I. Introduction

II. Statement of Problem

III. Strong LOT Model

IV. 7th Inning Stretch

V. Unanswered Questions

VI. Competing Models

VII. Conclusion

Catcher of the Fly

Introduction

Janel Krenz

Favorite Baseball Team:Milwaukee Brewers

Ivan Lau

Favorite Baseball Team:New York Yankees

Lori Naiberg

Favorite Baseball Team:Chicago Cubs

Ben Rahn

Favorite Baseball Team:

UW-Stout Blue Devils

Chad Seichter

Favorite Baseball Team:Oakland Athletics

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.

Statement of Problem

Background

• Old Theory– Complex calculations– Solved the problem in 3 dimensions– Fielders run straight path

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

Aboufadel’s Mathematical Model

H = Home PlateB = Position of BallF = Position of FielderB*= Projection of Ball onto FieldI = Fielders Image of the Ball

I* = Unique Perpendicular

• 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)

Strong LOT Model

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

 

• 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

bf

fb

bf

bf

yy

xxp

xx

yy

p

1

(3)

• 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.

pyxpx bb )1( 2

(4)12

p

pyxx bbi

Subtract yb from both sides and then multiply both sides by p. bb xxpyxp2

pyxxxp bb2

pyxxxp bb2

Add ybp and x to both sides, factor and divide and we get equation (4).

p

xxypx bb

)(

• Since F, B, and I are collinear, we have

• And zf = 0 and zi = qxi

fb

fb

fi

fi

xx

zz

xx

zz

)(

0

)(

0

fb

b

fi

i

zx

z

xx

z

(4.5)

• Combine equation (4) and (5) to get equation (6):

fbibfibi xzxzxqxxqx

biibibf xqxxzqxzx )(

ib

bbif qxz

qxzxx (5)

Plug in qxi for zi

and cross multiply.

biibfifb xqxxzxqxxz

Factor out, solve, and we get equation (5).

)()( fb

b

fi

i

xx

z

xx

qx

• This gives us the x-coordinate of the fielder.

11 2

2

ppyx

qzp

pyxqxz

bbb

bbbb

bbb

bbbbf pyxqpz

pyxqxzx

12

ib

bbbbf qxz

qxz

p

pyxx

12

(6)

Substitute (4) in for xi.

Multiply through and solve for xf and we get equation (6). .

• Now to find the y-coordinate of the fielder

• Solve Equation (3) for yf

fbbf xxpypy

p

pyxxy bfbf

Add pyb to both sides, divide both sides by p and we to get yf.

bf

fb

yy

xxp

• Combining yf and equation (6) and solving we get:

• This would give us the y-coordinate of the fielder.

(7) bbb

bbbbf pyxqpz

pyxqypzy

12

• 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.

bf

bbf

bb

b

xx

pyxpx

pyx

zq

)()1( 2

(8)

Ttb

Ttbb

bbb

Ttbbb

bbbbTtf

x

pyxqpyxqx

pyxqpz

pyxqxzx

)()(

)()1(

))((2

(9)

•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

00

0|

tf

f

tbf

fbt y

x

yy

xxp

(10)

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).

bf

bbf

bb

b

tt xx

pyxpx

pyx

zq

1lim|

2

00

|0

22

0 ''1'

1lim

tbb

b

bb

b

t pyxpz

ppyxz

(11)

To determine q, we use equation 8 and L’Hopital’s rule.

7th Inning Stretch

1. Who won last year’s World Series?

2. What two professional baseball players broke the homerun record in 1999?

Baseball Trivia

IT’S PEANUT TIME!!!

ENJOY!!!

1. New York Yankees!

2. Sammy Sosa and Mark McGwire!

Answers:

Unanswered Questions

1. If p and q are not constant, there is no unique path.

2. If p and q are not constant, there might be a shorter path.

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?

Competing Models

OAC Model(Optical Acceleration Cancellation)

• Straight running path• Constant speed

Problems:• Complex calculations• This model identifies the projection as a planer

optical projection.

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.

Conclusion

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!!!

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.