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 • 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 • 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
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 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 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?