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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 = s 0 + v 0 t + ½at 2 s = final position s 0 = initial position v 0 = 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 v avg = Δs / t = (s – s 0 ) / t Assuming constant acceleration v avg = (v + v 0 ) / 2 Combine the two: (s – s 0 ) / t = (v + v 0 ) / 2 Δs = ½ (v + v 0 ) t

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Math in Basketball Δs = ½ (v + v 0 ) t By definition: Acceleration a = Δv / t = (v – v 0 ) / t Solve for final velocity: v = v 0 + at Substitute velocity into Δs equation above Δs = ½ ( (v 0 + at) + v 0 ) t s – s 0 = ½ ( 2v 0 + at ) t = v 0 t + ½at 2 s = s 0 + v 0 t + ½at 2 Ta Da!

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Math in Basketball Displacement Function s = s 0 + v 0 t + ½at 2 Break into x and y components (s x ): x = x 0 + v 0x t + ½at 2 (s y ): y = y 0 + v 0y t + ½at 2 Displacement Vectors: s sysy sxsx

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Math in Basketball (s x ): x = x 0 + v 0x t + ½a x t 2 (s y ): y = y 0 + v 0y t + ½a y t 2 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: –a x = 0, x 0 = 0 –a y = -32 ft/sec 2 (s x ): x = v 0x t (s y ): y = y 0 + v 0y t + (-16)t 2

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Math in Basketball (s x ): x = v 0x t (s y ): y = y 0 + v 0y t + (-16)t 2 Next, want component velocity in terms of total velocity (s x ): x = v 0 cosθt (s y ): y = y 0 + v 0 sinθ t + (-16)t 2 v vxvx vyvy θ v 0x = v 0 cos θv 0x = v 0 cos θ v 0y = v 0 sin θv 0y = v 0 sin θ Exercise!

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Math in Basketball (s x ): x = v 0 cosθt (s y ): y = y 0 + v 0 sinθ t + (-16)t 2 Don’t know time… Solve x equation for t and plug into y t = x / (v 0 cosθ ) …into y equation… y = y 0 + v 0 sinθ [ x / (v 0 cosθ ) ] + (-16)[ x / (v 0 cosθ ) ] 2 y = y 0 + x tanθ + (-16)[ x 2 / (v 0 2 cos 2 θ ) ] 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 = y 0 + x tanθ + (-16)[ x 2 / (v 0 2 cos 2 θ ) ] Distance: x = 14 ft Initial height: y 0 = 7 ft (where ball released) Final height: y = 10 ft Angle: θ = 45 Find required velocity: v 0 7 = 10 + (14)tan(45) – 16[ 14 2 / (v 0 2 cos 2 (45)) ] 7 = 10 + 14 – 3136 / (0.5 v 0 2 ) 17 = 6272 / v 0 2 V 0 = 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 downward spin. Drops as it approaches plate For years, debated whether curve balls actually curved or 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 F Magnus Force = KwVC v K = Magnus Coefficient w = spin frequency V = velocity C v = 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: (s x ): x = x 0 + v 0x t + ½at 2 (s y ): y = y 0 + v 0y t + ½at 2 Use the same manipulation to get: y = y 0 + x tanθ + (-16)[ x 2 / (v 0 2 cos 2 θ ) ] Let’s compare velocity (v 0 ) and angle ( θ ) …solve for v 0

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Math in Baseball y = y 0 + x tanθ + (-16)[ x 2 / (v 0 2 cos 2 θ ) ] Solved for v 0 (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 2010 Season Stats SPLITSGABRH2B3BHRRBIBBSOSBCSAVGOBPSLGOPS Season8238820258210.348.516.6961.212 Career6192146271631158293394208303133?.358.500.858 Last 7 days6184620144200.333.455.6111.066 Projected162466162 410 10116241200.348.516.6961.212

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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 vs Shooting

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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 A B Player Goal θ Able to cut off a significant amount of shots like this Where should goalie stand to best defend a shot?

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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 A B Player θ 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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