Math and SportsMath and SportsPaul MoorePaul Moore

April 15, 2010April 15, 2010

Math in Sports?Math in Sports?Numbers EverywhereNumbers Everywhere– Score keepingScore keeping– Field/Court measurementsField/Court measurementsSports StatisticsSports Statistics– Batting Average (BA)Batting Average (BA)– Earned Run Average (ERA)Earned Run Average (ERA)– Field Goal Percentage (Basketball)Field Goal Percentage (Basketball)Fantasy SportsFantasy SportsPlaying SportsPlaying Sports– GeometryGeometry– PhysicsPhysics

OutlineOutlineReal World ApplicationsReal World Applications– BasketballBasketball

Velocity & angle of shotsVelocity & angle of shots

Physics equations and derivationPhysics equations and derivation

– BaseballBaseballPitchingPitching

Home run swingsHome run swings

StatsStats

– SoccerSoccerAngles of defense/offenseAngles of defense/offense

– Math in EducationMath in Education

Math in BasketballMath in BasketballScore KeepingScore Keeping– 2 point, 3 point shots2 point, 3 point shots– Free throwsFree throws

94’ by 50’ court94’ by 50’ courtBasket 10’ off the groundBasket 10’ off the groundBall diameter 9.5”Ball diameter 9.5”Rim diameter 18.5”Rim diameter 18.5”3 point line about 24’ from 3 point line about 24’ from basketbasket

Think of any ways math can be used in Think of any ways math can be used in basketball?basketball?

Math in BasketballMath in BasketballBasketball ShotBasketball Shot

At what velocity should a foul shot At what velocity should a foul shot be taken?be taken?

Assumptions/Given:Assumptions/Given:– DistanceDistance

About 14 feet (x direction) from FT About 14 feet (x direction) from FT line to middle of the basketline to middle of the basket

– HeightHeight10 feet from ground to rim10 feet from ground to rim

– Angle of approachAngle of approachClose to 90 degrees as possibleClose to 90 degrees as possibleMost are shot at 45 degreesMost are shot at 45 degrees

– Ignoring air resistanceIgnoring air resistance

Math in BasketballMath in BasketballHeavy Use of Kinematic EquationsHeavy Use of Kinematic Equations– Displacement:Displacement:

s = ss = s00 + v + v00t + ½att + ½at22

s = final position s = final position

ss00 = initial position = initial position

vv00 = initial velocity = initial velocity

t = timet = time

a = accelerationa = acceleration

This is 490….where did this equation come from?This is 490….where did this equation come from?

Math in BasketballMath in BasketballBy definition: Average velocityBy definition: Average velocity

vvavgavg = = ΔΔs / t s / t

= (s – s= (s – s00) / t) / t

Assuming constant accelerationAssuming constant acceleration

vvavgavg = (v + v = (v + v00) / 2) / 2

Combine the two:Combine the two:

(s – s(s – s00) / t = (v + v) / t = (v + v00) / 2) / 2

ΔΔs = ½ (v + vs = ½ (v + v00) t) t

Math in BasketballMath in BasketballΔΔs = ½ (v + vs = ½ (v + v00) t) t

By definition: AccelerationBy definition: Acceleration

a = a = ΔΔv / t v / t

= (v – v= (v – v00) / t) / t

Solve for final velocity:Solve for final velocity:

v = vv = v00 + at + at

Substitute velocity into Substitute velocity into ΔΔs equation aboves equation above

ΔΔs = ½ ( (vs = ½ ( (v00 + at) + v + at) + v00) t) t

s – ss – s00 = ½ ( 2v = ½ ( 2v00 + at ) t + at ) t

= v= v00t + ½att + ½at22

s = ss = s00 + v + v00t + ½att + ½at22Ta Da!

Math in BasketballMath in BasketballDisplacement FunctionDisplacement Function

s = ss = s00 + v + v00t + ½att + ½at22

Break into x and y componentsBreak into x and y components(s(sxx): x = x): x = x00 + v + v0x0xt + ½att + ½at22

(s(syy): y = y): y = y00 + v + v0y0yt + ½att + ½at22

Displacement Vectors:Displacement Vectors:

ssy

sx

Math in BasketballMath in Basketball(s(sxx): x = x): x = x00 + v + v0x0xt + ½at + ½axxtt22

(s(syy): y = y): y = y00 + v + v0y0yt + ½at + ½ayytt22

Need further manipulation for use in our real Need further manipulation for use in our real world applicationworld application

Often will not know the time (like in our example Often will not know the time (like in our example here) or some other variablehere) or some other variable

Here:Here:– aaxx = 0, x = 0, x00 = 0 = 0

– aayy = -32 ft/sec = -32 ft/sec22

(s(sxx): x = v): x = v0x0xtt

(s(syy): y = y): y = y00 + v + v0y0yt + (-16)tt + (-16)t22

Math in BasketballMath in Basketball(s(sxx): x = v): x = v0x0xtt

(s(syy): y = y): y = y00 + v + v0y0yt + (-16)tt + (-16)t22

Next, want component velocity in terms of total Next, want component velocity in terms of total velocityvelocity

(s(sxx): x = v): x = v0 0 coscosθθtt

(s(syy): y = y): y = y00 + v + v00sinsinθθ t + (-16)t t + (-16)t22

v

vx

vy

θ

•vv0x0x = v = v00cos cos θθ

•vv0y0y = v = v00sin sin θθExercise!

Math in BasketballMath in Basketball(s(sxx): x = v): x = v0 0 coscosθθtt

(s(syy): y = y): y = y00 + v + v00sinsinθθ t + (-16)t t + (-16)t22

Don’t know time…Don’t know time…

Solve x equation for t and plug into ySolve x equation for t and plug into yt = x / (vt = x / (v0 0 coscosθθ ) )

……into y equation…into y equation…

y = yy = y00 + v + v00sinsinθθ [ [ x / (vx / (v0 0 coscosθθ ) ) ] + (-16)[ ] + (-16)[ x / (vx / (v0 0 coscosθθ ) ) ]]22

y = yy = y00 + x tan + x tanθθ + (-16)[ x + (-16)[ x22 / (v / (v0022

coscos22θθ ) ) ]]

We know initial y, initial x, final x, and our angleWe know initial y, initial x, final x, and our angle

Now we have a usable equation!Now we have a usable equation!

Math in BasketballMath in Basketbally = yy = y00 + x tan + x tanθθ + (-16)[ x + (-16)[ x22 / (v / (v00

22 coscos22θθ ) ) ]]

Distance: x = 14 ftDistance: x = 14 ft

Initial height: yInitial height: y00 = 7 ft (where ball released) = 7 ft (where ball released)

Final height: y = 10 ftFinal height: y = 10 ft

Angle: Angle: θθ = 45 = 45

Find required velocity: vFind required velocity: v00

7 = 10 + (14)tan(45) – 16[ 147 = 10 + (14)tan(45) – 16[ 1422 / (v / (v0022coscos22(45)) ](45)) ]

7 = 10 + 14 – 3136 / (0.5 v7 = 10 + 14 – 3136 / (0.5 v0022))

17 = 6272 / v17 = 6272 / v0022

VV00 = 19.21 ft / sec = 19.21 ft / sec

Math in BasketballMath in BasketballPlayer must throw the ball about 19 feet per second at a 45 Player must throw the ball about 19 feet per second at a 45 degree angle to reach the basketdegree angle to reach the basket

This, of course, wouldn’t guarantee the shot will be madeThis, of course, wouldn’t guarantee the shot will be madeThere are other factors to consider:There are other factors to consider:– Air resistanceAir resistance– Bounce of the ball on the side of the rimBounce of the ball on the side of the rim

Math in BaseballMath in Baseball

What about in baseball?What about in baseball?– Any thoughts?Any thoughts?

So much physicsSo much physics– BattingBatting– Base runningBase running– Pitching Pitching

Math in BaseballMath in Baseball

““Sweet Spot” of hitting a baseballSweet Spot” of hitting a baseball– When bat hits ball, bat vibratesWhen bat hits ball, bat vibrates

–Frequency and Frequency and intensity depend on intensity depend on location of contactlocation of contact

–Vibration is really Vibration is really energy being energy being transferred from ball to transferred from ball to the bat (useless)the bat (useless)

Math in BaseballMath in BaseballSweet spot on bat where, when ball contacts, Sweet spot on bat where, when ball contacts, produces least amount of vibration…produces least amount of vibration…– Least amount of energy lost, maximizing energy Least amount of energy lost, maximizing energy

transferred to balltransferred to ball

Math in BaseballMath in BaseballPitching a Curve BallPitching a Curve Ball– Ball thrown with a downwardBall thrown with a downward

spin. Drops as it approachesspin. Drops as it approachesplateplate

For years, debated whether For years, debated whether curve balls actually curvedcurve balls actually curvedor it was an optical illusionor it was an optical illusion

With today’s technology,With today’s technology,it’s easy to see that they it’s easy to see that they do indeed curvedo indeed curve

Math in BaseballMath in BaseballCurve BallCurve Ball– Like most pitches, makes use of Magnus ForceLike most pitches, makes use of Magnus Force– Stitches on the ball cause drag when flying Stitches on the ball cause drag when flying

through the airthrough the air– Putting spin on the ball causes more drag on Putting spin on the ball causes more drag on

one side of the ballone side of the ball

Math in BaseballMath in BaseballFFMagnus ForceMagnus Force = = KwVCKwVCvv

K = Magnus K = Magnus Coefficient Coefficient w = spin frequency w = spin frequency V = velocityV = velocity

CCvv = drag = drag coefficientcoefficient

More spin = bigger More spin = bigger curvecurveFaster pitch = Faster pitch = bigger curve bigger curve

Math in BaseballMath in BaseballBattingBatting

90 mph fastball takes 0.40 seconds to get 90 mph fastball takes 0.40 seconds to get from the pitcher to the batterfrom the pitcher to the batter

If a batter overestimates by 0.013 second If a batter overestimates by 0.013 second swing will be early and will miss or foul swing will be early and will miss or foul ballball

What’s the best speed/angle to hit a ball?What’s the best speed/angle to hit a ball?

Math in BaseballMath in BaseballUse the same equations:Use the same equations:

(s(sxx): x = x): x = x00 + v + v0x0xt + ½att + ½at22

(s(syy): y = y): y = y00 + v + v0y0yt + ½att + ½at22

Use the same manipulation to get:Use the same manipulation to get:y = yy = y00 + x tan + x tanθθ + (-16)[ x + (-16)[ x22 / (v / (v00

22 coscos22θθ ) ) ]]

Let’s compare velocity (vLet’s compare velocity (v00) and angle () and angle (θθ))…solve for v…solve for v00

Math in BaseballMath in Basebally = yy = y00 + x tan + x tanθθ + (-16)[ x + (-16)[ x22 / (v / (v00

22 coscos22θθ ) ) ]]

Solved for vSolved for v0 0 (ft/sec)(ft/sec)

At a particular ballpark, home run distance is At a particular ballpark, home run distance is constantconstant– So distance (x) and height (y) are knownSo distance (x) and height (y) are known

20

2

0 cos)tan(

16

xyy

xv

Math in BaseballMath in BaseballGraphing solved function with known x and y Graphing solved function with known x and y compares velocity with angle of hitcompares velocity with angle of hit– shows a parabolic function with a minimum at 45 shows a parabolic function with a minimum at 45

degreesdegrees

When hit at a 45 degree angle, the ball requires When hit at a 45 degree angle, the ball requires the minimum home run velocity to reach the end the minimum home run velocity to reach the end of the ball parkof the ball park

Best angle is at 45 degreesBest angle is at 45 degrees

Exercise!

Math in BaseballMath in Baseball

)35(cos))35tan()500(203(

)500(16

cos)tan(

162

2

20

2

0

xyy

xv

775.133812.17895516.223

4000000 ft / sec

≈91.21 mph

Math in BaseballMath in BaseballPrevious examples do not incorporate drag or liftPrevious examples do not incorporate drag or lift

Graphs with equations including drag and lift:Graphs with equations including drag and lift:

Optimal realistic angle:Optimal realistic angle:about 35 degreesabout 35 degrees

Stats in BaseballStats in BaseballBaseball produces and uses more statistics than any other Baseball produces and uses more statistics than any other sportsport

Evaluating Team’s PerformanceEvaluating Team’s Performance

Evaluating Player’s PerformanceEvaluating Player’s Performance

Coaches and fantasy players use these stats to make Coaches and fantasy players use these stats to make choices about their teamchoices about their team

2010 Season Stats

SPLITS G AB R H 2B 3B HR RBI BB SO SB CS AVG OBP SLG OPS

Season 8 23 8 8 2 0 2 5 8 2 1 0 .348 .516 .696 1.212

Career 619 2146 271 631 158 2 93 394 208 303 13 3 ? .358 .500 .858

Last 7 days 6 18 4 6 2 0 1 4 4 2 0 0 .333 .455 .611 1.066

Projected 162 466 162 162 41 0 41 101 162 41 20 0 .348 .516 .696 1.212

Stats in BaseballStats in BaseballSome Important Stats:Some Important Stats:

BattersBatters– Batting Average (BA)Batting Average (BA)– Runs Batted In (RBI)Runs Batted In (RBI)– Strike Outs (SO)Strike Outs (SO)– Home Runs (HR)Home Runs (HR)

PitchersPitchers– Earned Run Average (ERA)Earned Run Average (ERA)– Hits Allowed (per 9 innings) (H/9)Hits Allowed (per 9 innings) (H/9)– Strikeouts (K)Strikeouts (K)

Stats in BaseballStats in BaseballBatting Average (BA)Batting Average (BA)

– Ratio between of hits to “at bats”Ratio between of hits to “at bats”– Method of measuring player’s batting Method of measuring player’s batting

performanceperformance– Format:Format:

.348.348

– ““Batting 1000”Batting 1000”

AtBats

HitsAVGBA )(

Exercise Exercise

≈ ≈ .294.294

Stats in BaseballStats in BaseballRuns Batted In (RBI)Runs Batted In (RBI)– Number of runs a player has batted inNumber of runs a player has batted in

Earned Run Average (ERA)Earned Run Average (ERA)– Mean of earned runs given up by a pitcher per Mean of earned runs given up by a pitcher per

nine inningsnine innings

Hits Allowed (H/9)Hits Allowed (H/9)– Average number of hits allowed by pitcher in a Average number of hits allowed by pitcher in a

nine inning periodnine inning periodpitchedinnings

allowedhitsH

_

)9_(9/

PitchedInnings

AllowedRunsEarnedERA

_

__9

SoccerSoccer““Soccer is a game of angles”Soccer is a game of angles”

Goaltending vsGoaltending vsShootingShooting

Angles in SoccerAngles in SoccerGoaltendingGoaltending

As a keeper, you want to give the shooter As a keeper, you want to give the shooter the smallest angle between him and the the smallest angle between him and the two posts of the goaltwo posts of the goal

A B

Player

Goal

θ

Able to cut off a Able to cut off a significant amount of significant amount of shots like thisshots like this

Where should goalie Where should goalie stand to best defend a stand to best defend a shot?shot?

Angles in SoccerAngles in SoccerPenalty KicksPenalty Kicks

This is why during penalty kicks, goalies are This is why during penalty kicks, goalies are required to stand on the goal line until the ball is required to stand on the goal line until the ball is touched. touched.

If they were able to approach the ball before, the If they were able to approach the ball before, the goalie would significantly decrease angle of goalie would significantly decrease angle of attackattack

A B

Playerθ

Goalie

Angles in SoccerAngles in SoccerMay think it best to stand in a position that May think it best to stand in a position that bisects goal line bisects goal line

Gives shooter more room between goalie Gives shooter more room between goalie and left post, than right postand left post, than right post

Angles in SoccerAngles in SoccerInstead would be better to bisect the angle Instead would be better to bisect the angle between shooter and two postsbetween shooter and two posts

Goalie should also stand square to the ballGoalie should also stand square to the ball

Angles in SoccerAngles in SoccerAs distance from goal increases, the angle As distance from goal increases, the angle bisection approaches the goal line bisection approaches the goal line bisectionbisection

Angles in SoccerAngles in SoccerShootingShooting

On the opposite end, shooter wants to On the opposite end, shooter wants to maximize angle of attackmaximize angle of attack

What path should they take?What path should they take?

http://illuminations.nctm.org/ActivityDetail.aspx?ID=158

Sports & Math EducationSports & Math EducationIncorporation and application of math in sports is Incorporation and application of math in sports is a creative, and wildly successful method of a creative, and wildly successful method of teaching mathematicsteaching mathematics

Professors, University of Mississippi taught Professors, University of Mississippi taught fantasy football to 80 student athletes. Before, fantasy football to 80 student athletes. Before, 38% received A’s on a pretest. After, 83% 38% received A’s on a pretest. After, 83% received A’s on a postestreceived A’s on a postest

http://www.fantasysportsmath.com/

Sports & Math EducationSports & Math EducationInnovative way to get students doing mathInnovative way to get students doing math

Even if some are not interested, they’re Even if some are not interested, they’re able to understand the practicality and able to understand the practicality and application of mathematical conceptsapplication of mathematical concepts

DiscussionDiscussionWhat sports did you all play?What sports did you all play?

Can you think of any other ways math is Can you think of any other ways math is involved in sports?involved in sports?

Do you think incorporating sports is an Do you think incorporating sports is an effective method of teaching effective method of teaching mathematics?mathematics?– Why or why not?Why or why not?