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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

### Math in Sports?

• Numbers Everywhere

• Score keeping

• Field/Court measurements

• Sports Statistics

• Batting Average (BA)

• Earned Run Average (ERA)

• Fantasy Sports

• Playing Sports

• Geometry

• Physics

### Outline

• Real World Applications

• Velocity & angle of shots

• Physics equations and derivation

• Baseball

• Pitching

• Home run swings

• Stats

• Soccer

• Angles of defense/offense

• Math in Education

• 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”

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

• 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

• 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?

• 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

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!

• 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

(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

(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

(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!

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

• 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

• Any thoughts?

• So much physics

• Batting

• Base running

• Pitching

### 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 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 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 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 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 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 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 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 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!

ft / sec

≈91.21 mph

### 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

• 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 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 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 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 is a game of angles”

• Goaltending vsShooting

### 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 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 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 Soccer

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

• Goalie should also stand square to the ball

### Angles in Soccer

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

### 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

• 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 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

• 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?