Math and Games.


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wicker container. Think about any ways math can be utilized as a part of b-ball? Math in ... toss the ball around 19 feet for every second at a 45 degree edge to achieve the wicker container ...
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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 estimations 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 & edge of shots Physics mathematical statements and inference Baseball Pitching Home run swings Stats Soccer Angles of guard/offense Math in Education

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Math in Basketball Score Keeping 2 point, 3 point shots Free tosses 94\' by 50\' court Basket 10\' off the ground Ball distance across 9.5" Rim measurement 18.5" 3 point line around 24\' from wicker bin Think of any ways math can be utilized as a part of b-ball?

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Math in Basketball Shot At what speed ought to a foul shot be taken? Suppositions/Given: Distance About 14 feet (x course) from FT line to center of the wicker bin Height 10 feet from ground to edge Angle of methodology Close to 90 degrees as could reasonably be expected 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 = last position s 0 = introductory position v 0 = starting speed t = time a = increasing speed This is 490… .where did this mathematical statement originate from?

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Math in Basketball By definition: Average speed v avg = Δ s/t = (s – s 0 )/t Assuming consistent quickening v avg = (v + v 0 )/2 Combine the two: (s – s 0 )/t = (v + v 0 )/2 Δ s = ½ (v + v 0 ) t

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Math in Basketball Δ s = ½ (v + v 0 ) t By definition: Acceleration a = Δ v/t = (v – v 0 )/t Solve for conclusive speed: v = v 0 + at Substitute speed into Δ s mathematical statement above Δ s = 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 segments (s x ): x = x 0 + v 0x t + ½at 2 (s y ): y = y 0 + v 0y t + ½at 2 Displacement Vectors: s y s x

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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 control for use in our genuine application Often won\'t know the time (like in our case 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, need part speed as far as aggregate speed (s x ): x = v 0 cos θ t (s y ): y = y 0 + v 0 sin θ t + (- 16)t 2 v y v 0x = v 0 cos θ v 0y = v 0 sin θ Exercise! θ v x

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Math in Basketball (s x ): x = v 0 cos θ t (s y ): y = y 0 + v 0 sin θ t + (- 16)t 2 Don\'t know time… Solve x mathematical statement for t and fitting into y t = x/(v 0 cos θ ) … into y comparison… y = y 0 + v 0 sin θ [ x/(v 0 cos θ ) ] + (- 16)[ x/(v 0 cos θ ) ] 2 y = y 0 + x tan θ + (- 16)[ x 2/(v 0 2 cos 2 θ ) ] We know starting y, beginning x, last x, and our point Now we have a usable mathematical statement!

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Math in Basketball y = y 0 + x tan θ + (- 16)[ x 2/(v 0 2 cos 2 θ ) ] Distance: x = 14 ft Initial tallness: y 0 = 7 ft (where ball discharged) Final stature: y = 10 ft Angle: θ = 45 Find required speed: 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 toss the ball around 19 feet for every second at a 45 degree edge to achieve the wicker container This, obviously, wouldn\'t ensure the shot will be made There are different elements to consider: Air resistance Bounce of the ball in favor of the edge

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Math in Baseball What about in baseball? Any contemplations? So much material science 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 power rely on upon area of contact Vibration is truly vitality being exchanged from ball to the bat (futile)

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Math in Baseball Sweet spot on bat where, when ball contacts, creates minimum measure of vibration… Least measure of vitality lost, boosting vitality exchanged to ball

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Math in Baseball Pitching a Curve Ball tossed with a descending twist. Drops as it methodologies plate For years, faced off regarding whether curveballs really bended or it was an optical deception With today\'s innovation, it\'s anything but difficult to see that they do in reality bend

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Math in Baseball Curve Ball Like most pitches, makes utilization of Magnus Force Stitches on the ball cause drag when flying through the air Putting turn on the ball causes more delay one side of the ball

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Math in Baseball F Magnus Force = KwVC v K = Magnus Coefficient w = turn recurrence V = speed C v = drag coefficient More turn = greater bend Faster pitch = greater bend

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Math in Baseball Batting 90 mph fastball takes 0.40 seconds to get from the pitcher to the hitter If a player overestimates by 0.013 second swing will be early and will miss or foul ball What\'s the best speed/edge to hit a ball?

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Math in Baseball Use the same comparisons: (s x ): x = x 0 + v 0x t + ½at 2 (s y ): y = y 0 + v 0y t + ½at 2 Use the same control to get: y = y 0 + x tan θ + (- 16)[ x 2/(v 0 2 cos 2 θ ) ] Let\'s look at speed (v 0 ) and edge ( θ ) … illuminate for v 0

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Math in Baseball y = y 0 + x tan θ + (- 16)[ x 2/(v 0 2 cos 2 θ ) ] Solved for v 0 (ft/sec) At a specific ballpark, grand slam separation is steady So remove (x) and stature (y) are known

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Math in Baseball Graphing unraveled capacity with known x and y contrasts speed and point of hit demonstrates an illustrative capacity with a base at 45 degrees When hit at a 45 degree edge, the ball requires the base grand slam speed to achieve the end of the ball stop Best edge 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 illustrations don\'t fuse drag or lift Graphs with mathematical statements including drag and lift: Optimal reasonable edge: around 35 degrees

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Stats in Baseball delivers and uses a bigger number of measurements than some other game Evaluating Team\'s Performance Evaluating Player\'s Performance Coaches and dream players utilize these details to settle on decisions about their group

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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 execution 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 surrendered by a pitcher for every nine innings Hits Allowed (H/9) Average number of hits permitted by pitcher in a nine inning period

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"Soccer is a round of points" Goaltending versus Shooting

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Angles in Soccer Goaltending As an attendant, you need to give the shooter the littlest edge amongst him and the two posts of the objective Able to remove a lot of shots like this Where ought to goalie stand to best protect a shot? Player θ A B Goal

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Angles in Soccer Penalty Kicks This is the reason amid extra shots, goalies are required to remain on the objective line until the ball is touched. On the off chance that they could approach the ball some time recently, the goalie would altogether diminish approach Player θ A B Goalie

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Angles in Soccer May think it best to remain in a position that cuts up objective line Gives shooter more space amongst goalie and left post, than right post

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Angles in Soccer Instead would be ideal to divide the point amongst shooter and two posts Goalie ought to likewise stand square to the ball

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Angles in Soccer As separation from objective builds, the edge cut methodologies the objective line cut

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Angles in Soccer Shooting On the inverse end, shooter needs to boost approach What way would it be advisable for them to take? http://illuminations.nctm.org/ActivityDetail.aspx?ID=158

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Sports & Math Education Incorporation and use of math in games is an innovative, and fiercely effective technique for showing arithmetic Professors, University of Mississippi taught dream football to 80 understudy competitors. Some time recently, 38% got An\'s on a pretest. After, 83% got An\'s on a postest http://www.fantasysportsmath.com/

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Sports & Math Education Innovative approach to get understudies doing math Even in the event that some are not intrigued, they\'re ready to comprehend the common sense and utilization of scientific ideas

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Discussion What sports did all of you play? Will you think about whatever other ways math is included in games? Do you think consolidating games is a successful technique for instructing arithmetic? Why or why not?

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