1 / 40

620 likes | 2.05k Views

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

Download Presentation
## Math and Sports

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**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?

More Related