Transcript The Mathematics of the Flight of a Golf Ball Mathematical Modeling 2008

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 Golf
٠
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:
x(t )  (v cos  )t
2
gt
y(t )  (v sin  )t 
2
v = launch velocity of the golf ball
g = gravitational acceleration 9.8 m/s/s
m = the launch angle in radians
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?
F  W  FD  FL
F
a 
m
F = Force
W = Weight of Ball
F d = Drag Force
FL = Lift Force
And Acceleration?
ax  
ay 
 S
2m
 S
2m
(v x  v y )(cd cos   cL sin  )
2
2
(vx  v y )(cd cos   cL sin  )  g
2
2
= density of air at sea level (1.225 kg/m3)
2
S=stream surface ( r ) 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)
Kinetic Equations of Motion
xnew  xold
v xn ew  v xo ld
y new  yold
v y n ew  v yo ld
1
 v x t  a x t 2
2
 a xo ld t
1
 v x t 
a x t 2
2
 a yo ld t
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)
100
10°
15°
80
20°
25°
30°
Height (Meters)
60
35°
40°
45°
40
50°
55°
60°
20
0
0
50
100
150
-20
Range (Meters)
200
250
Change in Launch Velocity (No Drag or Lift)
(When Angle = 30 degrees)
90
80
70
Height( (Meters)
60
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
50
40
30
20
10
0
0
100
200
300
-10
Range (Meters)
400
500
600
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.
ax  
 S
2m
(v x  v y )(cd cos   cL sin  )
2
2
• 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.
Change in Launch Angle
(When Velocity= 50 m/sec, Drag Coefficient= 0.22, Lift Coefficient= 0.16)
140
120
100
10°
15°
Height(Meters)
80
20°
25°
60
30°
35°
40°
40
45°
50°
20
55°
60°
0
-10
10
30
50
70
-20
Range(Meters)
90
110
130
150
Height vs. Angle Degree
(When Velocity = 50 m/sec, Drag Coefficient = 0.22, and Lift Coefficient = 0.16)
140
120
h() = 0.04742 - 0.7771 + 5.5722
R2 = 0.9998
Height (Meters)
100
80
60
40
20
0
0
10
20
30
40
Degree
50
60
70
Range vs. Angle Degree
(When Velocity = 50 m/sec, Drag Coefficient = 0.22, and Lift Coefficient = 0.16)
160
140
120
Range (Meters)
100
80
r() = -0.12412 + 9.7586 - 46.205
R2 = 0.9761
60
40
20
0
0
10
20
30
40
Degree
50
60
70
Change in Launch Velocity
(When Angle= 30 degrees, Drag Coefficient= 0.22, Lift Coefficient= 0.16)
70
60
50
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 (Meters)
40
30
20
10
0
0
50
100
150
-10
Range (Meters)
200
250
300
Height vs. Launch Velocity
(When Angle = 30 degrees, Drag Coefficient= 0.22, Lift Coefficient= 0.16)
70
h(v) = 0.0104v2 - 0.0708v + 2.7941
R2 = 1
60
Height (Meters)
50
40
30
20
10
0
30
40
50
60
Launch Velocity (Meters/Second)
70
80
90
Range vs. Launch Velocity
(When Angle = 30 degrees, Drag Coefficient= 0.22, Lift Coefficient= 0.16 )
300
r(v) = 3.6687v - 54.2
R2 = 0.997
250
Range (Meters)
200
150
100
50
0
30
40
50
60
Launch Velocity (Meters/Second)
70
80
90
Change in Drag Coefficient
(When Angle= 30 degrees, Velocity= 50 m/sec, Lift Coefficient= 0.16)
30
25
20
Height (Meters)t
0.21
0.22
15
0.23
0.24
0.25
10
5
0
0
20
40
60
80
-5
Range (Meters)
100
120
140
160
Height vs. Drag Coefficient
(When Angle = 30 degrees, Velocity = 50 m/sec, and Lift Coefficient = 0.16)
26.6
h(x) = 37x + 17.11
R2 = 1
26.4
26.2
Height (Meters)
26
25.8
25.6
25.4
25.2
25
24.8
0.205
0.21
0.215
0.22
0.225
0.23
Drag Coefficient
0.235
0.24
0.245
0.25
0.255
Range vs. Drag Coefficient
(When Angle = 30 degrees, Velocity = 50 m/sec, and Lift Coefficient = 0.16)
150
145
140
Range (Meters)
135
130
125
120
115
110
105
100
0.205
0.21
0.215
0.22
0.225
0.23
Drag Coefficient
0.235
0.24
0.245
0.25
0.255
Change in Lift Coefficient
(When Angle= 30 degrees, Velocity= 50 m/sec, Drag Coefficient= 0.22)
30
25
Height (Meters))
20
0.14
0.15
0.16
0.17
0.18
0.19
15
10
5
0
0
20
40
60
80
-5
Range (Meters)
100
120
140
160
Height vs. Lift Coefficient
(When Angle= 30 degrees, Velocity= 50 m/sec, Drag Coefficient= 0.22)
27
26.5
26
Height (Meters)
25.5
25
24.5
h(x) = -61.543x + 35.138
R2 = 0.9988
24
23.5
23
0.1
0.11
0.12
0.13
0.14
0.15
Lift Coefficient
0.16
0.17
0.18
0.19
0.2
Range vs. Lift Coefficient
(When Angle = 30 degrees, Velocity = 50 m/sec, and Drag Coefficient = 0.22)
138
136
134
Range (Meters)
132
130
128
r(x) = -315.51x + 180.26
R2 = 0.9736
126
124
122
120
118
0.1
0.11
0.12
0.13
0.14
0.15
Lift Coefficient
0.16
0.17
0.18
0.19
0.2
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