Baseball Trajectories: A Game of InchesJim HildenspergerKyle SpauldingDale Garrett

Baseball: Take Me out to the Ball Game

-Why is baseball considered a game of inches?

-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).

-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.

-In the MLB, you'll get a multimillion-dollar contract if you can hit a ball successfully anywhere near three out of 10 times.

-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.

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

-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

How do changes in the factors of batted ball’s Trajectory effect how far it goes?

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

vs Range

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Rang

e100 rad/s

200 rad/s

300 rad/s

400 rad/s

500 rad/s

600 rad/s

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

vs Range

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Results: Dataω

θ 100 rad/s 200 rad/s 300 rad/s 400 rad/s 500 rad/s 600 rad/s 10 81.27753454 92.25235616 102.9170612 113.802488 123.9316802 132.5787243 11 84.61554166 95.32714492 105.5330895 115.7791833 125.3205593 133.2975188 12 87.78774155 98.04325274 107.8248812 117.6429325 126.6199218 133.9507744 13 90.58896998 100.6182128 109.991161 119.2288125 127.5108407 134.3834756 14 93.04424081 102.8676455 111.8595648 120.5487555 128.4846391 134.5995588 15 95.57524999 104.8101868 113.4448087 121.7770441 129.0644082 134.7589415 16 97.77969915 106.6424246 114.9294655 122.7542745 129.5741771 134.706961 17 99.67677139 108.3666703 116.3157067 123.648573 130.01561 134.6020644 18 101.4679743 109.8126907 117.4418369 124.3038085 130.235218 134.4455965 19 103.1550631 111.1622225 118.3178339 124.8830255 130.2372085 134.0831408 20 104.7396378 112.2510999 119.1113376 125.2329394 130.3320902 133.5162919 21 106.0515163 113.2526623 119.8233736 125.5123533 130.0605556 132.90183 22 107.2716445 114.1678028 120.3006393 125.7221937 129.883876 132.2406184 23 108.2365358 114.9973724 120.7025667 125.7129846 129.4981458 131.37827 24 109.2793707 115.5882741 121.0298084 125.63846 128.9059846 130.4709265 25 110.0753248 116.1004505 121.283008 125.4992772 128.2600391 129.5193096 26 110.7892074 116.3854036 121.3161088 125.1491427 127.5608842 128.3690472 27 111.4213957 116.7439098 121.1347556 124.7376922 126.8091127 127.1759513 28 111.8239422 116.8806312 121.0310326 124.1204723 125.8564825 125.7857356 29 112.1512462 116.8025008 120.7166986 123.4447354 124.8532348 124.5088358 30 112.4034138 116.7962779 120.3369996 122.7107903 123.6519469 122.881407 31 112.5805315 116.579617 119.7531294 121.9189594 122.4018389 121.3684242 32 112.5450241 116.2947596 119.2446955 120.9282765 121.1033012 119.661192 33 112.5748154 115.9417196 118.5351652 119.8819983 119.7567494 117.9149702 34 112.3969669 115.3880021 117.628628 118.6407799 118.2164997 115.976267 35 112.1490647 114.9004372 116.7964329 117.4847863 116.629919 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.

0

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0 50 100 150 200 250 300 350 400 450 500

horizontal distance (ft)

he

igh

t (ft)

12 degrees

17 degrees

22 degrees

27 degrees

32 degrees

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

100 112.6 ≈106 31 ≈33

200 116.9 ≈113 28 ≈29

300 121.3 ≈122 26 ≈25

400 125.7 ≈130 22 ≈19

500 130.3 ≈137 20 ≈13

600 134.8 ≈143 15 ≈8

Results: Accuracy of Our Model

vs Range

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Rang

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600 rad/s

Conclusion

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. 

Optimal Conditions for Maximizing a Batted Ball’s Trajectory