Golf Ball Trajectory Model
After many years of research and testing, we have developed a mathematical model for the flight of a golf ball. We use this model to predict the flight based on initial golf ball parameters such as speed, launch angle, spin along with environmental conditions such as air temperature, air pressure, humidity, etc.
The model also predicts those initial conditions from the collision with a golf club swinging at various club head speeds.
To see trajectory diagrams of different clubs, click HERE.
Comparison of Tannar Golf Trajectory Model to Empirical Trackman Data
Trackman is a company that makes a golf ball monitor that uses Doppler Radar to track the golf ball as it flies through the air. It is used by the United States Golf Association, PGA Tour, Royal & Ancient Golf Association, golf club and golf ball manufacturing companies and many golf teaching professionals. It is known as the most accurate golf ball monitor on the market.
Trackman has been publishing some of its empirical results in its newsletters. In its January, 2010, Newsletter, it published PGA Tour and LPGA Tour average statistics for Driver trajectories (Page 6 of Jan 2010 Trackman Newletter, http://trackman.dk/getmedia/e388effb0686425b9824650d2e123c6e/January2010.aspx .
Below in the table you'll find the Trackman data as well as the values predicted by the Tannar Golf Trajectory Model which are outputs of a MS Excel Spreadsheet.
My model assumed a temperature of 25 degrees Celcius (77 F), no wind, near sea level. The Trackman data is accompanied by the note, " Please be aware that the location and weather conditions haven't been taken into consideration. Besides these reservations the data is based on a large number
of shots and give a good indication on key numbers for tour players."
Thus, the Trackman data would have been collected at various temperatures, altitudes and wind conditions. Some shots would have been with the wind, some into the wind, some in a crosswind. Note, that if air temperature is higher and altitude is higher, the ball will carry farther.



Clubhead 
Angle of 
Ball 
Smash 
Launch 
Spin 
Max 
Land 
Carry 



Speed 
Attack 
Speed 
Factor 
Angle 
Rate 
Height 
Angle 
Distance 



(mph) 
(deg) 
(mph) 

(deg) 
(rpm) 
(yd) 
(yd) 
(yd) 












PGA Tour Average 









Trackman 

112 
1.3 
165 
1.47 
11.2 
2685 
31 
39 
269 
Tannar Model 

112 
1.3 
167 
1.49 
10.3 
2374 
27 
33 
259 
























LPGA Tour Average 









Trackman 

94 
3.0 
139 
1.47 
14.0 
2628 
25 
36 
220 
Tannar Model 

94 
3.0 
140 
1.49 
14.5 
2043 
25 
33 
215 












As can been seen from the data, the Tannar Model predicts quite closely the golf ball parameters measured by the Trackman for the two club speeds of 112 mph and 94 mph.
December 13, 2008  LiveScience.com recently published a video titled The Secret of Golf Balls Revealed: Dimple Dynamics. The video features an extensive simulation based on research conducted by scientists at Arizona State University and the University of Maryland looking at the effects of air drag on the dimples on a golf ball. Click on the graphic below.
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 24% 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 1831 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), 357361 (1983).
5 P.W. Bearman and J.K. Harvery, "Golf Ball Aerodynamics," Aeronaut, Q. 27, 112122 (1976)
7 W.M. MacDonald and S. Hanzely, "The physics of the drive in golf," Am. J. Phys 59 (3) 213218 (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) 213218 (1991).
5 P.W. Bearman and J.K. Harvery, "Golf Ball Aerodynamics," Aeronaut, Q. 27, 112122 (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
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