Mathematical Exploration!!Modelling a Golf Shot!!Rationale!!!Golf is a sport that has interested me highly for a long time now and is also the reason why I took this topic for the coursework as I wanted to investigate the mathematics behind golfing. The first question is how accurately can a golf drive be modelled? What external factors need to be considered? How do the trajectories change when different forces are factored in? !!In this investigation, I start of with the foundations - before the ball flies where external forces need to be considered in. Firstly, I look at the impact of the golf club with the ball and therefore the launch velocity of the golf ball. Then I will proceed on the trajectories and will begin with the simple projectile motion as it provides a good introduction of flight paths. Furthermore, the drag force will be considered in. However, for this external factor, I kept it in a reasonably way because the air resistance, in reality, fluctuates drastically depending on the speed, direction, size of the ball but also the wind and other weather conditions. Lastly, I will be talking about the Magnus effect and how professional golfers use this technique to maximise distance. !!With the simple projectile motion, I investigated on the projectile angle and if 45 degrees was really the farthest distance. The solution was that indeed due to the fact that the horizontal component equals to the vertical component so that it reaches the maximum range. !!Further on I factored in the air resistance. At the end I plotted a graph to the table. I made a hypothesis and it was correct. The 30 degree projectile flew the farthest. The flight increased slow at first and increased faster later on. The angle was just 30 degrees, so the ball flew farther out of the three other angles because it flew closer to the ground and produced the highest velocity.!!!!!!!!!!!!!!

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The impact of the golf club with the golf ball!!The type of golf club and the mass of it together with the ball determine the kinetic energy and momentum induced upon impact. !!The mass of the club and ball are: mclub and mball!The velocities are then: Vclub and vball!As the kinetic energy for a rigid body is  : !1

!The conservation of energy is then: !!Additionally, the conservation of momentum tell us  : !2

!Doing some algebra let us find the solutions to these variables: !!

and !!!The ratio of the mass of the ball can be as small as it needs to be but will always be less than twice the speed of any club. !For instance, if vclub = 50 metres per second  (or 180 kilometres per hour), mclub = 0.195 3

kilograms  , mball = 0.0495 kilograms  . If we substitute these numbers in the formula to get Vball, 4 5

then we get 79.75 metres per second or 287.1 kilometres per hour. !!However, upon impact, the transfer of kinetic energy and momentum is not elastic because some is lost through heat and damage to the golf ball. !!For this instance, the ball launch speed equation is given as:! !!!Where CR is the coefficient restitution. The velocity of the golf ball depends on the mass of the club head and its velocity upon impact with the ball. For this investigation we assume a general head club weight of approximately 200 grams.  !6

!So, an elastic collision would equal 1 or when it equals 0, all kinetic energy has been lost. The coefficient restitution is usually around 0.78. Substitute it in the equation then we yield Vball of 72.04 metres per second or 259.344 kilometres per hour. !

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� Henderson, T. 2014. Kinetic Energy. [online] Available at: http://www.physicsclassroom.com/class/energy/u5l1c.cfm [Accessed: 23 Mar 2014].1

� Henderson, T. 2014. Momentum Conservation Principle. [online] Available at: http://www.physicsclassroom.com/class/momentum/u4l2b.cfm [Accessed: 2

23 Mar 2014].

� Rhodes, D. 2013. What Is the Average Golf Swing Speed? | LIVESTRONG.COM. [online] Available at: http://www.livestrong.com/article/208471-what-is-3

the-average-golf-swing-speed/ [Accessed: 23 Mar 2014].

� Golf, P. 2014. Golf Technology and Equipment - Weight. [online] Available at: http://www.purelygolf.com/E/Weight.htm [Accessed: 23 Mar 2014].4

� Elert, G. 2014. Mass of a Golf Ball. [online] Available at: http://hypertextbook.com/facts/1999/ImranArif.shtml [Accessed: 23 Mar 2014].5

� Tannar, K. 2014. Distance & Technology Part 1: Driver Head Weight. [online] Available at: http://probablegolfinstruction.com/PGI%20Newsletter/6

news02-12-04.htm [Accessed: 23 Mar 2014].

!

!The table above shows that, the smaller the mass of the golf club, the greater the ball speed. The ball speed reaches its maximum when the mass of the golf club is at 0.190 kg. The uncertainty of this is that all golfers have different swings and strengths so it varies depending on the person. The average mass is approximately at 200 grams as mentioned before. !!The contact collision between the golf club and the ball is around 1/2000th of a second. So the ball has not moved but the golf club has transferred a large amount of kinetic energy onto the ball and the compressed ball has stored it as potential energy. Once it leaves the tee, the ball becomes a spherical shape as the compressed ball releases the energy stored and is then converted back to kinetic energy.  !7

!Simple Projectile Motion!!The simple projectile motion is the projectile of a golf ball launched with initial conditions with the assumption on a flat ground factoring in the gravitational force but not the air resistance. !!It provides a good introduction to the flight trajectory of the golf ball as external forces aren’t factored in yet to simplify this investigation. The air resistance is very challenging to factor in as it varies depending on the speed, size and direction of the golf ball but this force will be focused later on. !!The projectile motion is given by Newtons laws of motion and in this situation, the 2nd law is the relevant one:!!F = ma (f and a are vector quantities) !F = 0i - mgj (gravity acts negative on the y-axis and m is the mass)!The initial velocity contains the horizontal (x) and the vertical (y) axis: !!!!!!

Golf club mass (kg)

Driver speed (ms

CR Ball speed (ms Ball speed/club speed ratio

0.110 66.02 0.721 78.36 1.19

0.130 59.64 0.741 75.20 1.26

0.150 55.61 0.757 73.46 1.32

0.170 52.02 0.770 71.31 1.37

0.190 49.33 0.780 69.66 1.41

0.210 47.09 0.789 68.17 1.44

0.230 44.84 0.798 66.34 1.48

�3� Arnold, D. 2014. The Science of a Drive. [online] Available at: http://www.mathaware.org/mam/2010/essays/ArnoldDrive.pdf [Accessed: 23 Mar 2014].7

(Where the subscript 0 = initial velocity; x and y = directions; i and j = vector direction)!!!!!The angle of initial velocity in the horizontal and vertical axis is given by: !!

! ! ! ! !!and!!!

The gravitational force acts as a vertical force that pulls the golf ball towards the ground over a period of time: ! !!!!To find the vertical displacement of the y axis, the next step is to differentiate the above equation which gives us:! !!!!!Then we solve for y through the separation of variables method. It solves partial differential equations:! !!!!!If it’s possible to integrate in a closed form, then it can be solved for y, then a solution to the problem is acquired:! !!!!For the initial velocity, y, t and c can be 0 as the golf ball is still on the ground and this will result in the following equation of:! !

!(1)!!!

To express y as a function of x, we eliminate t from this equation: (2)!And then substitute into equation (1).!!From equation (2) we first get: !!!!!!!!

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If we then substitute this into equation (1), we get: !

! ! ! ! ! !!!!!From the equation above we can get differentiate once to get the velocity of the ball:!!!!!!If we differentiate it another time, we can get the acceleration of the golf ball:!!!!!!Let’s substitute in some numbers to investigate the trajectory of a simple projectile. !!!The horizontal motion is: ! !!!Vertical motion: ! !!!!!!!!!!!!!!!!!!!!!!Figure 1: A general projectile motion   !8!

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(3)

(5)

(4)

� Nave, R. 2014. General Ballistic Trajectory. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/tra3a.gif [Accessed: 24 Mar 2014].8

Now let’s plug in some numbers into the horizontal and vertical motion equations. I used the angles of 30, 45 and 60 degrees.!The maximum range is said to be at 45 degrees because the horizontal component equals to the vertical component so that it reaches the maximum range  . !9

Here I want to examine if that is the case. !!!

!!The table above shows the different values for different angles. !So I plugged in the two equations: v0 = 72.04 ms-1 (which we calculated at the beginning)!I used 3 different angles for theta = 30°, 45° and 60°.!For time I used the initial time, when the ball is on the ground until the vertical component showed negative numbers, i.e.: when it landed on the ground again. !!!!!!!!

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time in seconds

x (distance) (30°)

y (height) (30°)

x (distance) (45°)

y (height) (45°)

x (distance) (60°)

y (height) (60°)

0 0 0 0 0 0 0

1 62.39 31.115 50.94 46.03 36.02 57.48

2 124.78 52.42 101.88 82.26 72.04 105.16

3 187.16 63.915 152.82 108.67 108.06 143.02

4 249.55 65.6 203.76 125.28 144.08 171.07

5 311.94 57.47 254.70 132.07 180.1 189.32

6 374.33 39.54 305.64 129.06 216.12 197.75

7 436.72 11.79 356.58 116.23 252.14 196.37

8 499.11 -25.76 407.52 93.60 288.16 185.19

9 458.46 61.15 324.18 164.19

10 509.40 18.90 360.2 133.38

11 560.34 -33.16 396.22 92.77

12 432.24 42.34

13 468.26 -17.89

� Henderson, T. 2014. Maximum Range. [online] Available at: http://www.physicsclassroom.com/mmedia/vectors/mr.cfm [Accessed: 24 Mar 2014].9

This is the graphical representation of the table above. The 45° angle flies the farthest and this is due to the fact that the horizontal component equals to the vertical component so that it reaches the maximum range. !When the ball is shot from a 30° or 60° angle, they reach different heights, yet land at the exact same distance.!At 60°, the golf ball has a higher vertical maximum so it doesn’t fly very far. Therefore the ball falls faster towards the ground. !At 30°, the golf ball flies closest to the ground but flies faster. Because of the launch angle, it goes forward instead of upwards. Therefore, both projectiles reach the same distance. !!Now, let’s see what the maximum heights of these projectiles are. !!ymax can be found through this equation:  !10!!y0 = 0 when the ball reaches it’s maximum height, vy = 0!!We want to get the vertical component so we solve for ymax. !!!! !!Now we substitute in and !!!!This will result in: !!!!!

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Hei

ght

0

50

100

150

200

Distance0 150 300 450 600

x (distance) (30°) x (distance) (60°) x (distance) (45°)

� Ducoff, N. 2014. Basic Equations and Parabolic Path - Boundless Open Textbook. [online] Available at: https://www.boundless.com/physics/two-10

dimensional-kinematics/projectile-motion/basic-equations-and-parabolic-path/ [Accessed: 24 Mar 2014].

Using this equation, the maximum height for 30° is: !!!The time it takes to reach the maximum height is, using this formula: !which I obtained by differentiating equation (1) and setting it to 0 because the velocity will be 0. !!!The time, in metres per seconds, takes: ! !!!!For 45° it is: ! and the time required: !!!For 60° it is: ! and the time required: !!!Now we want to find out how long the golf ball actually flies until it reaches the ground again. This can be calculated by doubling the tmax formula above because the time required to reach the maximum height, requires the same time to fall back towards the ground again. !!!!!!For 30° we take the tmax from above, and multiply by two: 3.67 x 2 = 7.34 seconds!For 45° it is: 5.19 x 2 = 10.38 seconds!For 60° it is: 6.36 x 2 = 27.24 seconds!!If we look back to the table and hence the graph above, we see that these numbers are precise and represent the data above. !!!Drag Force!!We just analysed the trajectory of a golf ball without air resistance. Now, we are going to factor it in. As the ball flies through the air, its speed is dragged back as it pushes against the air molecules. The third Newton’s law states that the air molecules apply an equal opposite force to the golf ball. !!One of the forces involved in this, is the gravitational force as used in the simple projectile motion: !!Fg = 0i - mgj (gravity acts negative on the y-axis and m is the mass)!!Then the force due to air resistance:!!Fdrag = -bv = -bvxi - bvyj (the drag force acts as a negative velocity and i and j are the vectors) !!Now we add the two equations together for the total force: !!Fg + Fdrag = -bvxi - (mg + bvy)j !!

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The initial velocity of the golf ball is on a x, y plane because gravity and the air resistance affect only the distance and the height as this investigation is on a 2-dimensional plane. Therefore, the z plane can be neglected. !!The equation of the motion is then: ! due to Newton’s second law!!!(where v = velocity; g = gravitational force; c = constant) !!Before we proceed, we need to find the terminal velocity.!The net force of a falling object is, in terms of the drag equation):   !11!At equilibrium -> Fnet = 0 ; Therefore: !!!

solving for v using simple algebra results in: !!!!-> Where p is the air density, A is the radius of the golf ball and Cd is the drag coefficient !!p = 1.1447 kgm-3 [  ] ; A = 0.00139m2 [  ]; Cd = 0.47   !12 13 14!We then substitute the numbers into the formula and we obtain the terminal velocity = 36.05 ms-1!!!Now we continue with the previous formula: divide by m and !!! ! ! ! ! substitute the terminal !! ! ! ! ! velocity in!!We obtain the horizontal component-> ! and vertical -> !!!Let’s work on the horizontal component first. So, now we integrate it:!!!!!!!Using the integration law of: ! ! !! !!!!!!

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� Nave, R. 2014. Air Friction. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/airfri.html [Accessed: 28 Mar 2014].11

� Guy, M. 2014. Google Answers: golf physics. [online] Available at: http://answers.google.com/answers/threadview/id/132351.html [Accessed: 29 Mar 12

2014].

� Wilkins, A. 2014. Golf Projectile. [online] Available at: http://nothingnerdy.wikispaces.com/file/view/Flight%20of%20golf%20ball%20-%20sample13

%20EE.pdf/99255043/Flight%20of%20golf%20ball%20-%20sample%20EE.pdf [Accessed: 29 Mar 2014].

� Weisstein, E. 2014. Drag Coefficient -- from Eric Weisstein's World of Physics. [online] Available at: http://scienceworld.wolfram.com/physics/14

DragCoefficient.html [Accessed: 29 Mar 2014].

We then obtain: ! !!!!!Where vx0 is the launch on the horizontal axis, therefore, the angle is cos(θ). We then substitute it in and using the log law, we power e as a base and we obtain the velocity function:!!!!!We then integrate it again to get the displacement function of the horizontal component:!!!!!!Now, we do the same procedure for the vertical component. We use the previous equation:!!!!!!And we integrate it:! !!!!!!To obtain this: ! !!!!!Using the same integration law as before, we obtain:!!!!!!Where vy0 is the initial velocity in the vertical axis, therefore, the angle is sin(θ). We then substitute it in and using the log law as before, we power e as a base and we obtain:!!!!!!We then integrate it again to get the displacement function of the vertical component. !!!!!!

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Now let’s plug in some numbers into the horizontal and vertical motion equations. I used the same angles used in the simple projectile motion of 30, 45 and 60 degrees.!My hypothesis is that out of those three angles, the 30 degrees will have the farthest flight path because due to the drag force, the flight will increase slow at first and increase drastically towards the end. As the angle is just 30 degrees, it will therefore fly closer to the ground and have the most velocity. !Here I want to examine if that is the case. !!!!!

!!!!!The table above shows the different values for different angles. !So I plugged in the two equations: v0 = 72.04 ms-1 (which we calculated at the beginning)!I used 3 different angles for theta = 30°, 45° and 60°.!For time I used the initial time, when the ball is on the ground until the vertical component showed negative numbers, i.e.: when it landed on the ground again. !For vt I used the number we calculated before, which was 36.05 ms-2!!!!!!!!!!!

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time in seconds

x (distance) (30°)

y (height) (30°)

x (distance) (45°)

y (height) (45°)

x (distance) (60°)

y (height) (60°)

0 0 0 0 0 0 0

1 54.62 27.05 44.60 40.11 31.54 50.13

2 96.23 39.06 78.57 62.07 55.56 79.73

3 127.92 39.62 104.45 70.22 73.86 93.69

4 152.07 31.47 124.16 67.83 87.80 95.74

5 170.46 16.66 139.18 57.43 98.41 88.71

6 184.47 -3.20 150.62 40.91 106.50 74.76

7 159.33 19.74 112.66 55.55

8 165.97 -4.97 117.36 32.33

9 120.93 6.05

10 123.66 -22.56

!This is the graphical representation of the table above. As predicted, at the angle of 30 degrees, the golf ball flies the farthest. The flight increases slow at first and increases faster later on. The angle is just 30 degrees, so the ball will tend to fly farther out of the three other angles because it flies closer to the ground and produces the highest velocity.!!This time, with the drag force factored in, the 45 degree projectile doesn’t fly the farthest as it was with the simple projectile motion because the angle is too high to achieve the greatest distance. !At 60 degrees, the ball flies too high and the velocity is quickly lost in the vertical component instead of the horizontal.!!From research, the three projectiles in reality would be: 30° for a 6 Iron which flies 160 yards  . My 15

calculations are close, as the distance is 179 on the x axis. !The 45° would be a 9 Iron which flies >130  . The graph is slightly over the real distance numbers. !16

However, the 60° is slightly above 120 on the x axis but in reality, the Lob Wedge has an angle of >60°  .!17!Also, if we compare the two graphs of the simple projectile motion and the one with the drag force, we notice that the projectile flies a lot less. According to my investigation with these variables I used, at 30°, it flies around 2.45 times less, at 45°, it flies around 3.16 times less and at 60°, it flies around 3.63 times less. !!The reason is probably due to the fact that this is a theoretical calculation and hasn’t factored in the wind velocity or any other weather conditions. Also, in reality, the distance is highly variable. A professional golfer hits any golf club farther than the average golfer. !

�12

Hei

ght

0

25

50

75

100

Distance0 50 100 150 200

x (distance) (30°) x (distance) (45°) x (distance) (60°)

� Aubuc.hon, V. 2014. Golf Club Comparison Chart - Loft and Distance. [online] Available at: http://www.vaughns-1-pagers.com/sports/golf-club-data.htm 15

[Accessed: 29 Mar 2014]

� Aubuc.hon, V. 2014. op. cit. Golf Club Comparison Chart - Loft and Distance.16

� Aubuc.hon, V. 2014. op. cit. Golf Club Comparison Chart - Loft and Distance.17

Magnus Effect!!Golf balls have dimples on the surface as it causes the drag force to transition from laminar flow to turbulent  . The laminar flow is the ideal flow type because the liquid particles travel in a straight 18

line have a constant velocity whereas the turbulent flows has a dissimilar velocity  . !19

However, on the golf ball, the turbulent flow is able to remain on the dimples for a longer time and creates a less pressure drag which results in further distance to travel  . !20!A professional golfer will strike the ball with a backspin that exerts a vertical force on the ball that keeps it in the air for a longer time. This is also known as the Magnus effect.! !If it’s spinning clockwise or with topspin, the force is down and when it’s spinning anti-clockwise the force is up.!!When the ball is spinning, the air is flying around the ball from the front to the back, as the ball is moving forward. The ball is spinning in the same direction as the airflow at the top of the ball but in the opposite direction at the bottom [  ][  ].!21 22

Air is dragged around the top of the ball downwards towards the back because of friction between the air and the ball surface, but at the bottom of the ball the airflow in the ball opposite direction the air comes to a halt very soon and instead is being deflected upwards[  ][  ].!23 24

Net result: air is deflected downwards and due to Newton’s third law, the air exerts an equal force on the ball which is upwards.!

!The general formula is  : !25

!Where (v)ω is the angle velocity of the golf ball multiplied by the velocity. !

!!!!!!

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� Arnold, D. 2014. The Science of a Drive. [online] Available at: http://www.mathaware.org/mam/2010/essays/ArnoldDrive.pdf [Accessed: 23 Mar 2014].18

� Sleigh, P. 2014. Laminar and Turbulent Flow. [online] Available at: http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section4/laminar_turbulent.htm 19

[Accessed: 29 Mar 2014].

� Sleigh, P. 2014. op. cit. Laminar and Turbulent Flow. 20

� Fitzpatrick, R. 2014. The Magnus force. [online] Available at: http://farside.ph.utexas.edu/teaching/329/lectures/node43.html [Accessed: 29 Mar 2014].21

� Hosch, W. and Liesangthem, G. 2014. Magnus effect (physics). [online] Available at: http://www.britannica.com/EBchecked/topic/357684/Magnus-effect 22

[Accessed: 29 Mar 2014].

� Fitzpatrick, R. 2014. op. cit. The Magnus force.23

� Hosch, W. and Liesangthem, op. cit. G. 2014. Magnus effect24

� Fitzpatrick, R. 2014. op. cit. The Magnus force.25

Conclusion!!This mathematical exploration was about modelling a golf shot. Before even considering the forces on the ball, I calculated the impact force between the club and the golf ball. That number was then used for the simple projectile motion. Then I figured out the time it took to reach the maximum and hence, until it reached the ground again. Furthermore, I factored the biggest challenge - the air resistance. First, I got the terminal velocity which I then substituted into the horizontal and vertical components which I derived from. !!With the simple projectile motion, I investigated on the projectile angle and if 45 degrees was really the farthest distance. The solution was that indeed due to the fact that the horizontal component equals to the vertical component so that it reaches the maximum range. !!I also found out that the 30 and 60 degree projectiles reach a different vertical number, however, land at the exact same spot. !At 60°, the golf ball has a higher vertical maximum so it doesn’t fly very far. Therefore the ball falls faster towards the ground. !At 30°, the golf ball flies closest to the ground but flies faster. Because of the launch angle, it goes forward instead of upwards. Therefore, both projectiles reach the same distance.!!Further on I factored in the air resistance. At the end I plotted a graph to the table. I made a hypothesis and it was correct. The 30 degree projectile flew the farthest. The flight increased slow at first and increased faster later on. The angle was just 30 degrees, so the ball flew farther out of the three other angles because it flew closer to the ground and produced the highest velocity.!!This time, with the drag force factored in, the 45 degree projectile didn’t fly the farthest as it was with the simple projectile motion because the angle was too high to achieve the greatest distance. !At 60 degrees, the ball flew too high and the velocity was quickly lost in the vertical component instead of the horizontal. !!If we compare the two graphs I plotted, according to my investigation with these variables I used, at 30°, it flies around 2.45 times less, at 45°, it flies around 3.16 times less and at 60°, it flies around 3.63 times less. !!A professional golfer will use the Magnus Effect to maximise the distance. He will strike the ball with a backspin that exerts a vertical force on the ball that keeps it in the air for a longer time. Air is dragged around the top of the ball downwards towards the back because of friction between the air and the ball surface, but at the bottom of the ball the airflow in the ball opposite direction the air comes to a halt very soon and instead is being deflected upwards.!!This mathematical exploration has uncertainties as I calculated using partly variables from online sources and partly from numbers I calculated myself, hence it wasn’t as accurate. Also, I didn’t factor in the wind velocity and other weather conditions so the reality golf shots are therefore longer or shorter dependent on environmental factors. For instance, the warmer the temperature, the further the golf ball flies. Also, it depends on the golfer, as a professional golfer hits a golf ball farther than the average golfer. This is due to their hip rotation which produces a so called lag that increases the club speed drastically. !

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References!Arnold, D. 2014. The Science of a Drive. [online] Available at: http://www.mathaware.org/mam/

2010/essays/ArnoldDrive.pdf [Accessed: 23 Mar 2014].!

Aubuchon, V. 2014. Golf Club Comparison Chart - Loft and Distance. [online] Available at: http://www.vaughns-1-pagers.com/sports/golf-club-data.htm [Accessed: 29 Mar 2014].!

Ducoff, N. 2014. Basic Equations and Parabolic Path - Boundless Open Textbook. [online] Available at: https://www.boundless.com/physics/two-dimensional-kinematics/projectile-motion/basic-equations-and-parabolic-path/ [Accessed: 24 Mar 2014].!

Elert, G. 2014. Mass of a Golf Ball. [online] Available at: http://hypertextbook.com/facts/1999/ImranArif.shtml [Accessed: 23 Mar 2014].!

Fitzpatrick, R. 2014. The Magnus force. [online] Available at: http://farside.ph.utexas.edu/teaching/329/lectures/node43.html [Accessed: 29 Mar 2014].!

Golf, P. 2014. Golf Technology and Equipment - Weight. [online] Available at: http://www.purelygolf.com/E/Weight.htm [Accessed: 23 Mar 2014].!

Guy, M. 2014. Google Answers: golf physics. [online] Available at: http://answers.google.com/answers/threadview/id/132351.html [Accessed: 29 Mar 2014].!

Henderson, T. 2014. Momentum Conservation Principle. [online] Available at: http://www.physicsclassroom.com/class/momentum/u4l2b.cfm [Accessed: 23 Mar 2014].!

Henderson, T. 2014. Maximum Range. [online] Available at: http://www.physicsclassroom.com/mmedia/vectors/mr.cfm [Accessed: 24 Mar 2014].!

Henderson, T. 2014. Maximum Range. [online] Available at: http://www.physicsclassroom.com/mmedia/vectors/mr.cfm [Accessed: 24 Mar 2014].!

Hosch, W. and Liesangthem, G. 2014. Magnus effect (physics). [online] Available at: http://www.britannica.com/EBchecked/topic/357684/Magnus-effect [Accessed: 29 Mar 2014].!

Nave, R. 2014. General Ballistic Trajectory. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/tra3a.gif [Accessed: 24 Mar 2014].!

Nave, R. 2014. Air Friction. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/airfri.html [Accessed: 28 Mar 2014].!

Probablegolfinstruction.com. 2014. Distance & Technology Part 1: Driver Head Weight. [online] Available at: http://probablegolfinstruction.com/PGI%20Newsletter/news02-12-04.htm [Accessed: 23 Mar 2014].!

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Rhodes, D. 2013. What Is the Average Golf Swing Speed? | LIVESTRONG.COM. [online] Available at: http://www.livestrong.com/article/208471-what-is-the-average-golf-swing-speed/ [Accessed: 23 Mar 2014].!

Sleigh, P. 2014. Laminar and Turbulent Flow. [online] Available at: http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section4/laminar_turbulent.htm [Accessed: 29 Mar 2014].!

Tannar, K. 2014. Distance & Technology Part 1: Driver Head Weight. [online] Available at: http://probablegolfinstruction.com/PGI%20Newsletter/news02-12-04.htm [Accessed: 23 Mar 2014].!

Weisstein, E. 2014. Drag Coefficient -- from Eric Weisstein's World of Physics. [online] Available at: http://scienceworld.wolfram.com/physics/DragCoefficient.html [Accessed: 29 Mar 2014].!

Wilkins, A. 2014. Golf Projectile. [online] Available at: http://nothingnerdy.wikispaces.com/file/view/Flight%20of%20golf%20ball%20-%20sample%20EE.pdf/99255043/Flight%20of%20golf%20ball%20-%20sample%20EE.pdf [Accessed: 29 Mar 2014].!

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