Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Superior
University, Lahore, PK, 7-8 April, 2016.
TRAJECTORY COMPARISON OF A ROUGH
GOLF BALL WITH A SMOOTH BALL
M.Haris Shafique*
College of Aeronautical Engg,
National University of Sciences
& Technology, Pakistan
[email protected]
Syed Irtiza Ali Shah*
College of Aeronautical Engg,
National University of Sciences &
Technology, Pakistan
[email protected]
Muneeb Iqbal*
College of Aeronautical
Engg, National University of
Sciences & Technology,
Pakistan
Shehryar Kharal*
College of Aeronautical
Engg, National University of
Sciences & Technology,
Pakistan
Aezaz Adeeb*
College of Aeronautical Engg,
National University of Sciences
& Technology, Pakistan
Abstract - Flight path of a smooth sphere and a
dimpled golf ball has been of great concern and has been
analyzed for centuries. In present work, it is concluded that
the dimpled golf ball out drive a smooth sphere, it means
that the trajectory of the dimpled golf ball is larger in
distance as compared to the trajectory of a smooth sphere.
Here spin of the ball was neglected which causes the very
important side force (also known as a Magnus effect) and it
was assumed that ball was hit without causing any spin.
The ball follows the equation of motion for plane motion (x,
z). It was considered that ball was struck with initial
velocity Vo and θo. Furthermore air at sea-level and a finite
range of initial conditions were taken which means that the
initial condition were assumed which could easily be
achieved. Equations of motion were integrated and solved
simultaneously to get trajectories for smooth sphere and
actual/dimpled golf ball. Comparison was made between
both the trajectories obtained. The primary objective of
this research was to compare the flight motion in terms of
range, which is between the height and the distance and
also considering velocity and time, of smooth sphere and
dimpled golf ball. A system of differential equations was
our special concern and interpreted to aid in the
discussion. After conducting the research, results indicate
that the dimpled golf balls have better performance than
the smooth spheres because in dimpled golf balls we have
turbulence and the separation points over the surface fall
back to the rear side of the ball due to which we get larger
distance covered by the ball and that is why the dimpled
golf balls are preferred over smooth balls as around the
world.
Keywords - Smooth Golf Ball, Rough sphere, Dimpled Golf
Ball, Equations of motions, Trajectories ,Magnus Effect,
Ordinary Differential Equations, Runge Kutta Method, NonLinear Differential Equations.
I.
INTRODUCTION
Golf is being played since 80BC. Initially people used to
hit stone which with the passage of time revolutionized and
replaced by a small wooden ball. New techniques and
methodologies were adapted and people moved to rubber ball
with feathers. Presently,t here are many materials used in golf
balls but the majority can be broken down into three
categories: rubbers, ionomers and urethanes.
Today’s golf ball didn’t always have dimples on it, it was
William Taylor; an engineer by profession, who first came up
with the idea of dimpled ball. He found that if the golf ball is
dimpled, it would be more aerodynamic, allowing golfers to
have better control for farther drives. Different people and
industries presented different designs of golf ball afterwards. A
typical golf ball have dimples ranging from 250 to 450.
Everyone has his own ideology for designing a ball with depth,
number and pattern of dimples. Till now the design of a golf
ball is considered as one of the most typical design and
actually the world has progressed so far in designing a golf ball
soon after it was discovered that the dimpled golf balls are far
better than the smooth golf balls. Table (1) shows shapes of
dimples for different balls. Each one of them concludes to
better golf ball aerodynamically in his own way. However,
research is still in its full bloom for being good to perfect.
Internationally, the United States Golf Association strictly
regulates the design and material used in golf ball
manufacturing. Table (2) represents an idea of the size and
mass of those golf balls.
Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Superior
University, Lahore, PK, 7-8 April, 2016.
Table (1) Ball Type VS Dimples Shape
This table shows different types of ball and their dimples
shape.
BALL TYPE
Callaway Big Bertha
PGF Optima TS PLUS+
Pinnacle Gold FX Long
Srixon AD333
Wilson Staff DX2 Soft
Distance
Top Flite D2 Distance
Taylor Made TP/Red LDP
DIMPLES SHAPE
Hexagonal
Circular
Circular
Circular
Circular
Circular within circular
Circular
Where;
Here;
F= Drag Force/ Air Resistance
= Co-efficient of Drag
= Density of Air
D= Diameter of the of the Sphere
And:
So the equation becomes by substituting the value of F:
Let us substitute a constant value k:
so
(C)
Considering equation (B):
Table (2) Ball type and its diameter and mass
This table shows different types of balls with their diameter in
millimeters and mass in grams.
BALL’S NAME
So the equation becomes by substituting the value of F:
MASS (g)
Callaway Big Bertha
DIAMETER
(mm)
42.67
PGF Optima TS PLUS+
Pinnacle Gold FX Long
Srixon AD333
42.68
42.67
42.67
45.4
45.8
45.4
Let us substitute a constant value k:
Wilson Staff DX2 Soft
Distance
Top Flite D2 Distance
Dunlop Progress
42.68
45.4
so
42.67
40.45
45.5
24.9
as :
II.
45.2
(D)
And:
RESEARCH METHODOLOGY
A golf ball’s pattern of flight is effected by different
aerodynamic factors. In our model we have designed a system
of differential equations to predict the trajectory of smooth ball
and dimpled golf ball. Microsoft Excel has been used to solve
the system of equations and we have designed a generalized
equation for both the axes in terms of time (t) and from there
we have predicted the expected distance and height over a
range of time (t). Throughout the flight interval velocity and
angle with which ball leaves the (x, z) plane effects the motion
of dimpled golf ball and the smooth sphere. After solving and
getting the result, trajectories of rough sphere/dimpled golf ball
and smooth ball have been drawn for the purpose of
comparison.
III.
Where;
MATHEMATICAL APPROACH
Some of the governing equations of motion which are
implemented are given as:
(A)
(B)
Consider equation (A);
IV.
CALCULATION FOR THE VALUE OF K
For smooth ball
The value of co-efficient of drag is taken from the graph given
in the problem as this is the value where it is almost constant
for smooth ball.
As;
Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Superior
University, Lahore, PK, 7-8 April, 2016.
For dimpled golf ball
The value of co-efficient of drag is taken from the graph
given in the problem as this is value where it is almost constant
for dimpled golf ball.
So the equation becomes:
After solving for
.
Assumptions - The following assumptions were made to
simplify the dimpled ball model:
As;
1. No wind.
2. No change in air conditions due to altitude or humidity.
3. No changes in Barometric pressure.
4. No Magnus effect.
From this equation it is easy to see that the velocity of the
ball in the x direction stays constant throughout the duration of
the flight. Note that throughout these systems, the initial
coordinates of the ball will be (0, 0).
The equations can only be solved by Runge Kutta
Method which is the solution of Non-Linear Differential
Equations. The simultaneous solution for the above equations
is possible by using any of the Soft wares such as Matlab,
Mathcad, and C++ etc.
From another reference we have derived the equations by
a simpler method which is also known as the equation of
Classic Parabolic Motion.
Motion can be derived from a system of two differential
equations, in the x and z directions. Every force, excluding
gravity, is neglected. This assumes no wind, Drag Force,
Magnus Effect, Change in altitude, or changes in humidity.
From Newton’s Second Law of Motion, and the fact that
gravity is the only force acting on the ball.
As no force is acting in the x-direction, so;
So the equation becomes:
after solving for
Now:
Fig. 1. Range of a smooth ball
This figure shows the range of the smooth ball is terms of a graph
which is between height and the distance.
The calculated trajectories were then plotted to show the
effects of angular acceleration and aerodynamic drag. From the
graph shown in Fig (D) ahead we can clearly see that the
distance covered by the rough sphere/dimpled golf ball is
larger as compared to the smooth ball. This is because of the
Drag Force especially. The drag force arises because the ball
has to push its way through the air, and the air exerts an equal
and opposite force on the ball. The details are more
complicated than one might expect, as indicated in Fig (B). At
very low speeds, air flows smoothly around the ball from the
front to the back. Smooth flow is classified as laminar flow. In
an ideal situation there is no friction between the air and the
ball, the pressure at the front of the ball is the same as the
pressure at the rear of the ball, so there is no drag force. In
reality, air flowing over the surface of the ball is slowed down
by friction with the surface of the ball until it comes to rest at a
point known as the separation point. As a result, the flow of air
separates from the ball at the separation point rather than
following the surface around to the rear of the ball. The air
pressure drops in a region at the back of the ball known as the
wake, causing air outside the wake to flow back into the rear
side of the ball in a turbulent manner, but the net result is that
the force on the front of the ball is larger than the force at the
back of the ball, so the ball experiences a backwards drag force
that increases as the ball speed increases.
Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Superior
University, Lahore, PK, 7-8 April, 2016.
If the surface of the ball is dimpled, then the flow of air
around the surface can become turbulent well before the air
speed drops to zero. In that case, air near the surface mixes
with air further away from the surface that is moving faster
than air right at the surface. The speed of the air at the surface
is therefore increased by the turbulence, so the surface air takes
longer to slow to a stop, and the separation point moves closer
to the rear side of the ball. The low pressure wake is therefore
narrower, so the force of that low pressure region on the ball is
decreased and the force of the higher pressure region outside
the wake is increased. As a result, the drag force is reduced
compared with that on a smooth ball which results in the larger
distances
covered
as
compared
to
the
smooth ball.
Table (3) Data analysis for the trajectory - This table shows
the height and the distance of the ball at different velocity,
angle and time.
X (yards)
Y (yards)
V (mph)
Angle (deg)
Time (sec)
0
31.59716629
62.8956608
93.91680844
124.6793959
155.1997762
185.4919875
0
16.96600421
31.21289299
42.76246181
51.63476552
57.84849476
61.42137506
150
143.6003266
137.9431871
133.0850348
129.0807993
125.9792774
123.8179704
0
0.5
1
1.5
2
2.5
3
215.5678932
62.3705819
122.6181346
245.4373495
60.71316068
122.3810065
275.1084018
56.46643712
123.0860655
304.5875075
49.64840237
124.6917712
333.8797778
362.9892296
40.27805818
28.37571086
127.1385977
130.353649
391.9190358
13.96320766
134.2558823
415.9764354
0
137.9857234
30
26.39076602
22.48322799
18.28692272
13.82529863
9.137580221
4.278840099
0.682362629
5.668852954
10.60150583
15.40633668
20.02048158
-24.3959566
28.50085438
31.71334205
3.5
4
4.5
5
5.5
6
6.5
6.918140702
Fig. 2. Flow Comparison of Smooth Ball and Dimpled Ball
This figure shows the comparison of the flow over the smooth ball and the
dimpled golf ball.
V.
IMPLEMENTATION
Equations of motion were integrated and with initial
conditions specified above were implemented. Calculations
and mathematics involved are shown, the equations can be
solved using the MatLAB, Mathcad, C++ etc. As far as it is
concerned the solution is not possible without using any if
such soft wares.
VI. RESULTS
Following are the tables taken from MS Excel and it
includes the graphs of trajectories.
Fig.3. Trajectory of a smooth ball
This figure shows the range of the smooth ball with respect to the height and
the distance covered.
VII. RECOMMENDATIONS
This work can be more significant by using more types of
balls with different series of Reynolds Number and and using
other values for the velocities and time. One can use CFD
(Computational Fluid Dynamics) analysis for the better
understanding and explanation of the flow effect over the
surfaces of both the balls.
Proceedings of the 3rd International Conference on Engineering & Emerging Technologies (ICEET), Superior
University, Lahore, PK, 7-8 April, 2016.
VIII. COMPARISON
In this part we can easily judge that which ball is better
either the dimpled golf ball of the smooth ball. Here from the
graph we can see that the dimpled golf ball flies more than
twice the distance of the smooth ball which means that
dimpled golf balls are better in this aspect.
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Fig. 4. Trajectory comparison of Dimpled golf ball Vs Smooth ball.
From the trajectories shown in Fig (D) we can have an idea of
how the dimpled golf ball differs from smooth ball, for a
general equation the trajectory of Classic Parabolic Motion is
shown.
IX. CONCLUSION
From all the research we conclude that the dimpled golf
balls are far better than the smooth ball in terms of farther
drives. More over the modern world prefers dimpled golf balls
as they are aware of its more efficient aerodynamic
properties.As mentioned we can have an idea of how the
turbulence effects in increasing the speed for a dimpled golf
ball as for a dimpled golf ball the separation points shift
towards the rear side of the ball so that lesser drag occurs at the
dimpled golf ball which in result gives us a longer drive as
compared to the smooth golf ball where the separation points
don’t shift which results in lesser distance covered. For this
work we have integrated the equations of motion and set a
generalized form of equation for both the axes and from there
we have obtained the graph which are shown.
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image
obtained
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http://wings.avkids.com/Book/Sports/instructor /golf-01.html
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