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The Science of Baseball pp 153–202Cite as

The Ball in Flight Model

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Abstract

This chapter is independent of the rest of this book. It derives equations for the three forces that affect the flight of the ball: the force of gravity, the drag force and the lift force due to the Magnus effect (the force due to a spinning object moving in an airflow). The lift and drag forces depend on air density. Altitude and weather affect air density, which in turn affects how far a batted baseball or softball travels. This chapter shows that air density is inversely related to altitude, temperature and humidity, and is directly related to barometric pressure. Regression analysis is used to show the relative importance of each of the four factors (altitude, temperature, humidity and barometric pressure) and to look for interactions between them. As shown by this model, on a typical July afternoon in a major-league baseball stadium, altitude is easily the most important factor, explaining 80% of the variability. This is followed by temperature (13%), barometric pressure (4%) and relative humidity (3%). A simple linear algebraic equation presented in this chapter predicts air density well. A different model shows how the batted-ball’s range depends on both the drag force and the Magnus force and considers the relative importance of the drag and Magnus forces. As asides, this chapter shows that a home run ball might go 26 feet farther in San Francisco then in Denver and it answers the question, “Can a tennis ball be thrown farther that a baseball?”

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Authors and Affiliations

  1. Systems and Industrial Engineering, University of Arizona, Tucson, AZ, USA

    A. Terry Bahill

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  1. A. Terry Bahill
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Appendix. Weather Data for Major-League Baseball Stadiums

Appendix. Weather Data for Major-League Baseball Stadiums

City/State

Team name

Altitude of home plate, feet

Altitude of home plate, m

Average daily high temperature in July °F

Average daily high temperature in July °C

SVP, mm Hg

Relative Humidity on an average July afternoon, percent

Average sea level corrected Barometric Pressure, inch Hg

Average sea level corrected barometric pressure, Pisa

Average sea level corrected barometric pressure, mm Hg

Average absolute pressure, mm Hg. Not sea-level corrected.

Air density, kg/m3

Arizona

Diamondbacks

1061

323

104

40

55

20

29.81

14.64

757

730

1.077

Atlanta

Braves

942

287

90

32

36

59

29.99

14.73

762

738

1.111

Baltimore

Orioles

36

11

87

31

33

53

29.99

14.73

762

761

1.154

Boston

Red Sox

16

5

82

28

28

57

29.93

14.70

760

760

1.164

Chicago

Cubs

601

183

84

29

30

60

29.93

14.70

760

745

1.135

Chicago

White Sox

595

181

84

29

30

60

29.93

14.70

760

745

1.135

Cincinnati

Reds

490

149

86

30

32

58

30.00

14.74

762

748

1.136

Cleveland

Indians

653

199

81

27

27

57

30.05

14.76

763

746

1.145

Colorado

Rockies

5186

1581

88

31

34

34

29.98

14.72

761

638

0.967

Detroit

Tigers

578

176

84

29

30

54

30.06

14.76

763

748

1.141

Houston

Astros

21

6

94

34

41

63

29.97

14.72

761

760

1.133

Kansas City

Royals

857

261

90

32

36

64

29.97

14.72

761

739

1.111

Los Angeles

Angels

146

45

84

29

30

52

29.94

14.70

760

756

1.155

Los Angeles

Dodgers

501

153

84

29

30

52

29.94

14.70

760

747

1.140

Miami

Marlins

5

2

91

33

37

63

30.00

14.73

762

762

1.143

Milwaukee

Brewers

618

188

81

27

27

64

30.00

14.74

762

746

1.144

Minnesota

Twins

827

252

83

28

29

59

29.96

14.71

761

740

1.130

New York

Mets

12

4

84

29

30

55

30.02

14.74

763

762

1.163

New York

Yankees

33

10

84

29

30

55

30.00

14.73

762

760

1.160

Oakland

Athletics

0

0

73

23

21

55

29.95

14.71

761

761

1.188

Philadelphia

Phillies

0

0

86

30

32

54

30.01

14.74

762

762

1.157

Pittsburgh

Pirates

726

221

83

28

29

54

30.00

14.73

762

742

1.134

Saint Louis

Cardinals

438

134

90

32

36

60

29.98

14.73

762

750

1.128

San Diego

Padres

15

5

76

24

23

67

29.88

14.68

759

758

1.175

San Francisco

Giants

8

2

68

20

18

65

29.99

14.73

762

762

1.201

Seattle

Mariners

17

5

75

24

22

49

30.04

14.75

763

763

1.187

Tampa Bay

Rays

44

13

90

32

36

64

30.08

14.77

764

764

1.149

Texas

Rangers

543

166

96

36

44

53

29.97

14.72

761

747

1.112

Toronto

Blue jays

268

82

80

27

26

55

30.00

14.73

762

755

1.161

Washington

Nationals

6

2

88

31

34

53

30.00

14.73

762

761

1.152

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Bahill, A.T. (2018). The Ball in Flight Model. In: The Science of Baseball. Springer, Cham. https://doi.org/10.1007/978-3-319-67032-4_7

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  • DOI: https://doi.org/10.1007/978-3-319-67032-4_7

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