As anyone who has played a round of golf will attest to, the sport is based
around many fundamental principals of physics. These basic laws are involved
with every aspect of the game from how a player swings the club to how the ball
moves through the air on its way toward the pin. It is the challenge that
physics presents to the golfer that has allowed the game, and equipment used, to
develop so drastically over the past one hundred years. The first golf balls
used were called featheries. They were made with a horsehide cover packed with
wet goose feathers. When the balls dried they became extremely hard. The major
flaw with the featheries was that they could not be used when the conditions
were wet because they would soften again.[5] Despite the flaw of the featheries,
they remained the only ball used up until the middle of the 19th century when
the revolutionary gutta-percha ball was invented. The new ball, sometimes
referred to as a “guttie”, was molded from the warmed, dried gum of the
sapodilla tree.[5] These balls were cheap to manufacture and opened up the game
of golf to a more diverse socio-economic group. This in turn made the game of
golf very popular, which led to dramatic improvements in golf balls in the next
decades. In 1900 a unique event occurred. Some claim that it can be called the
first professional sports endorsement. The Spalding Company paid England’s
Harry Vardon a considerable sum of money to come to the United States to
demonstrate what he could do in winning tournaments using the latest ball
design. He won the U.S. Open using the new rubber-wound Haskell ball.[5] This
led to another major revolution in the design of the golf ball. Not only was
this ball cheap to manufacture, but also it could be hit farther than any other
ball previously used. The Haskell ball was such a success that it was not until
1968 that the two-piece balls of today emerged in the market. Obviously a lot of
time, effort, research, and money were put forth into the development of the
golf ball, as it is manufactured today. The reason for this ongoing process is
to help a golfer use some laws of physics to his advantage (i.e. placing spin on
the ball to create lift) while finding a work around for other physical
properties that can be detrimental to a players golf game (i.e. drag which
causes the ball to slow down and fly closer to the ground). When examining the
physics, which surrounds the game of golf, one must carefully consider all
aspects of the game, not just the golf ball or even just the equipment being
used. The stroke is by far the most important aspect to any participants round
of golf. Among the scientific community, an event, such as the golf stroke, is
thought of as a dynamic process using the physical principals of mechanics based
on Newton’s Laws of motion. The stroke is actually three separate events; the
swing of the club, the impact of the club head with the ball, and the flight of
the ball toward the target. It is the sum of these three parts that makes a
successful stroke. Before delving into the details of the golf stroke, it is
important for one to consider the general concepts of motion that control the
swing of the golf club. Two men are most influential in this area of study,
Galileo Galilee and Isaac Newton. It is the principles of these two men that
will be used during the discussion of the physics of golf. A brief explanation
of momentum, moment of inertia, torque, centripetal force, and centrifugal force
can be located in “Appendix 4”. These terms were derived from the
experiments and research of first Galileo, and then expanded upon by Newton.
Although neither of these two men are solely responsible for all of the physical
principals presented in this paper, Galileo and Newton were two of the most
influential men in these areas of study. When a scientist attempts to explain
something, he or she always develops a model to work with. In the case of the
golf stroke, it has become evident that comparing such an action to the snapping
of a whip lends itself nicely to a deeper understanding. The model appropriate
to the study of a whip, such as a bullwhip, would be a large number of small
rods with flexible connections. This is important to understanding how the whip
works. At the start of the motion, as the hand moves the handle of the whip, the
momentum of the whip increases. The hand exerts a force on the whip handle for a
time, producing, according to Newton’s Second Law, an increase of momentum.
This force moving the whip handle a few feet also does work on the whip, giving
it kinetic energy. When the hand stops, the whip exerts a force on the hand, and
this force in turn decreases the momentum of parts of the whip. Thus, momentum
is not conserved because a force acts and there is no displacement because the
hand remains still. During the stroke, successive parts of the whip are stopped,
and the kinetic energy of these parts is fed into the successively smaller and
smaller sections of the whip. The kinetic energy of a body depends on its mass
and the square of its velocity according to the equation KE = ½ m v2.
Therefore, at the start of the stroke, the total mass of the whip is moving with
a moderate speed. Toward the end of the stroke, a much smaller mass must be
moving at a much higher speed to have the same kinetic energy. This is shown to
be true by the cracking of the whip, or the sonic shockwave the tip of the whip
sends out. Although it may not seem possible, a human swinging a golf club works
in a very similar manner to the whip. First, one must consider where the energy
for the stroke comes from. In the whip it obviously came from the muscles in the
arm. However, when swinging a golf club, much more energy is required, in fact
it has been estimated that the amount of energy transferred into the golf ball
during impact is about two horsepower.[1] Because muscle generates approximately
1/8th horsepower per pound, it would take about 32lbs [2] of fully loaded muscle
to generate enough energy to produce two horsepower. If however the muscle is
not suitably loaded, then more then 32lbs of muscle would be needed. If that
seems to the reader to be a lot of muscle, their assumption is correct; that is
a lot of muscle. The average person does not have that much muscle in their
arms. Instead they must rely on the much larger muscles in their back and legs.
The person uses their body to transfer the energy from these muscles into their
arms. The explanation of how this is done can be found in “Appendix 3” of
this paper. It shows a graph of the five torques which work on the arms during
the swing. This is the first aspect of how the whip works; the transferring of
energy. When interviewed, several professional golfers, including Sam Snead,
Tommy Amour, Cary Middlecoff and Frank Beard, although unable to give the
scientific reasons behind their down stroke, stated in one form or another, that
“the left shoulder pulls the left arm”[3]. The scientific explanation of
what they stated is that as the horizontal pull of the left shoulder on the left
arm produces a positive angular acceleration to help with the downswing.[3] This
shows clearly that the energy is transferred from the body into the arms and
subsequently down the shaft of the golf club and into the ball. The way this
energy was calculated was through the use of a computer program. It was setup so
that it gave the total kinetic energy of the arms and the club and the kinetic
energy of each of them separately. This can be seen by curves A, B, and C in
appendix 1 (please refer to the explanation at the bottom of the graph for an
explanation of the curves). A fourth curve, D, was also graphed. This curve
shows that work done by the golfer as a function of the downswing angle as he
applies the torque on his arms. To skip ahead to the point, the total kinetic
energy of the system when the club head makes contact with the ball comes 71%
from the work TS * a(i), 13% from the decrease in the potential energy of the
system, and 16% from the work down on the system in the shift of the golfer
toward the target. [5] The total kinetic energy is very important to ones game
of golf. According to the conservation of momentum principal, with any given
club and any given ball, the speed of the ball depends directly on the speed of
the club head. Therefore it is necessary to use the large muscles of the body to
generate the necessary club head speed (about 100mph) needed to hit the ball far
enough in order to approach the possibility of playing par golf. The chart below
demonstrates how ones game would be affected if they were not able to generate
enough club head speed. Assuming that the golfer is able to sink each of his
puts, the first example reveals that if the golfer were only able to drive the
ball 160 yards, he would lose 15 strokes because of his lack of distance off the
tee. As his driving distance increases, the number of strokes the golfer would
loose decreases until he is able to drive the ball 230 yards (or hit the ball
with a club head traveling about 100mph). Yards 160 170 180 190 200 210 220 230
Stroke Lost to par 15 12 9 7 5 3 1 0 The physics surrounding a game of golf is
not just based on the swing as shown above. While 50% of the game of golf is the
stroke used to hit the ball, the other 50% of the game is how the ball travels
through the air toward the pin. Because the flight of the ball cannot be
controlled with the same precision by the golfer that he can control his swing
with, many developments have been made toward creating an ideal golf ball. Just
looking back as few as 50 years one can see the tremendous affect physics has
played on the design of the golf ball. First, it was discovered that worn golf
balls tended to stay in the air longer because their uneven surface caused a
greater spin as the ball passed through the air at a high velocity. Later it was
determined that dimples on the golf ball serve the same purpose, and not only
that, improve on the affect first observed by the ware and tare on the original
golf balls. In the past 5 years, golf balls are being manufactured with three
different sized dimples placed in strategic locations on the ball. This allows
the ball to remain in the air as long as possible while sacrificing as little
energy to overcoming drag as possible. As demonstrated by any golfer who can hit
a ball in a straight line, the aerodynamic forces at work on a golf ball are
what make the flight of the ball so unique. If one were to stand behind a golfer
and watch the flight of the golf ball, that person would not see a parabolic
arch as one might expect. Instead, the ball will appear to climb in a straight
line for a few seconds and then begin to fall back to earth slowly. According to
Newton’s First Law (a body continues in a straight line at a constant speed
unless a force acts on it) the observed path of the ball does not seem possible.
As the designer of the golf ball would be quick to point out, it is the
aerodynamic force on the dimpled, spinning, ball, traveling at a high speed,
that was balancing the vertical force of gravity which caused non uniform motion
in the path of the balls flight. British scientist, P.G. Tait, performed the
first experiments done with the aerodynamics of a golf ball in 1887.[2]
Professor Tait showed through his studies the importance of spin on the flight
of the golf ball. He states that in his youth he was taught, “all spin is
detrimental”.[2] He practiced vigorously to hit a ball virtually spin free.
After completing his research, Tait wrote, “I understand it now, too late by
35 years at least”.[2] What Tait was referring to was the importance of spin
on a golf ball. He and his son performed experiments where, “we fastened one
end of a long untwisted tape to the ball and the other to the ground, and
induced a good player to drive the ball (perpendicularly to the tape) into a
stiff clay face a yard or two off, we find the tape is always twisted; no doubt
to different amounts by different players—say from 40 to 120 or so turns per
second. The fact is indisputable.”[2] Professor Tait clearly states that a
ball driven with spin about a horizontal axis with the top of the ball coming
toward the golfer has a lifting force on it that keeps the ball in the air much
longer than would be possible without spin. What the scientist was observing was
the competing affects of lift and drag. While it is possible to generate
equations and solutions for different swings and velocities and come up with an
optimum ratio of lift to drag, it has been stated that it is better for the
individual golfer to discover this for himself because not every swing is the
same. Research has shown that a larger spin produces a larger drag, which makes
the ball slow down more rapidly and thus decreases the distance it travels, but
a larger spin produces more lift, which keeps the ball in the air for a longer
time and thus allows it to fly father. An experienced golfer knows that the
force of lift will superceded the force of drag, however it is left up to the
individual to find their own balance between these two forces. The next logical
step in the explanation of the physics surrounding a game of golf is to relate
the two aspects just discussed. The following text is an explanation of what
happens between the time when the energy of the swing is transferred into the
club and the flight of the ball; or more specifically how the collision between
the club head and the ball transfers spin and energy into the ball. First, the
collision must be considered. During the collision between the club head and the
ball, several things happen. The club head is slowed down, and the ball is sent
off with a high speed at some angle above the horizontal with a high rate of
spin.[1] This all happens in less than a thousandth of a second while the club
head moves less than an inch.[1] Such a short time makes it extremely difficult
to observe what is happening during the collision. The force between the ball
and the club head averaged over the time of the collision is greater than
3000lb[1] and high speed photography has shown the ball to be considerably
flattened against the club head.[1] The elastic properties of the ball come into
place at this point because it is those properties that allow the ball to be
compressed and then spring away from the face of the club at a high velocity.
Although there is no scientific proof of what exactly happens at the point of
impact, through the use of physics, several educated guesses can be made.
Momentum is conserved: Since the club head is at the end of a somewhat flexible
shaft, one may, to a fair degree of approximation, assume that the club head in
its horizontal motion at the bottom of the swing acts as a free body.[4]
Therefore, the horizontal momentum of the club head before the collision must be
the same as the sum of the horizontal momentum of the club head and the ball
after the collision. It is important to note that the vertical momentum is not
conserved because the arms and shoulders pull up with a force on the club head.
The collision is inelastic: By simply holding a golf ball and feeling how firm
it is, one can easily see that is not perfectly elastic. Therefore, some
mechanical energy must be lost. Newton was the first to experiment with this
property of collisions. He found that the ratio of the speed with which the ball
leaves the floor to that with which the ball approaches the floor to be
practically a constant over a large range of speeds.[ ] The constant is called
the coefficient of restitution. For a perfectly elastic ball the coefficient is
one. For a ball that does not rebound at all, the coefficient would be zero.
This also applies to collisions that happen at an angle such as with the face of
the golf club and the ball. The ball slides and rolls on the clubface: Consider
this example as an explanation for how a golf ball generates its spin from the
uneven surface of the face of a golf club. As anyone who has thrown a bowling
ball will attest to, the ball slides down the alley at first. Since there is
some amount of friction between the ball and the lane, the ball slows down,
which allows the ball to begin rolling. After the ball has traveled some
distance it no longer slides and just purely rolls. The same is true for the
face of the golf club. As the collision occurs the ball begins to slide toward
the top of the face of the club. However, because the friction force between the
ball and clubface is so great it quickly begins to spin (roll) off the top of
the club. This generates the tremendous amount of spin necessary to keep the
ball a loft for drives at or above 230 yards. It is these three factors together
that the quantity known as effective loft is derived from. The effective loft of
any club is given as EL = L + a(i) - B(i) – Y. L is the loft of the club a(i)
and B(i) are angles that are dependant upon each swing and each person
performing the swing and Y is the back swing angle of the arm. From effective
loft of the club, one can estimate the components of drag and lift on the golf
ball. The following table expresses the variations that are possible during the
swing. B(0) - B(i) - EL + Spin + Lift + B(0) + B(i) + EL - Spin - Lift - Y - B(i)
+ EL - Spin - Lift - Y + B(i) - EL + Spin + Lift + TS + B(i) - EL + Spin + Lift
+ Al + B(i) - EL + Spin + Lift + As an example, the third line reveals that when
the back swing angle of the arms is decreased, the effective loft is decreased,
the spin is decreased, and the lift is decreased. As one can see through the
material presented above, the golf swing is a multi-stage process. It is not
simply the swing, or the transfer of energy, or the flight of the ball that is
subject to the laws of physics. The first aspect of the golf stroke, which is
based upon physical principals, is the downswing of the golf club. The golfer
must do two things in order to have a successful shot. He must first generate
enough energy to hit the ball a significant distance. And then he must transfer
this energy into the golf club. The energy is derived from the muscles in the
golfers body. As was previously stated it takes at least 32lbs of muscle to
generate the necessary two horsepower for hitting the golf ball. Most of this
energy comes from the legs and back of the individual. Then, the golfer uses his
body and arms, along with the shaft of the golf club like a whip. Just as a whip
transfers energy from its large mass at the handle down to the tip causing a
dramatic acceleration, the golfer transfers the energy through his body into the
shaft of the golf club, which flexes. When the golfer snaps his wrist at the
point of impact, all of the energy is transferred into the club head allowing it
to achieve a velocity of 100mph or even greater. At the point of impact, more
physical properties take over. As the club comes in contact with the ball, two
important factors are most prevalent. First, the ball is semi- elastic and
therefore the ball flattens somewhat when it comes in contact with the face of
the club. This allows the ball to spring away at a tremendous velocity, which is
also based on the principal of conservation of momentum. The other important
factor that happens at impact is the generation of spin. At first, the ball
begins to slide up the face of the club toward the top, however, because of the
large coefficient of friction; the ball stops sliding and begins rolling. This
action gives the ball a rotation around its horizontal axis, which creates lift
and drag. Lift and drag are the final aspects of how physics relates to golf. As
the ball spins, it creates lift by disturbing the flow of air around the ball.
The dimples help greatly with this. However, drag is also produced, which
threatens to pull the ball back toward the earth. It is the job of the golfer
and the golf ball manufacturer to generate enough lift either through the swing
or the dimpled design of the golf ball so that the upward lifting force
counteracts the downward forces of gravity and drag. As anyone who has played a
round of golf has observed, the spin created with modern clubs and ball design
more then compensates for drag and gravity and allows the ball to stay aloft for
a long time. Because of the unique challenges that physics present during a game
of golf, it will be a long time before anyone is able to master the game. In an
endeavor to improve scores many miracle products have claimed to lower ones
score, however it is evident that only those ideas and products, which have a
basis in science, have stayed on the market. The golf ball is a prime example of
this. It has made dramatic changes from being made of dried goose feathers to
the two piece dimpled design of today. All of the improvements on the ball were
based around trying to give the golfer and edge in lowering his score and
working around some of the laws of physics, which prevent him from reaching
perfection. Appendix 4 The following terms will be defined based on their
relevance to the physics of golf: momentum, moment of inertia, torque,
centripetal force, and centrifugal force Momentum: Newton’s first law defines
a property of a body called inertia, which describes what happens to a body when
no force acts on it; the inertia of a body is said to be measured by its mass.
When acted upon by a constant unbalanced force, the body will experience
acceleration proportional to the mass of the body. The mass of a body is
proportional to its weight. Momentum is then defined as the mass of a body
multiplied by its velocity. Like velocity, momentum, has a direction as well as
magnitude, making it a vector quantity. From the definition of momentum, for
constant mass the rate of change of momentum is the product of the mass and its
acceleration. Newton’s second law suggests that an unbalanced force on a body
is associated with its acceleration. For the purpose of this paper, Newton’s
second law states that the mass of a body multiplied by its acceleration is
proportional to the force acting on it, and the acceleration is in the direction
of the force. The way in which momentum applies to golf is through the transfer
of momentum from the golf club to the golf ball. Before the collision, the club
head is moving at a speed of 100mph along the horizontal. After the collision,
for a club without loft, the ball is moving off at a high velocity, and the club
head continues in the follow-through at a somewhat reduced velocity. For a club
without loft, these velocities will also be horizontal. The momentum is such
that the total momentum before the collision is equal to that of the momentum of
the club head after the collision plus the momentum of the ball. Moment of
Inertia: The linear acceleration of a body when acted upon by a constant force
depends on its mass, which as already stated is quantity proportional to its
weight. The larger the mass is, the smaller the acceleration will be for a given
force. Similarly, when a constant torque acts on a body, its angular
acceleration will depend on the mass of the body and on how the mass is
distributed in the body. The combination of mass and its distribution in the
body is called its “moment of inertia”. When the axis of rotation of the
body is chosen such that more of the mass is far from the axis, the moment of
inertia will be larger. Thus the moment of inertia will depend of the choice of
axis. This concept is easily demonstrated with the help of a golf club. When the
club is help at the grip end between two fingers and let hang so that the shaft
is along a vertical line, it is very easy to rotate the club along a vertical
axis. But when the club is held near the center of the shaft, where it balances
between the same two fingers, the same torque produces a much smaller angular
acceleration. The moments of inertia in the two cases differ by a factor of 10.[
] This same affect can be observed when a club is first waggled about the grip
in the usual way and then waggled while holding the head. Torque: Torque is the
term used to describe twist in a quantitative manner. Two factors, the amount of
force applied and the distance over which it is applied determine torque. The
size of the torque is found by multiplying the size of the force by the length
of the lever arm, the lever arm being the shortest distance from the line along
which the force acts to the axis about which the body may rotate. The force must
be in a plane perpendicular to the axis of the rotation. Centrifugal Force: This
force can be observed when a golf ball is placed on the dashboard of an
automobile just inside the windshield and is observed while the vehicle travels
around turns. One will notice that the ball will always roll to the outside of
the curve and rolls more quickly the tighter the turn. Actually, the ball does
not accelerate; it appears to accelerate since there is no centripetal force to
make it turn in the same path as the car. Its motion is the result of a lack of
centripetal force rather than the result of an outwardly directed force being
applied to the ball. Centripetal Force: According to Newton’s Second Law, the
centripetal force on a body moving in a circle is proportional to the mass of a
body multiplied by its centripetal acceleration. The centripetal acceleration
increases with the radius of the circle on which it moves and with the square of
the angular velocity of the motion. Appendix 1 The following curves were drawn
based on the information gathered and analyzed with the use of a computer. The
curves are calculations for the energies present during a swing. Curve A shows
the total kinetic energy as it develops throughout the swing. Curve B shows how
the kinetic energy of the arms varies throughout the downswing. Curve C shows
how the kinetic energy of the club alone varies throughout the swing. Curve D
shows the work done by the golfer as he applies the torque by his arms to the
system. (graph taken from source #5) Appendix 2 (original drawings but concept
from source #5) These drawings illustrate the forces on a golf ball during its
flight. The first set of pictures shows how the air moves around the ball during
its flight. The first pictures show that when there is some spin, the air
pressure around the ball is changed because of the turbulence created by the
rotation. The picture below that shows how the air would move if there were no
spin. The other two pictures demonstrate how using spin can change the flight of
the ball. For example, the top picture is showing that a ball spinning on a
vertical axis in a clockwise direction will travel to the right because of the
airflow around the ball. The bottom picture on that side is illustrating another
example of how air can flow around a ball with no spin. The other two
illustrations show a three-dimensional (on the top) and a two-dimensional (on
the bottom) view of some of the vectors involved with the flight of a golf ball.
The illustrations show the effective loft of the club, the lifting vector as
well as the path of the golf club and the angle at which the face points.
Appendix 3 This is a graph of the five-torques acting on the arms as they vary
throughout the downswing. Curve A shows the constant torque TS of the golfer on
the system. Curve B shows the torque that depends mainly on the acceleration of
the wrist-cock-angle. The torque represented by curve C depends mainly on the
square of the velocity of the wrist-cock-angle. Curves D and E show the torques
resulting from action of gravity and the golfers weight shift respectively. The
torque T shows how the sum of the five-torques on the arms varies during the
downswing and becomes very large just prior to the club colliding with the ball.
(graph from source #1)

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