Math and sports

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:
2
s = s0 + v0t + ½at
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
2
= v0t + ½at
2

s = s0 + v0t + ½at

!
a
D
a
T


Math in Basketball
Displacement Function
2
s = s0 + v0t + ½at

Break into x and y components
2
(sx): x = x0 + v0xt + ½at
2
(sy): y = y0 + v0yt + ½at

Displacement Vectors:

sy

s

sx


Math in Basketball

2
(sx
(sx): x = x0
x0 + v0x
v0xtt + ½ax
½axt
2
(sy
(sy): y = y0
y0 + v0y
v0ytt + ½ay
½ayt

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
(sx): x = v0x
v0xtt
2
(sy
(sy): y = y0
y0 + v0y
v0ytt + (-16)t


Math in Basketball

(sx
(sx): x = v0x
v0xtt
2
(sy
(sy): y = y0
y0 + v0y
v0ytt + (-16)t

Next, want component velocity in terms of total velocity

vy

v

ise!
c
r
e
Ex

v0x = v0cos θ
•v

(sx
): x = v0
(sxθ
v0 cosθ
cosθt
2

(sy
(sy): y = y0
y0 + v0
v0sinθ
sinθ t + (-16)t

vx

•v0y = v0sin θ


Math in Basketball
(sx
(sx): x = v0
v0 cosθ
cosθt
2
(sy
(sy): y = y0
y0 + v0
v0sinθ
sinθ t + (-16)t

Don’t know time…
Solve x equation for t and plug into y
t = x / (v0
(v0 cosθ
cosθ )
…into y equation…
2

y = y0
y0 + v0
v0sinθ
sinθ [ x / (v0
(v0 cosθ
cosθ ) ] + (-16)[ x / (v0
(v0 cosθ
cosθ ) ]
2
2
2
y = y0
y0 + x tanθ
tanθ + (-16)[ x / (v0
(v0 cos θ ) ]

We know initial y, initial x, final x, and our angle
Now we have a usable equation!


Math in Basketball
2
2
2
y = y0
y0 + x tanθ
tanθ + (-16)[ x / (v0
(v0 cos θ ) ]
Distance: x = 14 ft
Initial height: y0

y0 = 7 ft (where ball released)
Final height: y = 10 ft
Angle: θ = 45
Find required velocity: v0
v0
2
2
2
7 = 10 + (14)tan(45) – 16[ 14 / (v0
(v0 cos (45)) ]
2
7 = 10 + 14 – 3136 / (0.5 v0
v0 )
2
17 = 6272 / v0
v0
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:
2
(sx): x = x0 + v0xt + ½at
2
(sy): y = y0 + v0yt + ½at

Use the same manipulation to get:
2
2
2
y = y0

y0 + x tanθ
tanθ + (-16)[ x / (v0
(v0 cos θ ) ]

Let’s compare velocity (v0) and angle (θ)
…solve for v0


Math in Baseball
2
2
2
y = y0
y0 + x tanθ
tanθ + (-16)[ x / (v0
(v0 cos θ ) ]

Solved for v0 (ft/sec)

2

16 x
v0 =
2
( y0 − y + x tan θ ) cos θ

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 θ ) cos θ
(3 − 20 + (500) tan(35)) cos 2 (35)

4000000
=
= 17895.812 = 133.775 ft / sec
223.516
≈91.21 mph