The Science of Baseball The Ball in Flight Model - PowerPoint Presentation

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The Science of Baseball The Ball in Flight Model - PPT Presentation

A Terry Bahill Emeritus Professor of Systems Engineering University of Arizona terrysiearizonaedu 20172018 Bahill Reference Terry Bahill The Science of Baseball Modeling BatBall Collisions and the Flight of the Ball ID: 783582

2018 ball air 2017 ball 2018 2017 air bahill density spin range mph baseball 2012 launch feet 2018

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Slide1

The Science of BaseballThe Ball in Flight Model

A. Terry BahillEmeritus Professor ofSystems EngineeringUniversity of Arizonaterry@sie.arizona.edu©, 2017-2018, Bahill

Slide2

Reference

Terry Bahill, The Science of Baseball: Modeling Bat-Ball Collisions and the Flight of the Ball, Springer Nature, NY, NY, 2018Chapter 7

Slide3

10/14/2018

Slide4

Purpose

To derive equations for the 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). To show how altitude, temperature, barometric pressure and relative humidity affect air density and consequently how air density affects the flight of the ball. A home run ball might go 26 feet farther in Denver than in San FranciscoCan a tennis ball be thrown farther that a baseball?

10/14/2018

Slide5

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Slide6

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Slide7

Major league baseball, 2017

Speed of the pitch at the pitcher’s release point

Pitch spin rate, absolute values

Number of pitches

Type of pitch

average, mph

standard deviation

average, m/s

average, rpm

std. dev.

4-seam fastball

93.6

2.3

41.8

2169

363

10,000

2-seam fastball

92.7

2.4

41.4

2148

321

3,000

Slider

3.1

745!

346

4,000

Changeup

85!

3.5

1714

419

2,000

Curveball

3.8

1286!

461

1,800

Slide8

Gravity-induced and spin-induced drop

Pitch speed and type

Spin rate (rpm)

Duration of flight (msec)

Drop due to gravity (ft)

Spin-induced vertical drop (ft)

Total drop (ft)

95 mph fastball

-1200

404

2.63

-0.91

1.72

90 mph fastball

-1200

426

2.92

-0.98

1.94

85 mph slider

+1400

452

3.29

+0.74

4.03

80 mph curveball

+2000

480

3.71

+1.40

5.1175 mph curveball+20005134.24+1.465.70

10/14/2018

Slide9

Simulated fastballs thrown with (top) a four-seam grip and (bottom) a two-seam grip. Videos of these simulated fastballs are available at

http://sysengr.engr.arizona.edu/baseball/index.html.10/14/2018

Slide10

10/14/2018

10Pitch variation

Batters say that the ball hops, drops, curves, breaks, rises, sails or tails away.

The pitcher might tell you that he throws a fastball, screwball, curveball, drop curve, flat curve, slider, backup slider, change up, split fingered fastball, splitter, forkball, sinker, cutter, two-seam fastball or four-seam fastball. This sounds like a lot of variation.

However, no matter how the pitcher grips or throws the ball, once the ball is in the air its motion depends only on gravity, its velocity and its spin.

Slide11

10/14/2018

11Velocity and spin

Once the ball is in the air, its motion depends only on gravity, velocity and spin, which are vectors with magnitude and direction.

*These pitch characteristics are described by a linear velocity vector and an

angular velocity vector,

each with magnitude and direction.

The magnitude of the linear velocity vector is called

pitch speed

and the magnitude of the angular velocity vector is called

spin rate.

Slide12

10/14/2018

12Dave Baldwin said,

“If a major league baseball pitcher* describes the flight of one of his pitches; he usually illustrates the trajectory using his pitching hand, much like a kid or a pilot demonstrating the yaw, pitch and roll of an airplane.

The hand used as an analog in this way is a gestural example of a somatic metaphor.”

Slide13

10/14/2018

13Angular right-hand rule

Slide14

10/14/2018

14Coordinate right-hand rule

Slide15

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15SaD Sid

Thumb → Spin axis

Index finger

→

irection of motion

Middle finger

→

nduced

eflection

Slide16

Spin axis for common pitches

For a right-handed pitcher

Slide17

Left-handed pitcher

Slide18

Batter’s view of a slider thrown by a right-handed pitcher

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Slide19

Forces acting on a spinning ball flying through the air

10/14/2018

Slide20

Forces acting on the ball

Slide21

10/14/2018

Slide22

10/14/2018

Slide23

Magnus force

Kutta-Joukowski Lift Theorem

Slide24

10/14/2018

Slide25

Second derivation

An experimental equation for the lift force

Slide26

10/14/2018

Type of launch

Initial Speed (mph)

Spin rate (rpm)

Spin Parameter, SP

*Reynolds Number, Re, times 10

-5

Fastball

2200

0.20

1.685

Slider

2000

0.20

1.540

Curveball

2300

0.25

1.431

Change-up

1700

0.17

1.540

Knuckle ball

0.00

1.178

Home run, initial value

2000

0.18

1.776

Home run. hitting ground

1760

0.28

0.996

Slow line drive

2500

0.25

1.540

Fast line drive

100

1800

0.16

1.812

Extreme pop-up

6000

0.74

1.268

NCAA softball pitch

1200

0.21

1.538

*The Reynolds number will be discussed in the next section.

Slide27

10/14/2018

Experimental data give

Remember we started with

So put the above lift coefficient into this equation and we get

Slide28

Drag force

Slide29

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Slide30

Home run, my model

Slide31

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Slide32

Sensitivity analysis

Parameters

Nominal values

Nominal values increased by +1%

Altered range, ft

Change in range, ft

Sensitivity values

Range, ft

384.87

Batted-ball speed, mph

91.9

92.819

389.56

4.69

469

Ball diameter, in.

2.90

2.9336

382.59

-2.28

-228

Drag coefficient, C

0.4

0.404

383.16

-1.71

-171

Ball weight, oz

5.125

5.1763

386.18

1.31

131

Air density, r, kg/m

1.0582

1.0688

383.65

-1.22

-122

1.2

1.212

385.37

0.50

Ball spin, rpm

-2000

-2020

385.37

0.50

Ball spin, rpm

-2000

-1980

384.37

-0.50

-50

Launch angle, degrees

34.34

384.39

-0.48

-48

Launch height, feet

3.03

384.90

0.03

Slide33

Interactions

On the left side, when the launch angle increases, the range goes up. These curves do not have the same shape. The curve for the2000 rpm spin rate has a steeper drop on the right side. This is the effect of the interaction. The difference in spacing of the lines is not the effect of the interaction. That is merely the dependence of the batted-ball speed on spin rate.

Slide34

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Slide35

10/14/2018

When the ball’s spin axis is not horizontal, the Magnus force should be decomposed into a force lifting the ball up and a lateral force pushing it sideways.

where VaSa is the

angle from the

ertical

xis to the

xis.

The magnitude of the lateral force is

Slide36

Effect of angle SaD

SaD

is the angle from the

xis

to the

irection of motion

Slide37

Gravity-induced and spin-induced drop

Pitch speed and type

Spin rate (rpm)

Duration of flight (msec)

Drop due to gravity (ft)

Spin-induced vertical drop (ft)

Total drop (ft)

95 mph fastball

-1200

404

2.63

-0.91

1.72

90 mph fastball

-1200

426

2.92

-0.98

1.94

85 mph slider

+1400

452

3.29

+0.74

4.03

80 mph curveball

+2000

480

3.71

+1.40

5.1175 mph curveball+20005134.24+1.465.70

10/14/2018

Slide38

Effects of Air Density on a Spinning Ball in Flight

Slide39

Air densityAir density is inversely related to

altitude temperature humidity Air density is directly related to barometric pressure 10/14/2018

Slide40

Humid air is lighter than normal air

“Boy it’s humid. Feel how heavy the air is.”

Each cubic meter of air contains the same number of molecules, about 1025

.Air is composed of nitrogen and oxygen, with atomic weights of 14 and 16, respectively.

Both of these gasses are diatomic, N

and O

, yielding molecular weights of 28 and 32

Introduce water vapor, H

O, molecular weight of 18.

The nitrogen and oxygen molecules with molecular weights of 28 and 32 will be displaced by water with a molecular weight of 18.

Thus, humid air is less dense than regular air.

10/14/2018

Slide41

Altitude (feet above sea level)

Temp (°F)

Relative Humidity (percent)

Baro. pressure (inch Hg)

Air density (kg/m3)

Air density, percent change from midlevel

Low altitude

29.92

1.16

9.4

Low temperature

2600

29.92

1.09

2.9

Low humidity

2600

29.92

1.06

0.7

Low barometric pressure

2600

855029.331.04-2.0Lowest density

5200

100

29.33

0.91

-14.0

Midlevel

2600

29.92

1.06

0.0

Highest density

30.51

1.22

15.5

High barometric pressure

2600

30.51

1.08

2.0

High humidity

2600

29.92

1.05

-0.7

High temperature

2600

100

29.92

1.03

-2.9

High altitude

5200

29.92

0.97

-8.6

10/14/2018

Slide42

Ramberg’s simple equation

Air density = = 1.045 + 0.01045{- 0.0035 (Altitude - 2600) - 0.2422 (Temperature - 85)

- 0.0480 (Relative Humidity - 50)

+ 3.4223 (Barometric Pressure - 29.92)} Air density

is in kg/m

Altitude

is in feet,

Temperature

is in degrees Fahrenheit,

Relative Humidity

is in percent and

Barometric Pressure

is in inches of Hg.

This linear equation explains 99.3 percent of the variation of air density across the 81 data points*.

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Slide43

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Slide44

Ramberg’s least squares analysis

Altitude explains 80% of the variability Temperature explains 13% Barometric pressure accounts for 4%Relative humidity accounts for 3%10/14/2018

Slide45

Air density at some typical baseball stadiums

Altitude (feet above sea level)

Temperature

(

°F

)

average daily maximum in July

Relative humidity, on an average July afternoon

Average barometric pressure in July (inch of Hg)

Air density (kg/m

)

Denver

5190

34 %

29.98

0.96

Phoenix

1086

104

20 %

29.81

1.07

Houston

63 %

29.97

1.11Minneapolis8158359 %29.961.11Seattle107549 %30.041.18San Francisco

65 %

29.99

1.19

10/14/2018

The red arrows compare Phoenix to San Francisco for home run distance

Slide46

Batted-ball range varies inversely with air density*

Air Density (kg/m

)

Range (feet)

0.9

408

1.0

392

1.1

377

1.2

363

Denver

Phoenix

San Francisco

Slide47

Range as a function of air density

Air Density (kg/m3)

Range (ft)

Home run

Pop up

Line drive

1.3

372

266

1.2

382

268

1.1

394

269

1.0

406

271

0.9

418

272

0.8

432

104

Slide48

Pitch variations with air density

Air Density (kg/m

)

Fastball released at 93 mph

Curveball released at 79 mph

Speed at the plate (mph)

Height above the plate (ft)

Speed at the plate (mph)

Height above the plate (ft)

1.3

83.5

3.18

71.3

1.84

1.2

84.2

3.08

71.9

1.97

1.1

84.9

2.98

72.5

2.1

1.0

85.7

2.93

73.1

2.240.986.52.8673.72.39

0.8

87.3

2.81

74.6

2.52

10/14/2018

Slide49

A tale of two cities

City

Altitude (feet above sea level)

Average daily high temperature (°F)

Average relative humidity

Average barometric pressure (inch of Hg)

Average air density (kg/m

)

Denver

5190

34 %

29.98

0.96

San Francisco

65 %

29.99

1.19

10/14/2018

City

Air density (kg/m

)

Computed range in feet for a home run ball

Computed range in meters for a home run ball

Denver0.96423

129

San Francisco

1.19

399

122

Slide50

Variable

Value in SI units

Value in US customary units

Default state

Midlevel

Altitude

792 m

2600

Temperature

29.4 °C

85°F

Relative Humidity

50 %

Barometric pressure

760 mm Hg

29.92 inch Hg

Air density

Dynamic viscosity of air

Kinematic viscosity of air

Diameter of a baseball

0.07366 m

2.9 in

Mass of a baseball

0.145 kg

5.125 oz

Launch speed

43 m/s

97 mph

Launch angle

34 degrees

Launch spin

-209 rad/s

-2000 rpm

Reynolds number

Spin parameter

Magnus coefficient

1.2

Drag Coefficient

0.4

10/14/2018

Slide51

Which can be thrown farther a baseball or a tennis ball?

Slide52

10/14/2018

Slide53

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Slide54

Lift and drag coefficients

Slide55

10/14/2018

Parameter

Baseball, nominal values

Baseball, mass increased by 10%

Baseball, mass +10% and reduced launch velocity

Tennis ball

Softball

Bocce ball

Women’s shot put

ball weight, oz

5.125

5.637

2.03

6.75

32.45

141

ball mass, kg

0.15

0.16

0.06

0.19

0.92

4.00

launch speed, mph

105855530.81

launch speed, m/s

42.5

41.6

46.9

38.0

24.6

13.8

ball diameter, in

2.90

2.51

3.84

4.21

4.04

ball diameter, m

0.07

0.06

0.10

0.11

0.10

drag coefficient

0.38

0.56

0.4

air density

0.002

air density

1.045

lift coefficient

0.7

0.75

0.8

ball spin, rpm

-2000

-2200

-1800

-1200

-12

ball spin, rad/s

-209

-157

-63

-0.6

launch angle, degrees

launch height, feet

launch height, m

1.5

flight duration, seconds

5.26

5.24

5.15

5.49

5.13

3.02

2.06

range, ft

372

384

374

250

297

186

range, m

113

117

114

20.67

Slide56

Summary lines from previous table

Parameter

Baseball

Heavy baseball

Tennis ball

Softball

Bocce ball

Women’s shot put

Launch speed, mph

105

Range, feet

372

374

250

297

186

10/14/2018

A baseball can be thrown farther than a tennis ball.

A heavy baseball can be thrown slightly farther than a regular baseball.

But how is that possible?

Slide57

Which can be thrown farther a heavy ball or a light ball?

The sensitivity analysis suggested that a heavier ball would go farther10/14/2018

Slide58

Sensitivity analysis

Parameters

Nominal values

Nominal values increased by +1%

Altered range, ft

Change in range, ft

Sensitivity values

Range, ft

384.87

Batted-ball speed, mph

91.9

92.819

389.56

4.69

469

Ball diameter, in.

2.90

2.9336

382.59

-2.28

-228

Drag coefficient, C

0.4

0.404

383.16

-1.71

-171

Ball weight, oz

5.125

5.1763

386.18

1.31

131

Air density, r, kg/m

1.0582

1.0688

383.65

-1.22

-122

Magnus

coefficient

1.2

1.212

385.37

0.50

Ball spin, rpm

-2000

-2020

385.37

0.50

Ball spin, rpm

-2000

-1980

384.37

-0.50

-50

Launch angle, degrees

34.34

384.39

-0.48

-48

Launch height, feet

3.03

384.90

0.03

Slide59

Technical note:

The numbers in this book should not be compared between tables, because the tables have different purposes. Therefore, they may have used different versions of the model and different parameters.

10/14/2018

Slide60

The nominal range of the home run ball in Table 7.6 is

385 feet: the air density was 1.0582 kg/m3, the drag coefficient was 0.4 and the Magnus coefficient was 1.2. Table 7.11 gave the nominal range for six different air densities: 0.8, 0.9, 1.0, 1.1 and 1.2 kg/m3. Table 7.14 gave the nominal range for two air densities: 0.96 and 1.19 kg/

m3. The nominal range of the home run ball in Table 7.16 is 372

feet, but that is for an air density of 1.045 kg/m3, a drag coefficient 0.38 and a Magnus coefficient of 0.7.

10/14/2018

Slide61

Numerical values are not as important as the comparisons that are made using the numerical values in the tables.

10/14/2018

Slide62

Which can be thrown farther a heavy ball or a light ball?2

If the balls were launched with the same velocity, then the heavier ball must have been given more energy. Therefore, it will have more momentum and it will take more force and time to slow it down. 10/14/2018

Slide63

Which can be thrown farther a heavy ball or a light ball?3

The only terms in our equations that depend on mass are the acceleration terms. At the beginning of motion, the ball with the bigger mass has smaller accelerations:Both of these will be smaller for the heavier ball. Which means that the horizontal and vertical velocities will not slow down as fast. This will make the heavier ball go farther.

10/14/2018

Slide64

Which can be thrown farther a heavy ball or a light ball?4

Physics says …In the simulation, we increased the mass of the baseball by 10% and we kept the launch speed the same. The heavier ball traveled 384 instead of 372 feet. 10/14/2018

Slide65

Which can be thrown farther a heavy ball or a light ball? 5

Physiology says …The force-velocity relationship of muscle says that our muscles will produce a lower velocity for a heavier load than for a lighter load. The 10% heavier baseball will be launched at 93 mph instead of 95 mph. This reduced launch velocity will reduce the range from 384 to 374 feet.

Slide66

Which can be thrown farther a heavy ball or a light ball?6

Increasing the baseball’s mass by 10% increased the range by 12 feet. However, the subsequent reduction in launch velocity caused by the force-velocity relationship of muscle decreased the range by 10 feet. Therefore, if a human is throwing balls of about the same mass, then the heavier ball might go slightly farther.

10/14/2018

Slide67

Summary

Slide68

10/14/2018

Slide69

Air densityAir density

= = 1.045 + 0.01045{- 0.0035 (Altitude - 2600) - 0.2422 (Temperature - 85)

- 0.0480 (Relative Humidity - 50)

+ 3.4223 (Barometric Pressure - 29.92)}

10/14/2018

Slide70

10/14/2018

Slide71

Air density in some typical baseball stadiums

Altitude (feet above sea level)

Temperature

(

°F

)

average daily maximum in July

Relative humidity, on an average July afternoon

Average barometric pressure in July (inch of Hg)

Air density (kg/m

)

Denver

5190

34 %

29.98

0.96

Phoenix

1086

104

20 %

29.81

1.07

Houston

63 %

29.97

1.11Minneapolis8158359 %29.961.11Seattle107549 %30.041.18San Francisco

65 %

29.99

1.19

10/14/2018

The red arrows compare Phoenix to San Francisco for home run distance

Slide72

10/14/2018

Air Density (kg/m

)

Range (feet)

0.9

408

1.0

392

1.1

377

1.2

363

Denver

Phoenix

San Francisco

Slide73

FinallyA baseball can be thrown farther than a tennis ball.

If a human is throwing balls of about the same mass, then the heavier ball might go slightly farther.10/14/2018

Slide74

10/14/2018

Slide75

10/14/2018

75Seminar materials

Balls skewered on boltsa baseball

a bat

Slide76

10/14/2018

The Science of Baseball The Ball in Flight Model - PowerPoint Presentation

The Science of Baseball The Ball in Flight Model - PPT Presentation

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