Baseball Trajectories:
of Inches
Jim Hildensperger
Kyle Spaulding
Dale Garrett
A Game
Baseball: Take Me out to the Ball Game
-Why is baseball considered a game of inches?
-Importance of the pitcher batter confrontation
-According to USA Today, hitting a baseball is the absolute hardest
thing to do in sports. “Considering that a major-league pitch can reach
speeds more than 95 mph, hitters have only 0.4 seconds to find the ball, decide
where the ball is going and swing the bat.
-Yale University physics professor Robert Adair explains that it
takes 0.15 seconds for humans to voluntarily blink their eyes in
response to visual signals.
-In the MLB, you'll get a multimillion-dollar contract if you can
hit a ball successfully anywhere near three out of 10 times.
-On average, there is approximately 1 home run hit during a full
MLB game. This stat is based on the number of home runs allowed
divided by total number of batters faced, times the average
number of batters per game in that specific league (generally
around 38 batters a game).
Factors Effecting the Trajectory of a Batted Ball
-Initial Velocity (Vo, meters per second):
The velocity a ball leaves the bat after contact
-Spin (w, radians per second): The spin rate in which a ball spins as it flies through the air
-Air Temperature (T, degrees Fahrenheit): The average temperature of where the
ball is hit, when it is hit
-Altitude (Y, feet): The measured altitude of the stadium where the baseball is hit
-Angle of Contact (θ, degrees): The angle at which the ball leaves the bat after contact
-Lift and Drag forces (FL and FD, Newton): The Forces acting upon the ball as it
flies through the air
Hypothesis
How do changes in the factors of batted ball’s
Trajectory effect how far it goes?
-The distance a batted ball travels increases as the ball’s rotation rate increases
-The optimum angle of a batted ball depends on its spin rate
-A batted ball travels farther in hotter temperatures and higher altitudes
Newton’s Laws
• 1st Law- An object in a state of
motion tends to remain in
motion unless an external force
is applied to it.
• 2nd Law-The relationship
between an object's mass m, its
acceleration a, and the applied
force F is F = ma. In this law
the direction of the force vector
is the same as the direction of
the acceleration vector.
• 3rd Law-For every action there is
an equal and opposite reaction.
Force Diagram of a Baseball
Calculating the Initial Forces
Approximating velocities
Approximating Distance
Our modified model
• Here it is
Results: Range and Spin Rate
• The range of the ball
increases as the spin rate
increases.
• With spin rate of 100
rad/sec, maximum range
is 112 meters
• With spin rate of 300
rad/sec, maximum range
is 121 meters
• With a spin rate of 600
rad/sec, maximum range
is 134 meters
• Range is maximized when
the ball is spinning its
fastest
Results: Spin Rate and Angle of Contact
• A ball with a slower spin
rate requires a greater
angle of contact to reach
its maximum range.
• With spin rate of 600
rad/sec, maximum range
occurs when angle of
contact is 15º
• With spin rate of 100
rad/sec, maximum range
occurs when angle of
contact is 31º
Results: Data
ω
θ
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
100 rad/s
81.27753454
84.61554166
87.78774155
90.58896998
93.04424081
95.57524999
97.77969915
99.67677139
101.4679743
103.1550631
104.7396378
106.0515163
107.2716445
108.2365358
109.2793707
110.0753248
110.7892074
111.4213957
111.8239422
112.1512462
112.4034138
112.5805315
112.5450241
112.5748154
112.3969669
112.1490647
200 rad/s
92.25235616
95.32714492
98.04325274
100.6182128
102.8676455
104.8101868
106.6424246
108.3666703
109.8126907
111.1622225
112.2510999
113.2526623
114.1678028
114.9973724
115.5882741
116.1004505
116.3854036
116.7439098
116.8806312
116.8025008
116.7962779
116.579617
116.2947596
115.9417196
115.3880021
114.9004372
300 rad/s
102.9170612
105.5330895
107.8248812
109.991161
111.8595648
113.4448087
114.9294655
116.3157067
117.4418369
118.3178339
119.1113376
119.8233736
120.3006393
120.7025667
121.0298084
121.283008
121.3161088
121.1347556
121.0310326
120.7166986
120.3369996
119.7531294
119.2446955
118.5351652
117.628628
116.7964329
400 rad/s
113.802488
115.7791833
117.6429325
119.2288125
120.5487555
121.7770441
122.7542745
123.648573
124.3038085
124.8830255
125.2329394
125.5123533
125.7221937
125.7129846
125.63846
125.4992772
125.1491427
124.7376922
124.1204723
123.4447354
122.7107903
121.9189594
120.9282765
119.8819983
118.6407799
117.4847863
500 rad/s
123.9316802
125.3205593
126.6199218
127.5108407
128.4846391
129.0644082
129.5741771
130.01561
130.235218
130.2372085
130.3320902
130.0605556
129.883876
129.4981458
128.9059846
128.2600391
127.5608842
126.8091127
125.8564825
124.8532348
123.6519469
122.4018389
121.1033012
119.7567494
118.2164997
116.629919
600 rad/s
132.5787243
133.2975188
133.9507744
134.3834756
134.5995588
134.7589415
134.706961
134.6020644
134.4455965
134.0831408
133.5162919
132.90183
132.2406184
131.37827
130.4709265
129.5193096
128.3690472
127.1759513
125.7857356
124.5088358
122.881407
121.3684242
119.661192
117.9149702
115.976267
114.153913
Results: Maximizing Angle of Contact
• Spin Rate= 400
rad/sec
• Temperature= 56º F
• Elevation= 0
• Wind= 0
• The angle of contact
that maximizes the
ball’s range (125
meters) is 22º.
Results: Range and Altitude
• Elevation affects the range of
a batted baseball.
• Range can increase as much
as 10 meters from an
elevation of 5 to 5205 feet.
• Major League Baseball
ballparks range in altitude
from Dolphin Stadium at 5
feet above sea level, to Coors
Field at 5198 feet above sea
level.
• According to our model, there
is a distinct advantage to
playing at stadiums with
higher altitudes.
Results: Range and Air Temperature
• Temperature also
has a significant
impact on the range
of a ball.
• A ball hit in 92º F
weather travels up
to 6 meters farther
than a ball hit in
32º F weather.
Results: Accuracy of Our Model
• In comparing our results with those obtained by Watts and Baroni
(1989), the maximum ranges and optimum angles of contact for
varying spin rates are quite similar.
• For a spin rate of 300 rad/sec the maximum ranges and optimum
angles are nearly identical.
• For other spin rates, the results are not as similar, however they are
still reasonably close.
• The maximum range between models differs by as much as 9 meters
and the angle of contact differs by no more than 7º.
Spin Rate
(Rad/sec)
Maximum Range
(meters) Obtained by
Our Model
Maximum Range
(meters) Obtained by
Watts and Baroni
Maximum Angle of
Contact (degrees)
Obtained by Our
Model
Maximum Angle of
Contact (degrees)
Obtained by Watts
and Baroni
100
200
300
400
112.6
116.9
121.3
125.7
≈106
≈113
≈122
≈130
31
28
26
22
≈33
≈29
≈25
≈19
500
130.3
≈137
20
≈13
600
134.8
≈143
15
≈8
Results: Accuracy of Our Model
Conclusion
Optimal Conditions for Maximizing a Batted Ball’s Trajectory
After analyzing the estimated trajectories of batted baseballs using a modified version
of Professor Nathan’s model, it is apparent that a ball’s range is significantly
affected by spin rate, air temperature, and altitude. As each of these
parameters is increased, the ball’s range increases. Also, the optimum angle for
maximizing the ball’s range is dependent on the spin rate. As the spin rate
of the ball is increased, the angle required to maximize the ball’s range decreases. If
baseball truly is a “game of inches,” such changes in range caused by varying spin
rates, angles of contact, air temperatures, and altitudes are great enough to
significantly alter the outcome of a game.