The Mathematics of Sports:
Myths and Misses
Keep your eye on the ball.
Keep Your Eye on the Ball:
Curveballs, Knuckleballs and
Baseball Fallacies
Robert Watts and Terry Bahill
Keeping Your Eye on the Ball
Keeping Your Eye on the Ball
From trigonometry,
Tan(u) = d/2.
From calculus, if the pitch
has speed 90 mph the rate
of change of u is given by
264
u’ = 4  d 2
Keeping Your Eye on the Ball
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The ball crosses home plate when d =0
For a 90 mph fastball, u’ = 66
radians/second at this time
Unfortunately, most humans can only
track objects at the rate of about 3
radians per second.
So, keeping your eye on the ball in this
case is physically impossible
When Can You Keep
Your Eye on the Ball?
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If the ball is moving slower than 6 ft/s, or
about 4 mph.
At 90 mph, until the ball is 9 feet away;
batter would have started swing already.
“Miles and miles …”
The Longest Golf Shot
Start with Newton’s law F = ma.
Gravity is the only force on the moon.
Acceleration due to gravity on the
moon is about 1/6 that on Earth.
Calculus shows that distance (and
height) of trajectory will be 6 times
the equivalent trajectory on earth.
How far could you hit a 6-iron with one
arm wearing a spacesuit?
Shepard’s estimate:
300 yards
“Miles and miles …”
The Longest Golf Shot
Mikhail Tyurin from International Space Station,
November, 2006. NASA estimate: 1 million miles
Russian estimate: 1 billion miles
The Curve Ball is Just an Illusion
Life Magazine, 1941: "a baseball is so heavy an object
that the pitcher‘s spinning action appears to be
insufficiently strong to appreciably change its course."
Magnus Force
The Curve Ball is Just an Illusion
Life Magazine, 1949: ok, so it actually does curve,
but it does not break sharply like baseball players
say that it does.
Baseball pitches do not hop up or drop down;
this is an illusion caused by the inability of the
batter to watch the ball all the way to the plate.
Curve balls mostly break down and fastballs
(cutters) can move left-to-right. It’s all in the
angle of the spin.
Myth: the wall is fine !
Field Goal
A field goal from a short distance and to the side
must go through the uprights at an angle.
Coach’s decision: kick a field goal from 20 yards from the
side hash mark, or take an intentional 5-yard penalty to
back up and improve the angle.
Field Goals
For a kick from the
hash mark, can you
improve the angle
by backing up?
µ
 = t an¡
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1
¶
 + 
¡ t an ¡

µ
1
¶
 ¡ 

Calculus problem: find x to maximize the
angle A.
For high school, P=11’8” and H=26’8” and
the max A occurs at x=24’.
This is still 2 yards in the endzone!
To get the best angle, don’t back up!
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How much does the angle change?
Degrees
Yards
To hit it twice as hard,
swing twice as hard !
200 mph !
100 mph !
TV show “Sport Science” on HSN:
“Arm speeds” : Hockey 75 mph
Golf
76 mph
“Stick speeds”: Hockey 80 mph
Golf 150 mph
Golf collision “lively” with COR = 0.83.
Hockey collision a complicated dance.
“It wasn’t my fault; the green is bad.”
Stats from the 2007 PGA Tour:
Distance
Percent Made
5 feet
80.9
8 feet
53.1
12 feet
32.1
25 feet
10.6
Stats for the Tru-Roller, 12 feet:
Surface
Pool table
Percent Made
100
PGA course
73
After round
30
The culprit:
footprints !
How Bad Is It?
Werner and Greig estimate a standard deviation
of 2% of the distance. For a 12-foot putt, this
is 2.88 inches, wider than the hole!
Sometimes you get lucky!
Before you leave:
Are there any questions?