Report

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.