to follow a path that keeps p and q constant

  16. For this hypothesis, HI* has a slope of p, so it follows that B*F has a slope of –1/p • Equation (3) is true at every point in time (3)

  17. The equation of HI* is y = px and the equation of B*F is y = yb-(x-xb)/p and the point I* is determined by the intersection of these two lines. Set them equal and solve. • Subtract yb from both sides and then multiply both sides by p. • Add ybp and x to both sides, factor and divide and we get equation (4). (4)

  18. Since F, B, and I are collinear, we have • And zf = 0 and zi = qxi (4.5)

  19. Plug in qxi for zi and cross multiply. • Factor out, solve, and we get equation (5). • Combine equation (4) and (5) to get equation (6): (5)

  20. Substitute (4) in for xi. • Multiply through and solve for xf and we get equation (6). . • This gives us the x-coordinate of the fielder. (6)

  21. Now to find the y-coordinate of the fielder • Solve Equation (3) for yf • Add pyb to both sides, divide both • sides by p and we to get yf.

  22. Combining yf and equation (6) and solving we get: • This would give us the y-coordinate of the fielder. (7)

  23. Then solving equation 6 for q we get: • What we now have, for every time t > 0 and for every trajectory B, is a relationship between (xf, yf) and (p, q). If we know p and q, we can solve for the fielder’s xf and yf, and if we know the fielder’s positions xf and yf then we can solve for p and q. (8)

  24. Proof that the fielder will intersect the ball. (T = time when ball hits ground) • Using equation (6) The same method is used to show that yf = yb (9)

  25. As a consequence of the Strong LOT Model, since p and q are constant, you can calculate them. Since equation 3 (which determines the slope of HI*) is true for all t, it is true when the batter hits the ball (t=0). (10)

  26. To determine q, we use equation 8 and L’Hopital’s rule. (11)

  27. 7th Inning Stretch

  28. Baseball Trivia • Who won last year’s World Series? • What two professional baseball players broke the homerun record in 1999?

  29. IT’S PEANUT TIME!!! ENJOY!!!

  30. Answers: • New York Yankees! 2. Sammy Sosa and Mark McGwire!

  31. Unanswered Questions

  32. If p and q are not constant, there is no unique path. If p and q are not constant, there might be a shorter path.

  33. 3. If the fielder establishes the LOT model, can he run straight to the destination point? 4. How fast does the fielder have to be?

  34. Competing Models

  35. OAC Model(Optical Acceleration Cancellation) • Straight running path • Constant speed Problems: • Complex calculations • This model identifies the projection as a planer optical projection.

  36. Robert Adair’s Model • Adair’s Model focuses on the path of a fly ball. • A fielder runs laterally so that the ball goes straight up and down from his or her view. • The lateral alignment and monitoring of up and down ball motion requires information that is not perceptually available from the fielder’s vantage.

  37. Conclusion

  38. Wrapping It Up • Next time you are out on the field, don’t forget to use the Strong LOT Model!!! • Remember to keep p and q constant!!! • Follow these two hints and you will NEVER miss a fly ball again!!!

  39. Sources • A. Aboufadel. “A Mathematician Catches a Baseball”. American Mathematical Monthly. December 1996. • M. McBeath, D. Shaffer, and M. Kaiser. “How Baseball Outfielders Determine Where to Run to Catch Fly Balls”. Science. April 28, 1995. • P. Hilts. “New Theory Offered on How Outfielders Snag Their Prey”. The New York Times. April 28, 1995. • J. Dannemiller, T. Babler, and B. Babler. “On Catching Fly Balls”. Science. July 12, 1996.

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