The Mathematics of the Flight of a Golf Ball

Mathematical Modeling

Isabelle Boehling, John A. Holmes High

Wen Huang, Junius H. Rose High

2008

Outline

• Questions• Background Information• The Math Side of the Flight of a Golf Ball• Modeling the Flight Path

– Using VPython– Data

• Summary

Problem

How do the different launch angles and launch velocities of hitting a golf ball affect that path it takes? What are the effects of lift and drag on the distance and height of a golf ball of flight?

The Game of Golf٠ Clubs with wooden or metal heads are used to hit a small, white ball into a number of holes (9 or 18) in progression, situated at a variety of distances over a course

٠ Objective: to get the ball into each hole in as few strokes as possible.

٠ Exact origins are uncertain: open to debate as being Chinese, Dutch, or Scottish.

٠ Most acknowledged golf history idea is that the sport began in Scotland in the 1100s.

Hitting a Golf Ball

• Club is swung at the motionless ball wherever it has come to rest (side stance.)

• Putts and short chips: played without much movement of the body

• Full swing: complex rotation of the body aimed at accelerating the club head to a great speed.

http://www.all-about-lady-golf-clubs.com/images/how_to_hit_a-golf_ball_above_your_feet.jpg

Basic Parabolic Flight

• Without taking dimples, drag, and lift into consideration, the flight of a golf ball would be a simple parabola.

http://www.golf-simulators.com/physics.htm

The Math Behind It

• Calculating the distance traveled or the range at time t :

• Calculating the height at time t:

v = launch velocity of the golf ballg = gravitational acceleration 9.8 m/s/s

m = the launch angle in radians

tvtx )cos()(

2)sin()(

2gttvty

Now With Dimples...

• The dimples are meant to give the ball more lift and less drag when the ball is in the air.

• Create laminar flow so the ball will fly farther

• Because of the dimples, the turbulence boundary layer is separated at a later point

http://www.golfjoy.com/golf_physics/images/drag.gif

Magnus Effect

• Upward push due to the dimpled drag on the air at the top and bottom parts of the golf ball

• Pressure difference causes the ball to lift and stay in the air for a longer time.

• Spinning ball has a whirlpool of rotating air around it

• Circulation generated by mechanical rotation

http://www.symscape.com/files/images/curveball_1.img_assist_custom.jpg

How Do We Account For Force?

LD FFWF

m

Fa

F = Force W = Weight of Ball

F d = Drag ForceFL = Lift Force

And Acceleration?

)sincos)((2

22 Ldyxx ccvv

m

Sa

gccvvm

Sa Ldyxy

)sincos)((

222

= density of air at sea level (1.225 kg/m3)

S=stream surface ( ) where r=20.55 mm

m= mass of the ball (0.050 kg)

= angle with respect to the horizontal

v= velocity

cd=drag coefficient

cl=lift coefficient

g=gravitational pull (g=weight/mass)

2r

Kinetic Equations of Motion

tavv

tatvxx

oldoldnew xxx

xxoldnew

2

2

1

tavv

tatvyy

oldoldnew yyy

xxoldnew

2

2

1

Our Code

-Incorporated the equations on the previous slide into the code (the text enclosed by the red circle)

-The rest of the code defines the parameters for the sliders which enables us to vary the launch angle and velocity as well as the drag and lift coefficient.

VPython Model (Change in Angle)

-Pink slider from the picture on the left changes the launch angle of the golf ball

-Picture on the right shows the path of the golf ball at different angles

-Data was recorded and compiled in graphs

VPython Model (Change in Velocity)

-Green slider from the picture on the left changes the launch velocity (in meters/second) of the golf ball

-Picture on the right shows the path of the golf ball launched at different velocities

-Data was recorded and compiled in graphs

Change in Launch Angle (No Drag or Lift)(Velocity=50 m/sec)

-20

0

20

40

60

80

100

0 50 100 150 200 250

Range (Meters)

Hei

gh

t (

Met

ers)

10°

15°

20°

25°

30°

35°

40°

45°

50°

55°

60°

Change in Launch Velocity (No Drag or Lift)(When Angle = 30 degrees)

-10

0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500 600

Range (Meters)

Hei

gh

t( (

Met

ers)

40 m/s

45 m/s

50 m/s

55 m/s

60 m/s

65 m/s

70 m/s

75 m/s

80 m/s

What About Drag and Lift?• Drag:

– Comes mainly from air pressure forces.

• occurs when the pressure in front of the ball is significantly higher than that behind the ball.

• Lift:– how high the ball flies and

how quickly it stops after landing.

• Bernoulli’s Principle:– Pressure and density are

inversely related (a slow moving object exerts more pressure than a fast moving one.) http://www.ralphmaltby.com/assets/39/Golf_Ball_Flight_Principles.jpg

How Do We Account For That?• The acceleration equation calls for a drag and lift

coefficient.

• There is not a defined number for the lift and drag coefficient.

• According to the US patent for golf balls, the drag coefficient usually falls between 0.21 and 0.255 and the lift coefficient usually falls between 0.14 and 0.19.

)sincos)((2

22 Ldyxx ccvv

m

Sa

Change in Launch Angle(When Velocity= 50 m/sec, Drag Coefficient= 0.22, Lift Coefficient= 0.16)

-20

0

20

40

60

80

100

120

140

-10 10 30 50 70 90 110 130 150

Range(Meters)

Hei

gh

t(M

eter

s )

10°

15°

20°

25°

30°

35°

40°

45°

50°

55°

60°

Height vs. Angle Degree(When Velocity = 50 m/sec, Drag Coefficient = 0.22, and Lift Coefficient = 0.16)

h() = 0.04742 - 0.7771 + 5.5722

R2 = 0.9998

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70

Degree

Hei

gh

t (M

eter

s)

Range vs. Angle Degree(When Velocity = 50 m/sec, Drag Coefficient = 0.22, and Lift Coefficient = 0.16)

r() = -0.12412 + 9.7586 - 46.205

R2 = 0.9761

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70

Degree

Ran

ge

(Met

ers)

Change in Launch Velocity(When Angle= 30 degrees, Drag Coefficient= 0.22, Lift Coefficient= 0.16)

-10

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

Range (Meters)

Hei

gh

t (M

ete

rs)

40 m/s

45 m/s

50 m/s

55 m/s

60 m/s

65 m/s

70 m/s

75 m/s

80 m/s

Height vs. Launch Velocity(When Angle = 30 degrees, Drag Coefficient= 0.22, Lift Coefficient= 0.16)

h(v) = 0.0104v2 - 0.0708v + 2.7941

R2 = 1

0

10

20

30

40

50

60

70

30 40 50 60 70 80 90

Launch Velocity (Meters/Second)

Hei

gh

t (M

ete

rs)

Range vs. Launch Velocity(When Angle = 30 degrees, Drag Coefficient= 0.22, Lift Coefficient= 0.16 )

r(v) = 3.6687v - 54.2

R2 = 0.997

0

50

100

150

200

250

300

30 40 50 60 70 80 90

Launch Velocity (Meters/Second)

Ran

ge

(Me

ters

)

Change in Drag Coefficient(When Angle= 30 degrees, Velocity= 50 m/sec, Lift Coefficient= 0.16)

-5

0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160

Range (Meters)

Hei

gh

t (M

ete

rs)t

0.21

0.22

0.23

0.24

0.25

Height vs. Drag Coefficient(When Angle = 30 degrees, Velocity = 50 m/sec, and Lift Coefficient = 0.16)

h(x) = 37x + 17.11

R2 = 1

24.8

25

25.2

25.4

25.6

25.8

26

26.2

26.4

26.6

0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25 0.255

Drag Coefficient

Hei

gh

t (M

ete

rs)

Range vs. Drag Coefficient(When Angle = 30 degrees, Velocity = 50 m/sec, and Lift Coefficient = 0.16)

100

105

110

115

120

125

130

135

140

145

150

0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 0.25 0.255

Drag Coefficient

Ran

ge

(Me

ters

)

Change in Lift Coefficient(When Angle= 30 degrees, Velocity= 50 m/sec, Drag Coefficient= 0.22)

-5

0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160

Range (Meters)

Hei

gh

t (M

ete

rs)) 0.14

0.15

0.16

0.17

0.18

0.19

Height vs. Lift Coefficient(When Angle= 30 degrees, Velocity= 50 m/sec, Drag Coefficient= 0.22)

h(x) = -61.543x + 35.138

R2 = 0.9988

23

23.5

24

24.5

25

25.5

26

26.5

27

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

Lift Coefficient

Hei

gh

t (M

ete

rs)

Range vs. Lift Coefficient(When Angle = 30 degrees, Velocity = 50 m/sec, and Drag Coefficient = 0.22)

r(x) = -315.51x + 180.26

R2 = 0.9736

118

120

122

124

126

128

130

132

134

136

138

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

Lift Coefficient

Ran

ge

(Me

ters

)

Conclusion• Results:

– With No Drag/Lift: after launch angle reached 40-45 degrees, the ball flew higher in the air and the range began to decrease (max range around 250 meters); the greater the launch velocity, the higher and further the ball would go.

– With Drag/Lift: after launch angle reached 40-45 degrees, the ball flew higher in the air and the range began to decrease (max range around 150 meters); the greater the launch velocity, the higher and further the ball would go.

– Change in Drag Coefficient: Range stayed around 133 meters

– Change in Lift Coefficient: the smaller the lift coefficient was, the higher the ball would fly and the larger the range would be.

Summary • Researched the distance, height, force, and acceleration

formulae

• Created VPython simulations with the final equations

• Recorded data from the changing VPython models

• Analyzed the data and transferred it to excel to create graphs of the path of the golf ball

Acknowledgements

We would like to thank our Mathematical Modeling professor, Dr. Russell Herman, our teacher, Mr. David Glasier, Mr. and Mrs. Cavender, all the staff here at SVSM UNCW, and our parents. Thanks for this opportunity!

Bibliography• Aerodynamic Pattern for a Golf Ball.

http://www.patentstorm.us/patents/6464601/claims.html

• Flight Dynamics of Golf Balls. http://www.golfjoy.com/golf_physics/dynamics.asp

• Golf Ball. http://en.wikipedia.org/wiki/Golf_ball

• The Pysics of Golf. http://www.golf-simulators.com/physics.htm

• Scott, Jeff. Golf Dimples and Drag. 2005. http://www.aerospaceweb.org/question/aerodynamics/q0215.shtml

• Tannar, Ken. Probable Golf Instruction. 2001-2008. http://probablegolfinstruction.com/science_golf_ball_flight.htm

• Werner, Andrew. Flight Model of a Golf Ball. http://www.users.csbsju.edu/~jcrumley/222_2007/projects/awwerner/project.pdf

• Wisse, Menko. Golf Ball Trajectory Simulation Applet. http://www.ecs.syr.edu/centers/simfluid/red/golf.html