Application of Mathematics in sports

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Mr. PUNDIKALA VEERESHA M Sc.
APPLICATION OF MATHEMATICS IN SPORTS
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SPORTS

• INTRODUCTION
• Sport (or sports) is all forms of usually
  competitive physical activity which, through
  casual or organized participation, aim to use
  maintain or improve physical ability and skills
  while providing entertainment to participants.
• Sports are usually governed by a set of rules
  or customs, which serve to ensure fair
  competition, and allow consistent adjudication of
  the winner. Winning can be determined by physical
  events such as scoring goals or crossing a line
  first, or by the determination of judges who are
  scoring elements of the sporting performance.

• HISTORY
• There are artifacts and structures that
  suggest that the Chinese engaged in sporting
  activities as early as 2000 BC. Monuments to the
  Pharaohs indicate that a number of sports,
  including swimming and fishing, were
  well-developed and regulated several thousands of
  years ago in ancient Egypt.
• A wide range of sports were already
  established by the time of Ancient Greece and the
  military culture and the development of sports in
  Greece influenced one another considerably.
  Sports became such a prominent part of their
  culture that the Greeks created the Olympic
  Games,.

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UNIT 2.1 ATHLETICS

• INTRODUCTION AND HISTORY
• Athletics is an exclusive collection of sporting
  events that involve competitive running, jumping,
  throwing, and walking. The most common types of
  athletics competitions are track and field, road
  running, cross country running, and race walking.
  The simplicity of the competitions, and the lack
  of a need for expensive equipment, makes
  athletics one of the most commonly competed
  sports in the world.

Athletics events were depicted in the Ancient
Egyptian tombs in Saqqara, with illustrations of
running at the HebSed festival and high jumping
appearing in tombs from as early as of 2250 BC.
The Tailteann Games were an ancient Celtic
festival in Ireland, founded around 1800 BC, and
the thirty-day meeting included running and
stone-throwing among its sporting events.
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• APPLICATIONS

1. If a runner ran one lap of the track in the
second lane, how far did she / he run?

• Solution
• You might think that the answer is 400 meters,
  since it is a 400 meter track. Both
  straight-aways are 100 meters but as you go
  further out from the inside rail on the
  semicircular turns, you run farther. To calculate
  the distance around the turns, we use the formula
  for the circumference (C) of a circle with radius
  (r) ?? 2???? since the two semicircles on the
  inside rail form a circle with a circumference of
  200 m, we can find the radius of the inside rail
  semicircles.
• ?? 2?????2002????
• 
  ??31.83

Now when you run in the second lane, the
semicircle turns have a radius that is one meter
longer or 32.83 m Thus, distance around both
turns in the second lane is ?? 2????2??
32.83 206.2 ??
.
Thus, a lap in the second lane has a total
distance of 406.2 m, 200 meters for the
straight-aways plus 206.2 m for the turns.
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• 2. A runner jogs ten laps of the track in the
  eighth lane. How far does he run?
• a) Measured in meters. b) Measured in miles.
• Solution
  
  
• a) Let us first determine the distance of one
  lap in the eighth lane. The straight-away are
  still 100 m each but the semicircular turns now
  have a radius that is 8 m more than the radius of
  the inside rail.
• 
  ??31.838 39.83 ??
• The distance around both turns in the eighth lane
  is as follows.
• 
  ?? 2???? 2??(39.83)250.1??
• Thus, a lap in the eighth lane is 450.1 m, 200
  meters for the straight-aways plus 250.1 m for
  the turns. Ten laps would be ten times that
  amount or 4501 m.
• b) To change 4501 meters into miles we can set up
  a proportion comparing miles to meters using the
  conversion, 6.2 mi 10,000 m.
• Let x the distance in miles
• 6.2 /
  10,000 ?? / 4501 ? 10,000 ?? 27,906.2
• 
  ?? 2.8
• By running ten laps in the eighth lane, the
  jogger ran about 2.8 miles.

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• 3. An archery target consists of five concentric
  as shown . The value for an arrow in each region
  starting from the inner circle is 9, 7, 5, 3, 1
  points. In how many ways could five scoring
  arrows exam 29 points?
• Solution
• We can set up Diophantine equations to model the
  problem.
• Let a the number of 9 point arrows
• b the number of 7 point arrows
• c the number of 5 point arrows
• d the number of 3 point arrows
• e the number of 1 point arrows.
  Since there are five scoring arrows,
• ?? ?? ?? ?? ??
  5.
• Since the five arrows score 29 points,
  
• 9?? 7?? 5?? 3?? 1 ?? 29
  
• We can now set up a table of values and
  systematically
• find all possible values that simultaneously
  satisfy both equations.
• We can start with the largest value for a and
  determine the
• values for the other variables making sure we use
  whole numbers and
• a total of five arrows.

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The maximum value for a is 3, since when a
reaches 4, 9??36, which is over the 29 total
points.
a(9) b(7) c(5) d(3) e (1)
3 0 0 0 2
2 0 2 0 1
2 1 0 1 1
1 2 1 0 1
1 2 0 2 0
1 1 2 1 0
1 0 4 0 0
0 4 0 0 1
0 3 1 1 0
0 2 3 0 0
There are ten ways to
attain a score of 29 with five arrows. The
examples in this section have shown you that math
can be used in analyzing sport stats and scoring.
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UNIT 2.2 BASKET BALL

• INTRODUCTION AND HISTORY
• Basketball is a sport played by two teams of
  five players on a rectangular court. The
  objective is to shoot a ball through a hoop 18
  ??????h???? (46 ????) in diameter and 10 ????????
  (3.0 ??) high mounted to a backboard at each end.
  Basketball is one of the world's most popular and
  widely viewed sports.
• In early December 1891, Canadian American Dr.
  James Naismith, a physical education professor
  and instructor at the International Young Men's
  Christian Association Training School (YMCA)
  (today, Springfield College) in Springfield,
  Massachusetts, (USA).
• In 1922, the Commonwealth Five, the first
  all-black professional team was founded. The New
  York Renaissance was founded in 1923.In 1939 the
  all-black New York Renaissance beat the all-white
  Oshkosh All-Stars in the World Pro Basketball
  Tournament.

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• APPLICATIONS

We know the equation
?? ?? -16 ?? 0 2 ???? ?? 2 ?? ?? 2
???????? ?? h 0
Where h 0 Height from which
the ball is thrown, ?? Angle at which the ball
is thrown, ?? 0 Speed at
which the ball is thrown, ?? Distance
that the ball travels, To help figure out the
velocity at which a basketball player must throw
the ball in order for it to land perfectly in the
basket.  When shooting a basketball you want the
ball to hit the basket at as close to a right
angle as possible.  For this reason, most players
attempt to shoot the ball at a 45angle.  To find
the velocity at which a player would need to
throw the ball in order to make the basket we
would want to find the range of the ball when it
is thrown at a 45 angle.  The
formula for the range of the ball is
?????????? ?? 0 2 sin(2??) 32


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But since for angle at which the ball is thrown
is ??45 (angle), we have ?????????? ?? 0 2
sin?(2??) 32 ?? 0 2 sin?(245) 32 ?? 0 2
32 Now, if a
player is shooting a 3 point shot, then he is
approximately 25 ???????? from the basket. If we
look at the graph of the range function we can
get an idea of how hard the player must throw the
ball in order to make a 3 point.
So, by solving the formula knowing that the range
of the shot must be 25????????, 25 ?? 0 2 32
? ?? 0 2 800 ?? 0 28.2843 So in order to
make the 3 point shot, the player must throw the
ball at approximately 28 ???????? per second,
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1. A basketball player realizes that in order to
be more competitive in salary negotiations, he
needs to complete at least 88 of his free throw
attempts by the end of the season. After 50
games, he has made 173 out of 202 attempted free
throws. If in the last 42 games of the season, he
expects to have 168 more free throws, how many of
those does he have to make in order to reach the
88 free throw mark? Solution Let
?? the number of free throws he needs to make
out of 168. 173 ?? the number of
free throws made 202168 the number of
free throws attempted 173 ?? 370
ratio of free throws made to attempted To
find x set that ratio equal to 88 0.88 and
solve. 173 ?? 370 0.88
173??325.6 ?
??152.6153 Thus, by making 153 out of his last
168 free throws, the player will reach the 88
mark.
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UNIT 2.3 BASEBALL

• INTRODUCTION AND HISTORY
• Baseball is a bat-and-ball game played between
  two teams of nine players who take turns batting
  and fielding. The offense attempts to score more
  runs than its opponents by hitting a ball thrown
  by the pitcher with a bat and moving
  counter-clockwise around a series of four bases
  first, second, third and home plate. A run is
  scored when the runner advances around the bases
  and returns to home plate.
• David Block discovered that the first recorded
  game of Bass-Ball took place in 1749 in Surrey,
  and featured the Prince of Wales as a player.
  William Bray, an English lawyer, recorded a game
  of baseball on Easter Monday 1755 in Guildford,
  Surrey.
• By the early 1830s, there were reports of a
  variety of un-codified bat-and-ball games
  recognizable as early forms of baseball being
  played around North America.

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• APPLICATIONS

• A professional baseball team has won 62 and lost
  70 games. How many consecutive wins would bring
  them to the 500 mark?
• Solution
• Let ?? The number of
  consecutive wins.
• 62 ?? The number of wins
• 62 70 ?? The total games played
• ( 62 ?? ) / (132 ??) Ratio of wins to
  total games
• Since the 500 mark is 50, set the ratio equal
  to 0.50and solve for x
• ( 62 ?? ) / ( 132 ?? ) 0.50
• 62 ?? 66
  0.50 ??
• 0.5 ?? 4
• ??
  8
• By winning 8 consecutive games,
  the team would reach the 500 mark

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• The Pythagorean Theorem
• There are many instances when distances in sports
  can be
• determined using the Pythagorean Theorem.
• This theorem states for a right triangle with
  legs a and b and hypotenuse c,

?? 2 ?? 2 ?? 2
2. How long is the throw from third base to first
base on a professional baseball diamond where the
bases are 90 feet apart? Solution A right
triangle is formed by the third base line and the
first base line with the throw from third to
first is the hypotenuse of the right triangle, if
d represents the distance from third base to
first base, using the Pythagorean Theorem, we get
the following ?? 2
?? 2 ?? 2 90 2 90
2 ?? 2 ?16200 ?? 2
127.3 ??
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UNIT 2.4 GOLF

• INTRODUCTION AND HISTORY
• Golf is a sport that many people enjoy and there
  are more math aspects of the game than most
  people can imagine. The winner was whoever hit
  the ball with the least number of strokes into a
  target several hundred yards away.
• A golf-like game is, apocryphally, recorded as
  taking place on 26 February 1297, in Leonean
  de-Vecht, where the Dutch played a game with a
  stick and leather ball was also played in
  17th-century Netherlands and that this predates
  the game in Scotland. There are also other
  reports of earlier accounts of a golf-like game
  from continental Europe.

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• APPLICATIONS

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• Professional golfers frequently drive a golf ball
  300 or more yards, but very few have an average
  distance of 300 or more yards for the year. For
  example, at one point in a season, Tiger woods
  had an average of 293.4 ?????????? out of 104
  officially measured drives. What does he have to
  average on his next 40 measured drives to bring
  his average for the yard up to 300 ???????????
• Solution
• To find the average distance, find the total
  yards of all his drives divided by the number of
  drives.
• Let ?? the average of the next 40
  drives.
• Total yards 104 drives at 293.4 yards plus 40
  drives at x yards
• 104(293.4) 40?? 30,513.6
  40??
• Number of drives 104 40 144
• Average total yards/number of drives
• (30,513.640??)/144
• Setting the average to 300 and solving for x, we
  get
• (30,513.6 40??)/144 300? 30,513.6 40??
  43,200
• ??
  317.16
• Thus, Tiger Woods must average 317.16 yards on
  his next 40 measured drives to average 300 yards
  for year.

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• Place-kicker scored a school record of 17 points
  with field goals (3 points each) and extra points
  (1 point each). How many different ways could he
  have scored the 17 points ?
• Solution
• We could use a guessing process and determine
  the different ways to score 17 points, for
  example, 4 field goals and 5 extra points or 5
  field goals and 2 extra points. However, if we
  are not careful in our analysis, we might miss
  some of the possible solutions.
• Let ?? the number of field goals
  scored.
• ?? the number of extra points
  scored.
• Since each field goal is 3 points and each extra
  point is one point, we get the following.
• 3?? 1?? 17
• There are some logical restrictions on x and y
  both must be whole numbers
• The maximum value for ?? 5 since 6 field goals
  would be 18 points and the maximum value for
  ??17.
• Thus, the solution to the problem becomes solving
  the equation
• 3?? 1??17 where x and y are whole numbers (
  0??5, ?????? 0 ??17)
• Consider a table for x and y and systematically
  find all solutions by assigning an integer for x
  from 0 to 5 and calculating the value for y.

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X (3 points) Y ( 1 points)
0 17
1 14
2 11
3 8
4 5
5 2
There are six ways in which the kicker could
score 17 points. Such an equation with integer
coefficients and more than one integer solution
is called a Diophantine Equation after the Greek
mathematician, Diophantus (c. 250 A.D).
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UNIT 2.5 FOOTBALL

• INTRODUCTION AND HISTORY
• Football refers to a number of sports that
  involve, to varying degrees, kicking a ball with
  the foot to score a goal. The most popular of
  these sports worldwide is association football,
  more commonly known as just football or
  soccer.
• The Ancient Greeks and Romans are known to have
  played many ball games, some of which involved
  the use of the feet. The Roman game harpastum is
  believed to have been adapted from a Greek team
  game known as ?p?s????? (Episkyros), which is
  mentioned by a Greek playwright, Antiphanes
  (388311 BC). These games appear to have
  resembled rugby football. The Roman politician
  Cicero (10643 BC) describes the case of a man
  who was killed whilst having a shave when a ball
  was kicked into a barber's shop.

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• APPLICATIONS
• 1. If a college football player kicks a
  field-goal when the ball was put in play 40 yards
  from the goal line, what is the maximum length of
  a field goal as measured along the ground?
• Solution
• To find the maximum length, we have to take the
  following into consideration
• 1. Field goals are kicked from a point 7 yards
  behind the scrimmage line.
• 2. The field goal is kicked from the right hash
  mark and cleared the left goal post.
• 3. From the diagram of the football field below,
  we see that
• The goal is 10 yards deep in the end zone
• The hash mark is 60 ft. from the side line
• The left goal post is 89.25 ft. from the side
  line.
• Let ?? the distance to the farthest goal post
  along the ground.

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The ball was kicked 7 yards behind the scrimmage
line and there are 40 yards to the goal line and
10 yards more to the end line where the goal post
is located. Thus, the perpendicular distance from
the point which the ball was kicked to the end
line is 7 40 10 57 yards or 171 feet. The
distance along the end line from the left goal
post to the perpendicular is 89.25 ft. 60 ft.
29.25 ft. The right triangle formed can be
solved using Pythagorean theorem.
?? 2
29.252 1712
?? 2 30096.5625
??173.5 ???? 57.8 ????
Thus, even though the ball was
kicked with a 40-yard scrimmage line, the maximum
distance of the kick as measured along the ground
is about 57.8 yards.
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• 2. A football player runs 40 yards in 4.2
  seconds, what is his speed
• In feet per second and
• Miles per hour? 
• Solution
• We are given the distance in yards but we need
  the distance in feet to represent the rate in
  feet per second. To convert yards to feet we can
  set up and solve a proportion comparing yards to
  feet. Using the conversion fact,1 ???? 3 ????,
  we get
• (d the distance in feet)
• 1 ????/ 3 ???? 40 ???? / ?? ????
• 1/3 40 / ??
• 1?? 3 (40)
• ?? 120 ????
• Thus, the football player ran a distance of 120
  ???? (?? 120) in 4.2 seconds (t 4.2) and the
  rate (r) is ?? ?? / ?? 120 ???? / 4.2 28.6
  ???? /??????

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(b) To find the rate in miles per hour, we can
change the distance, 120 ????, into miles and the
time, 4.4 ??????????????, into hours. As before,
we do this by setting up and solving proportions
using the conversion facts, 1 ???? 5280 ft and
1 h?? 3600 ??????. Distance
Let ?? the distance in
miles 1 ?????????? / 5280 ???? ??
?????????? / 120 ???? 1 /
5280 ?? / 120 ??
0.022727 ???? Time Let ?? the time in
hours 1 h?? / 3600 ?????? ?? h??
/ 4.4 ?????? 1 / 3600
?? / 4.4 ??
0.001222 h?? The rate (r) is ?? ?? / ??
0.022727 ???? / 0.001222 h??
?? 18.6 ????h
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