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
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
vy
v
•v0x = v0cos θ
•v0y = v0sin θ
θ
vx
(sx): x = v0 cosθt
(sy): y = y0 + v0sinθ t + (-16)t2
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 / (v02 cos2θ ) ]
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 / (v02 cos2θ ) ]
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 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
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 / (v02 cos2θ ) ]
Let’s compare velocity (v0) and angle (θ)
…solve for v0
Math in Baseball
y = y0 + x tanθ + (-16)[ x2 / (v02 cos2θ ) ]
Solved for v0 (ft/sec)
2
16 x
v0 
2
( y0  y  x tan  ) cos 
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
16 x 2
16(500 ) 2
v0 

2
( y0  y  x tan  ) cos 
(3  20  (500 ) tan( 35)) cos 2 (35)
4000000

 17895.812  133.775 ft / sec
223.516
≈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
2010 Season Stats
SPLITS
G
AB
R
H
2B
3B
HR
RBI
BB
SO
SB
CS
AVG
OBP
SLG
OPS
Season
8
23
8
8
2
0
2
5
8
2
1
0
.348
.516
.696
1.212
619
2146
271
631
158
2
93
394
208
303
13
3
?
.358
.500
.858
6
18
4
6
2
0
1
4
4
2
0
0
.333
.455
.611
1.066
162
466
162
162
41
0
41
101
162
41
20
0
.348
.516
.696
1.212
Career
Last 7 days
Projected
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”
Hits
BA( AVG ) 
AtBats
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
Earned _ Runs_ Allowed
ERA  9 
Innings_ Pitched
Hits Allowed (H/9)
– Average number of hits allowed by pitcher in a
nine inning period
(hits _ allowed  9)
H /9 
innings_ pitched
Soccer
“Soccer is a game of angles”
Goaltending vs
Shooting
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
Player
θ
A
B
Goal
Able to cut off a
significant amount of
shots like this
Where should goalie
stand to best defend a
shot?
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
A
θ
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?