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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

Math and Sports

Paul Moore

April 15, 2010

math in sports
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
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
math in basketball
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?
math in basketball5
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
math in basketball6
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?
math in basketball7
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

math in basketball8
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!

math in basketball9
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

math in basketball10
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

math in basketball11
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

math in basketball12
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!
math in basketball13
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

math in basketball14
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
math in baseball
Math in Baseball
  • What about in baseball?
    • Any thoughts?
  • So much physics
    • Batting
    • Base running
    • Pitching
math in baseball16
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)
math in baseball17
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
math in baseball18
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
math in baseball19
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
math in baseball20
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
math in baseball21
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?
math in baseball22
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
math in baseball23
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
math in baseball24
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!

math in baseball25
Math in Baseball

ft / sec

≈91.21 mph

math in baseball26
Math in Baseball
  • Previous examples do not incorporate drag or lift
  • Graphs with equations including drag and lift:
  • Optimal realistic angle:about 35 degrees
stats in baseball
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
stats in baseball28
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)
stats in baseball29
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
stats in baseball30
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
soccer
Soccer
  • “Soccer is a game of angles”
  • Goaltending vsShooting
angles in soccer
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

angles in soccer33
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

angles in soccer34
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
angles in soccer35
Angles in Soccer
  • Instead would be better to bisect the angle between shooter and two posts
  • Goalie should also stand square to the ball
angles in soccer36
Angles in Soccer
  • As distance from goal increases, the angle bisection approaches the goal line bisection
angles in soccer37
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
sports math education
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/
sports math education39
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
discussion
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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