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) Field Goal Percentage (Basketball) Fantasy Sports Playing Sports Geometry Physics Outline

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Presentation Transcript

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
• 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”
• 3 point line about 24’ from basket
• 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
• 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

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?