Flight Trajectories

   Last time I showed the flight trajectories for the Spalding TOP-FLITE XL^II, the TOP-FLITE II, and the TOP-FLITE PLUS ^II golf balls. These three golf balls appeared on the USGA Conforming List during the early 1990’s.

   The Top-Flite Plus II ball had a low trajectory at high ball velocity and then a higher trajectory at lower ball velocity resulting in long carry and little if any roll. The Top-Flite II ball had an initial high trajectory at high ball velocity which decreased slowly at lower ball velocities resulting in short carry and long roll. The Top-Flite XL^II ball had the highest trajectory and intermediate carry and roll.

   To measure golf ball trajectories, Spalding began using the United States Golf Association Test: Symmetry (1.5) in the 1980’s. Tests were carried out on an outdoor range with a driver to measure the carry distance, total distance, timed flight, and trajectory. Golf ball properties such as ball weight, compression, coefficient of restitution, shore D hardness, and cut resistance were recorded.¹

Effects of dimple diameter, depth, and volume

   In the late 1980’s Spalding began looking at the effect of dimple diameter and dimple depth on golf ball trajectories.^2,3,4 When there were multiple dimple sizes on the golf ball, the weighted average dimple diameter (wt av Diameter) and the weighted average dimple depth (wt av depth) were calculated.

   As it turned out, the total dimple volume (TDV) was also important in understanding golf ball trajectories. This is how we calculated the total dimple volume for the three Spalding golf balls. A golf ball dimple can be thought of as two spherical segments–one rotated and placed under the other. The volume (V) of one spherical segment can be calculated using the formula⁵ where (h) is one half the dimple depth (wt av depth) and (a) is equal to the dimple radius (wt av Diameter/2).

V = 1/6 ∏ h(3a²+h²)

   The volume of one dimple is thus twice the volume of one spherical segment (2V). The total dimple volume (TDV) of all the dimples on a golf ball equals the number of dimples on the ball times twice the volume of one dimple (2V). Table 1 shows the calculated properties for the three golf balls.

golf ball	dimples	coverage	wt av D	wt av d	TDV
			mm	mm	mm3
Top-Flite XLII	410	70%	3.56	0.247	504
Top-Flite II	422	73%	3.55	0.256	536
Top-Flite Plus II	422	78%	3.67	0.249	557

   In general, a golf ball with a large TDV (Top-Flite Plus^II) will have a lower trajectory and longer total distance. Likewise, a golf ball with a smaller TDV (Top-Flite II) will have a higher trajectory and long carry distance.⁶ The trajectory for the TOP-FLITE XL^II ball did not seem to correlate with the TDV. The TOP-FLITE XL^II, which had only 410 dimples, had medium TDV, but surprisingly the highest trajectory, and medium roll. Other factors need to be considered also.

Lift and Drag are important too

   The other factors are lift and drag.⁷ Lift results from a difference in pressure created by a distortion in the air flow resulting from back spin of the golf ball (Figure 1). Drag is the aerodynamic force component acting parallel to the ball flight direction (Figure 2).

Figure 1

Figure 2

   The study of golf ball aerodynamics has led to numerous ways to calculate and optimize the drag and lift coefficients. In one study⁸ a golf ball was disclosed that had low lift and low drag coefficients at high velocities, and high lift and high drag at low velocities. This was purported to mean that the ball would have lower lift during the ascent so the ball travels further. Furthermore, the ball would have higher lift during descent and thus maximizing the carry distance.

   In the late 1990’s the USGA^9,10 and golf ball manufacturers¹¹ began to use indoor test range technology instead of outdoor ranges to measure the lift and drag coefficients of golf balls to determine the trajectories of golf balls.

References

1. Michael J. Sullivan, US 4,884,814, Dec. 5, 1989, filed Jan 15, 1988.
2. Joseph F. Steifel, R.D. Nesbit, Terence Melvin, US 5,018,741, May 28, 1991, filed Jul. 24, 1989.
3. R. Dennis Nesbit, Joseph F. Stiefel, US 5,044,638, Sep. 3, 1991, filed Jun. 12, 1990.
4. Donald J. Bunger, Joseph F. Stiefel, US 5,060,953, Oct, 29, 1991, filed Jan. 18, 1991.
5. Weisstein, Eric W. “Spherical Cap.” From MathWorld-A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalCap.html
6. Jeffrey L. Dalton, Laurent Bissonnett, US 6,796,912, Sep. 28, 2004, filed Nov. 21, 2001.
7. Laurent C. Bissionnette, Jeffery L. Dalton, Steven Aoyama, US 6,729,976, May 4, 2004, filed Mar. 14, 2002.
8. Steven Aoyama, Douglas E. Jones, US 6,923,736, Aug 2, 2005, filed Jan 6, 2003.
9. Leonard F. Anfinsen, et.al., US 5,682,230, Oct. 28, 1997, filed Nov. 1, 1995.
10. Burton B. Lieberman, et.al., US 6,186,002, Feb. 13, 2001. filed Apr. 21, 1998.
11. Douglas Winfield, William Gobush, US 6,285,445, Sep. 4, 2001, filed Sep. 17, 1999.