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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 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

- 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

- 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 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

- Distance

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

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

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 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 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 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 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 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 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 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

- What about 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!

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?

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