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Math and Sports Paul Moore April 15, 2010 Math in Sports? Numbers Everywhere Score keeping Field/Court measurements Sports Statistics Batting Average (BA) Earned Run Average (ERA) Field Goal Percentage (Basketball) Fantasy Sports Playing Sports Geometry Physics Outline

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Math and Sports

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Math and Sports

Paul Moore

April 15, 2010


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Math in Sports?

  • Numbers Everywhere

    • Score keeping

    • Field/Court measurements

  • Sports Statistics

    • Batting Average (BA)

    • Earned Run Average (ERA)

    • Field Goal Percentage (Basketball)

  • Fantasy Sports

  • Playing Sports

    • Geometry

    • Physics


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Outline

  • Real World Applications

    • Basketball

      • Velocity & angle of shots

      • Physics equations and derivation

    • Baseball

      • Pitching

      • Home run swings

      • Stats

    • Soccer

      • Angles of defense/offense

    • Math in Education


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Math in Basketball

  • Score Keeping

    • 2 point, 3 point shots

    • Free throws

  • 94’ by 50’ court

  • Basket 10’ off the ground

  • Ball diameter 9.5”

  • Rim diameter 18.5”

  • 3 point line about 24’ from basket

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


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Math in Basketball

  • Basketball Shot

  • At what velocity should a foul shot be taken?

  • Assumptions/Given:

    • Distance

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

    • Height

      • 10 feet from ground to rim

    • Angle of approach

      • Close to 90 degrees as possible

      • Most are shot at 45 degrees

    • Ignoring air resistance


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Math in Basketball

  • Heavy Use of Kinematic Equations

    • Displacement:

      s = s0 + v0t + ½at2

      s = final position

      s0 = initial position

      v0 = initial velocity

      t = time

      a = acceleration

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


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Math in Basketball

  • By definition: Average velocity

    vavg = Δs / t

    = (s – s0) / t

  • Assuming constant acceleration

    vavg = (v + v0) / 2

  • Combine the two:

    (s – s0) / t = (v + v0) / 2

    Δs = ½ (v + v0) t


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Math in Basketball

Δs = ½ (v + v0) t

  • By definition: Acceleration

    a = Δv / t

    = (v – v0) / t

  • Solve for final velocity:

    v = v0 + at

  • Substitute velocity into Δs equation above

    Δs = ½ ( (v0 + at) + v0) t

    s – s0 = ½ ( 2v0 + at ) t

    = v0t + ½at2

    s = s0 + v0t + ½at2

Ta Da!


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Math in Basketball

  • Displacement Function

    s = s0 + v0t + ½at2

    Break into x and y components

    (sx): x = x0 + v0xt + ½at2

    (sy): y = y0 + v0yt + ½at2

    Displacement Vectors:

sy

s

sx


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Math in Basketball

(sx): x = x0 + v0xt + ½axt2

(sy): y = y0 + v0yt + ½ayt2

  • Need further manipulation for use in our real world application

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

  • Here:

    • ax = 0, x0 = 0

    • ay = -32 ft/sec2

      (sx): x = v0xt

      (sy): y = y0 + v0yt + (-16)t2


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Math in Basketball

(sx): x = v0xt

(sy): y = y0 + v0yt + (-16)t2

  • Next, want component velocity in terms of total velocity

    (sx): x = v0 cosθt

    (sy): y = y0 + v0sinθ t + (-16)t2

vy

v

  • v0x = v0cos θ

  • v0y = v0sin θ

Exercise!

θ

vx


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Math in Basketball

(sx): x = v0 cosθt

(sy): y = y0 + v0sinθ t + (-16)t2

  • Don’t know time…

  • Solve x equation for t and plug into y

    t = x / (v0 cosθ )

    …into y equation…

    y = y0 + v0sinθ [ x / (v0 cosθ ) ] + (-16)[ x / (v0 cosθ ) ]2

    y = y0 + x tanθ + (-16)[ x2 / (v02cos2θ )]

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

  • Now we have a usable equation!


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Math in Basketball

y = y0 + x tanθ + (-16)[ x2 / (v02cos2θ )]

Distance: x = 14 ft

Initial height: y0 = 7 ft (where ball released)

Final height: y = 10 ft

Angle: θ = 45

Find required velocity: v0

7 = 10 + (14)tan(45) – 16[ 142 / (v02cos2(45)) ]

7 = 10 + 14 – 3136 / (0.5 v02)

17 = 6272 / v02

V0 = 19.21 ft / sec


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Math in Basketball

  • Player must throw the ball about 19 feet per second at a 45 degree angle to reach the basket

  • This, of course, wouldn’t guarantee the shot will be made

  • There are other factors to consider:

    • Air resistance

    • Bounce of the ball on the side of the rim


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Math in Baseball

  • What about in baseball?

    • Any thoughts?

  • So much physics

    • Batting

    • Base running

    • Pitching


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Math in Baseball

  • “Sweet Spot” of hitting a baseball

    • When bat hits ball, bat vibrates

  • Frequency and intensity depend on location of contact

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


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Math in Baseball

  • Sweet spot on bat where, when ball contacts, produces least amount of vibration…

    • Least amount of energy lost, maximizing energy transferred to ball


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Math in Baseball

  • Pitching a Curve Ball

    • Ball thrown with a downwardspin. Drops as it approachesplate

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

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


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Math in Baseball

  • Curve Ball

    • Like most pitches, makes use of Magnus Force

    • Stitches on the ball cause drag when flying through the air

    • Putting spin on the ball causes more drag on one side of the ball


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Math in Baseball

  • FMagnus Force = KwVCv

  • K = Magnus Coefficient

  • w = spin frequency

  • V = velocity

  • Cv = drag coefficient

  • More spin = bigger curve

  • Faster pitch = bigger curve


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Math in Baseball

  • Batting

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

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

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


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Math in Baseball

  • Use the same equations:

    (sx): x = x0 + v0xt + ½at2

    (sy): y = y0 + v0yt + ½at2

  • Use the same manipulation to get:

    y = y0 + x tanθ + (-16)[ x2 / (v02cos2θ )]

  • Let’s compare velocity (v0) and angle (θ)…solve for v0


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Math in Baseball

y = y0 + x tanθ + (-16)[ x2 / (v02cos2θ )]

  • Solved for v0 (ft/sec)

  • At a particular ballpark, home run distance is constant

    • So distance (x) and height (y) are known


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Math in Baseball

  • Graphing solved function with known x and y compares velocity with angle of hit

    • shows a parabolic function with a minimum at 45 degrees

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

  • Best angle is at 45 degrees

Exercise!


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Math in Baseball

ft / sec

≈91.21 mph


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Math in Baseball

  • Previous examples do not incorporate drag or lift

  • Graphs with equations including drag and lift:

  • Optimal realistic angle:about 35 degrees


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Stats in Baseball

  • Baseball produces and uses more statistics than any other sport

  • Evaluating Team’s Performance

  • Evaluating Player’s Performance

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


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Stats in Baseball

  • Some Important Stats:

  • Batters

    • Batting Average (BA)

    • Runs Batted In (RBI)

    • Strike Outs (SO)

    • Home Runs (HR)

  • Pitchers

    • Earned Run Average (ERA)

    • Hits Allowed (per 9 innings) (H/9)

    • Strikeouts (K)


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Stats in Baseball

  • Batting Average (BA)

    • Ratio between of hits to “at bats”

    • Method of measuring player’s batting performance

    • Format:

      • .348

    • “Batting 1000”

  • Exercise

  • ≈ .294


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Stats in Baseball

  • Runs Batted In (RBI)

    • Number of runs a player has batted in

  • Earned Run Average (ERA)

    • Mean of earned runs given up by a pitcher per nine innings

  • Hits Allowed (H/9)

    • Average number of hits allowed by pitcher in a nine inning period


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Soccer

  • “Soccer is a game of angles”

  • Goaltending vsShooting


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Angles in Soccer

  • Goaltending

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

Able to cut off a significant amount of shots like this

Where should goalie stand to best defend a shot?

Player

θ

A

B

Goal


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Angles in Soccer

  • Penalty Kicks

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

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

Player

θ

A

B

Goalie


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Angles in Soccer

  • May think it best to stand in a position that bisects goal line

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


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Angles in Soccer

  • Instead would be better to bisect the angle between shooter and two posts

  • Goalie should also stand square to the ball


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Angles in Soccer

  • As distance from goal increases, the angle bisection approaches the goal line bisection


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Angles in Soccer

  • Shooting

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

  • What path should they take?

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


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Sports & Math Education

  • Incorporation and application of math in sports is a creative, and wildly successful method of teaching mathematics

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

  • http://www.fantasysportsmath.com/


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Sports & Math Education

  • Innovative way to get students doing math

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


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Discussion

  • What sports did you all play?

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

  • Do you think incorporating sports is an effective method of teaching mathematics?

    • Why or why not?


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