Scientific Model of Golf Ball Flight -- physics of golf, mathematical model

Science of Golf Ball Projectile Model

The purpose of this report is to provide scientific evidence regarding the scientific model used throughout this site. The flight of a golf ball when struck by a golf club like a Driver is very complex but now well understood, as I'll indicate in this report. Many reports have been written on the subject with far fewer verifiable experimental results. Check out my resources page for some references.

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Previous papers have explored the flight of a launched golf ball. Many of the most recent papers can be found in the three volumes of the World Scientific Congress of Golf1, 2,3. Erlichson4 analyzed golf ball trajectories taking into account aerodynamic lift and drag. Bearman and Harvey5 measured the lift and drag coefficients for both conventionally dimpled and hexagonally dimpled golf balls in wind tunnel tests and used these values to determine how the range depended on initial ball launch speed, angle and spin. MacDonald and Hanzely7 used the Bearman and Harvey coefficients to determine the angle of launch, which produces the largest carry of a golf ball. McPhee and Andrews8 also used the coefficients to determine the effects of sidespin on the golf ball. Smits9 measured coefficients of lift and drag over a larger range of ball speeds and also measured the rate of spin decay. A foundation of many of these papers is the class book, "Search for the Perfect Swing"10, which was one of the first scientific writings on many facets of the game of golf.

To develop my model of the flight of a golf ball, I mainly used the treatise of MacDonald and Hanzely7 and the combined coefficients of Bearman and Harvey5 and Smits9. The Smits coefficients apply to a larger range of spins necessary for application to all clubs but are slightly higher in the spin range of the Bearman and Harvey coefficients. Both methods used to determine the coefficients had pros and cons. I used an average of the two different coefficients in my model.

Williams11 provided data on the carry and drive of a British golf ball hit with a driving machine. He did not cite the loft of the driver used but it has been assumed that it was a standard 10o commonly used at the time. He found that the carry of a golf ball depended linearly on the launch speed by the equation C = 1.5v -103, where v is the speed in ft/s and C is in yards. I varied the ball speed in my model (without sidespin or wind). My model replicated the linearity of the dependency with a similar equation,

C = 1.45v - 87.5 (see Appendix A). "Search for the Perfect Swing"10 explains and cites that an American ball (used in my model) will carry 2-4% less than a British size ball. Comparison figures are recorded in the table below.

Ball Speed (ft/s)	180	200	220	240	260	280
British ball measured carry (yd)	167	197	227	257	287	317
My model's British ball(yd)	177	207	237	266	296	325
My model's American ball(yd)	174	203	232	261	290	319
Percentage Difference	5.8	5.0	4.3	3.5	3.1	2.5

On average, my model predicts the carry within 4 % of the Williams data. Since there was no recording of loft angle, launch angle, and atmospheric conditions, the reasons for the differences can only be speculated. It is difficult to find empirical data that has well documented conditions of launch. My model does predict, however, a linear dependency between launch speed and carry as determined by McPhee and Andrews8, MacDonald and Hanzely7 as well as most recently A.R. Penner12.

Using my model, I compared its predicted results to the results of McPhee and Andrews because they investigated the effects of sidespin (see Appendix B). The charts indicates balls launched at 200 ft/s as an angle of 16 degrees with a range of initial directions, on value of sidespin and cross wind speed. Comparing the results of the two models, the flight time difference is 0.3%, the range down the fairway difference is 22.3%, and the lateral deflection to the right difference is 4.5%. The range down the fairway is quite large because the McPhee model is flawed. According to the empirical results of Williams, a British ball launched at 200 ft/s at an angle of 10 degrees would carry 197 yards or 591 feet. An American ball would fly about 2% less distance or about 579 feet. McPhee's model predicts 588 feet, but when launched at an angle of 16 degrees. According to Erlichson4 and A.R. Penner12 who investigated how a golf ball's carry depends on the launch angle, the greater the launch angle the greater the carry up to a maximum carry at about 23. If one were to launch a ball at 200 ft/s at an angle of 16 degrees, it would go significantly farther than one launched at 10 degrees, thus the McPhee model predicts ranges down the fairway which are too small. From "Iron Byron" (mechanical robot the swings a club) tests which are reported by various golf club companies, support the ranges predicted by my model. For instance, Slazenger13 tested one of their balls and found that its range when projected at about 8 degrees at a speed of 158 mph (70.2 m/s or 230 ft/s) was about 276 yards or 828 feet (again no atmospheric conditions reported, but probably optimal, i.e. downwind).

My model does agree with McPhee model when comparing flight times and lateral displacements (due to slicing and wind). The spin and crosswind effects do not have a great of an effect on the lateral or vertical displacement as they do on the forward displacement because the forces are not as significant in those directions. Thus various models would not differ as much in those directions. The McPhee model, for instance, assumed the aerodynamic forces depended on the ball speed, while my model assumes the square of the ball speed as does the model of MacDonald and Hanzely7 and A.R. Penner12.

As additional support for the accuracy of my model, it can also model shots hit by all irons. Using spin rates for iron shots cited by D.C. Winfield14 and G. Tavares, M. Sullivan & D. Nesbitt15, my model also predicts the correct ranges of distances for iron shots.

There is not much empirical data available to quantify the effect that sidespin has on the golf ball. Companies with robots, including the USGA, seem to keep the data a secret unless paid large sums of money. Some golf club companies do cite small amounts of data from testing they have done. One such company is 13Adams Golf who has tested and compared drivers of other companies with the clubface 2 degrees open. Their website has a picture of sliced drives and cites "While the biggest names in golf, including the CallawayT HawkeyeT and Taylor Made FiresoleT were all an average 18-31 yards off the centerline, the SC Series Driver ." My model predicts that a driver hitting a ball at 72.4 m/s (260.6 km/h =162.9 mph=237.5 ft/s) at a trajectory of 8.5 degrees (very typical conditions cited by companies) would carry 252.5 yards. If the clubface were open 2o, the ball would travel 245.6 yards down the fairway but would also travel 24 yards to the right. This amount of lateral displacement is within the range cited my Adams Golf.

I have spent hundreds of hours developing the model and the spreadsheet for predicting a golf ball trajectory. I have researched the subject immensely, compared the model's predictions to other experts' predictions and to empirical data. I believe it is a fairly accurate model of the golf ball trajectory and can be used with confidence.

My Computer Model

The equations of motion used in my model are:

ax = -Bu(Cdux - Cl(uz sin(a) - uycos(a) )

ay = -g -Bu(Cduy - Cluxcos(a) )

az = -Bu(Cduz + Clux sin(a))

as also used by MacDonald and Hanzely7. Each equation represents the acceleration of the golf ball in the 3 possible directions:

x = down the fairway (always positive)

y = vertically upwards (up is positive, down is negative)

z = laterally sideways (right is positive, left is negative)

B = a constant dependent on the conditions of the air

u = relative velocity between the ball and the air (i.e. wind)

Cd = coefficient of drag which depends on the speed and spin of the ball

Cl = coefficient of drag which depends on the speed and spin of the ball

a = the angle between the vertical and the axis of rotation of the spinning ball

g = the acceleration due to gravity = 9.80 m/s2

Because the coefficients of drag and lift change as the ball is in flight, solving the equations using Calculus and the analysis of differential equations is too complex if not impossible. Fortunately, very accurate approximations can be made using a computer and a spreadsheet by calculating the accelerations after very small periods of time have been used. I used a time interval of 0.05 s. My predictions would be more accurate if I used a smaller interval of time, but the increase in accuracy would be much smaller than the inaccuracy in the measured values of the coefficients of drag and lift (for instance, a 0.05 interval predicts a maximum height of 35.8 m while a 0.01 s interval predicts 35.9 m, am 10 cm difference).

Accelerations, velocities and positions are calculated every 0.05 s. Various parameters such as launch speed and angle, wind, backspin, sidespin, etc., can all be varied. One can change one quantity such as wind speed, and determine how the wind in various directions can affect the end result.

1,2,3 Science and Golf I, II & III, Proceedings of the World Scientific Congress of Golf, 1990, 1994, 1998, edited by A.J Cochran and M.R. Farrally.

4 H. Erlichson, "Maximum projectile range with drag and lift, with particular application to golf," Am. J. Phys. 51 (4), 357-361 (1983).

5 P.W. Bearman and J.K. Harvery, "Golf Ball Aerodynamics," Aeronaut, Q. 27, 112-122 (1976)

7 W.M. MacDonald and S. Hanzely, "The physics of the drive in golf," Am. J. Phys 59 (3) 213-218 (1991).

8 J.J. McPhee and G.C. Andrews, "Effect of sidespin and wind on projectile trajectory, with particular application to golf", Am. J. Physics, Vol. 56, NO. 10, October 1988

9 A.J.Smits, "A new aerodynamic model of a golf ball in flight", Science and Golf II: Proceedings of the World Scientific Congress of Golf. Edited by A.J. Cochran and M.R. Farrally, 1994

10 A. Cochran and J. Stobbs, "The Search for the Perfect Swing," (Lippincott, New York, 1968)

7 W.M. MacDonald and S. Hanzely, "The physics of the drive in golf," Am. J. Phys 59 (3) 213-218 (1991).

5 P.W. Bearman and J.K. Harvery, "Golf Ball Aerodynamics," Aeronaut, Q. 27, 112-122 (1976)

9 A.J.Smits, "A new aerodynamic model of a golf ball in flight", Science and Golf II: Proceedings of the World Scientific Congress of Golf. Edited by A.J. Cochran and M.R. Farrally, 1994

11 D. Williams, "Drag force on a ball in

Curriculum Vitae: Expert's Qualifications

Ken Tannar, 601 Capri Road, Enderby, B.C. V0E 1V3, Canada

Graduated with a Bachelor of Science with majors in Physics and Mathematics from the University of British Columbia, Vancouver Canada, in 1982.

Have been teaching Physics 11 (Intro to physics in Grade 11, 2nd to last year or secondary school), Physics 12 (university preparation in physics in Grade 12, graduating year from secondary school), Calculus 12 (first year university/college Calculus) for the past 18 years.

Have spent the last 5 years researching the literature on the topic of Physics of Golf

(a subtopic of the Physics of Sports or the Science of Sports). Much of my research has centered on the organization called the Scientific Congress of Golf, which holds a conference every 4 years at St. Andrews, Scotland

See Footnotes for a listing of the resources I have used to do my research.

Designed a website called ProbableGolf Instruction.com that provides golfers with methods to improve their game without having to do any physical practice. My ideas teach golfers to apply some simple results of some very complex mathematics and physics to my better club selections and decisions on the course. Much of the information is provided for free, but golfers can purchase other information and services on-line. The basis of the information provided in the website are results of the research and development I have done on the physics and mathematics of golf.

I have developed a computer spreadsheet that simulates the path a golf ball travels through the air as well as the collision between the various golf clubs and the golf ball. My model takes into account variables such as clubhead speed, loft, ball speed, initial trajectory angle, open, square or closed clubface, backspin, sidespin, air temperature, humidity, density, etc. As will be demonstrated later in this report, my model takes into account the same variables as other researchers with comparable results.

I am a 1-handicap amateur golfer. I have been quite successful competitively winning dozens of tournaments throughout British Columbia. I have played in many B.C. Amateurs (one year I missed making the Provincial Team by 1 shot) and 3 Canadian Amateurs. Most recently I came 12th in the 2000 Canadian Champions of Club Champions sponsored by the Royal Canadian Golf Association. I have been Club Champion for the last 5 out of 7 years at the Salmon Arm Golf Club.

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