Brian Manzella
Administrator
Here is a some real data:
1
Optimization of Driving Distance –
Importance of Determining the Attack Angle
Fredrik Tuxen, ISG A/S, Vedbaek, DENMARK
ABSTRACT
The trajectory of a golf ball is determined by the following ball launch data: 1) Ball Speed, 2)
Vertical and horizontal launch angle, and 3) Spin rate and spin axis orientation. From this point,
the trajectory will evolve in a deterministic way influenced only by the golf ball aerodynamic
properties and the environmental properties - wind speed and direction, temperature, pressure, and
humidity. Many different combinations of golf swings and club builds can produce the same
launch data, but for only a few of these combinations are the obtained launch data optimal. While
there is consensus in the golf community that optimal driving distance is obtained from a straight
shot hit squarely in the center of the club face, there are, however, many variations in the
recommendations for optimal vertical launch angle and spin rate for a given club head speed. By
using the commercially available ball flight monitor TrackMan™ that measures the club delivery,
ball launch data, and ball flight trajectory, the parameters to obtain optimum carry can be studied.
As it turns out, the attack angle and speed of the club head at impact are essential in determining
the optimal ball launch data for a given golfer in both club fitting and instruction.
KEYWORDS
Club fitting, driver optimization, attack angle, TrackMan™.
INTRODUCTION
Regarding optimization of driving distance, some work has been produced by Quintavalla (2006).
Quintavalla studied the effect of variation of the club head speed for the different balls used on the
US PGA TOUR. Quintavalla found that 1 mph higher club head speed Vc results in 3.05 yards
(2.78 m) more carry/total for a Vc of 90 mph (40.2 m/s) and 2.3 yards (2.1 m) for a Vc of 120 mph
(53.6 m/s). The diminishing effect at higher club head speeds is caused by a slowly decreasing
COR for higher club head speeds. However, Quintavalla did not study the impact on launch data
and distance by changing the attack angle.
2
The Golf Shot Chain
It is convenient to divide up the golf shot in different stages. See the “Golf Shot Chain” (Figure 1).
Club and Ball collision
The collision between the club and the ball has been studied theoretically by Jorgensen (1994),
who introduced the D plane. Jorgensen taught that the initial direction of the ball Vb always will be
in the plane suspended by the club direction Vc and club orientation N, and that the spin axis is
oriented in the direction of the vector cross product Vc x N. The later is not true for off center hits
– in this case the spin vector is influenced by the gear effect spin vector.
Ball Flight Trajectory
Multiple people have studied the details that influence the ball flight trajectory. See Beasley (2002)
and Quintavalla (2002). However, all this work has been oriented toward providing a theoretical
model for estimating ball trajectories.
With the commercially available TrackMan™ system that accurately measures real time
trajectory and carry for every shot, the knowledge about which combinations of ball launch data
provide the longest carry distances has been widely spread.
Figure 1 The “Golf Shot Chain” is divided into 4 processes: (I) the swing, (II) the club/ball collision, (III) the ball flight
trajectory, and (IV) the bounce and roll of the ball. After each process is a “result”: the club delivery, the ball launch, the
carry landing and the final/total position of the ball
Figure 2 The club/ball collision: The COG of the club head moves at impact in the direction Vc, the vertical angle of Vc
relative to the horizon is the club head attack angle (AA). The club face orientation N is perpendicular to the club face at
the point of impact with the ball, the vertical angle relative to the horizon is the dynamic loft DL. The initial direction of
the ball Vb is inclined relative to the horizon by the vertical launch angle VLA.
3
Bounce and Roll
The details regarding bounce and roll will not be discussed in this paper. However, it is important
to note that turf conditions must be considered in order to maximize total distance.
METHODS
Two sets of experiments were carried out using a golf robot. In both cases, a TrackMan™ Pro
system was used to measure the club delivery, the ball launch, and the entire trajectory.
In Test 1, the attack angle was varied by having the tee position back, neutral, and forward.
When moving the tee, caution was taken to align the tee for a center hit. Two different drivers were
tested: a conventionally shaped driver (Driver 1) and a square shaped driver (Driver 2). To
indicate if there could be any ball specific issues, two different balls were used for each setup:
Titleist ProV1 (Ball A) and the R&A/USGA calibration ball (Ball B). Also, multiple vertical swing
planes vSP were tested (49, 54 and 59 degrees). The sample size was six shots for each setup.
Throughout this test, the club head speed was 48.8 m/s, which is slightly below the US PGA
TOUR average of 50.3 m/s (112.5 mph).
In Test 2, a driver with 3 different lofts (8, 10 and 13 degrees) was tested. Except for the loft
variable, all other club specifications were identical. Only ball B was used for this test. Throughout
this test, the club head speed was constant at 51.6 m/s (115.4 mph) and the attack angle was -3.0
deg. This attack angle setting is comparable to the 2007 season average from some of the world’s
top players, including Tiger Woods and Phil Mickelson (who both have a slightly higher club head
speeds than the test speed).
TrackMan™ Pro
The TrackMan™ Pro system is a Doppler radar based system that is placed 2-4 meters behind the
golfer and measures 3-dimensionally the club head position vector Xc(t) and velocity vector Vc(t)
during the bottom of the forward swing arc. TrackMan ™ also measures the entire ball flight
trajectory unit landing (position vector Xb(t) ) and velocity vector Vb(t). In addition, TrackMan™
measures the total spin rate ω(t) of the ball, independent of spin axis orientation.
From these fundamental measurements, several important measurement parameters can be
computed, including club head speed Vc, attack angle AA, and club path CP, which are determined
from the club head velocity vector Vc(t) at the point in time representing first contact with the ball.
Similarly, the initial ball speed Vb , vertical launch angle VLA, and the horizontal launch angle
HLA are derived from the ball velocity vector Vb(t) at the point in time when the ball leaves the
clubface. Also the launch spin rate ω is computed at point in time when the ball leaves the
clubface.
4
68
69
70
71
72
73
74
75
76
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
B
a
ll
S
p
ee
d
[m
/s
]
Ball A
Ball B
y = 0.75x + 8.61
0
2
4
6
8
10
12
14
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
V
e
r
t
ic
a
l
L
au
n
ch
A
ng
le
[d
eg
]
Ball A
Ball B
30
35
40
45
50
55
60
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
S
p
in
R
a
te
[
rp
s
]
Ball A
Ball B
170
180
190
200
210
220
230
240
250
260
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
C
a
r
ry
D
is
ta
n
ce
[m
]
Ball A
Ball B
RESULTS
Results for the attack angle test (Test 1) for Driver 1 and vertical swing plane of 49 degrees are
shown below in figure 3-6.
With attack angle the only variable and everything else regarding the club delivery constant, it is
expected that VLA to be the only ball launch parameter among the primary three launch data (Vb,
VLA and ω), to change significantly. This is because the angle between club direction Vc and club
orientation N remains unchanged. Consequently, it is expected that VLA would follow 1:1 with
changes in the attack angle.
Figure 3 and 5 shows indeed that Vb and ω are independent of the AA, and figure 4 shows that
VLA increases linearly with AA – but with a ratio less than 1:1.
When looking at the results for all the combinations tested, see Table 1, in the big picture it is
confirmed that Vb and ω are independent AA, and that VLA increases linearly with AA. When
studing the results more detailed there are deviations from the general picture. These deviations
are believed to be a result of non-ideal test situation. It should be realized that it is difficult to
conducts these experiments insuring that there will be no (or constant) influence of gear-effect on
Figure 3 - Ball Speed vs. Attack Angle
(Test 1, Dr. 1/49)
Figure 4 - Vertical Launch Angle vs. Attack Angle
(Test 1, Dr. 1/49)
Figure 5 - Spin Rate vs. Attack Angle
(Test 1, Dr. 1/49)
Figure 6 - Carry Distance vs. Attack Angle
(Test 1, Dr. 1/49)
5
6 7 8 9 10 11 12 13 14
Club Loft [deg]
Ball Speed
Vertical Launch Angle
Spin Rate
Carry Distance
80
79
78
77
76
75
74
73
72
71
70
B
a
l
l S
p
e
e
d
[m
/s
]
17
16
15
14
13
12
11
10
9
8
7
V
e
r
t
ic
a
l L
a
u
n
ch
A
ng
le
[d
eg
]
90
85
80
75
70
65
60
55
50
45
40
S
p
in
R
a
te
[r
p
s
]
270
265
260
255
250
245
240
235
230
225
220
C
a
rr
y
D
is
ta
n
ce
[m
]
spin rate and launch angle for the different attack angles. This is very difficult to insure even for
robot testing.
Table 1 - Average variations on Vb, VLA, ω, and carry distance when changing AA. The clubhead speed was 48.8 m/s.
∆AA ∆Vb ∆VLA ∆ω ∆Carry
49deg vSP, Driver 1, ball A 8.6 deg 0.0 m/s 6.5 deg -1.4 rps 40 m
49deg vSP, Driver 1, ball B 8.2 deg -0.1 m/s 6.3 deg -2.5 rps 47 m
54deg vSP, Driver 1, ball A 10.1 deg 0.2 m/s 7.3 deg -1.7 rps 39 m
54deg vSP, Driver 1, ball B 10.2 deg 0.4 m/s 7.4 deg -3.3 rps 43 m
49deg vSP, Driver 2, ball A 6.8 deg 0.6 m/s 6.5 deg -10.1 rps 14 m
49deg vSP, Driver 2, ball B 6.8 deg 0.5 m/s 6.6 deg -13.1 rps 16 m
54deg vSP, Driver 2, ball A 8.5 deg 1.3 m/s 6.3 deg -0.4 rps 15 m
54deg vSP, Driver 2, ball B 7.9 deg 1.8 m/s 6.2 deg 0.1 rps 16 m
59deg vSP, Driver 2, ball A 8.5 deg 1.8 m/s 7.4 deg -0.1 rps 19 m
59deg vSP, Driver 2, ball B 8.3 deg 2.3 m/s 7.8 deg 2.2 rps 21 m
Figure 6 demonstrates that the carry distance increases with an increase in attack angle, clearly
illustrating the advantage of a positive attack angle for driving distance optimization. While ball A
and B had identical launch conditions (Vb, VLA and ω), the carry distance is longer with ball A
than ball B. This illustrates the difference in aerodynamic properties of the two balls.
For the club loft test (Test 2), the results are shown in Figure 7.
Figure 7 Results from TrackMan™ for Test 2, where club loft was the only variable. This was accomplished by using
different lofted clubs of the exact same type (head design, shaft specifications, etc.).
6
From Figure 7, the following is observed:
• When increasing the club loft, a small reduction in ball speed and a more dramatic increase
in spin rate are observed.
• VLA increases very close to 1:1 with increasing club loft.
• The longest carry is obtained with a 10 deg lofted driver, which offered the best comprise
between high vertical launch angle and low spin rate for this club speed and attack angle
combination.
DISCUSSION AND APPLICATION
For a golfer, it is very difficult change one’s own club head speed. Also, the attack angle is
difficult for most golfers to change without intervention, such as instruction. So, the easiest thing
the golfer can do is change the club!
Table 2 - A generalization of the sensitivity of the primary launch conditions (Vb, VLA and ω) are listed as function of Vc,
AA and club loft. The data are averages from the experiments described in this paper and added with data from
Quintavalla (2006) regarding the club speed variation
Ball Speed, Vb Vertical Launch Angle, VLA Spin Rate, ω
Club Speed, Vc +1m/s 1.4 m/s -0.04 deg 0.69 rps
Attack Angle, AA +1deg 0.1 m/s 0.82 deg -0.42 rps
Club Loft +1 deg -0.25 m/s 0.68 deg 2.8 rps
From Table 2, some clear conclusions can be made. The club head speed and attack angle of a
golfer will offset the achievable combinations of his/her vertical launch angle and spin rate.
This means that depending on the club head speed and the attack angle of a golfer, the set of
optimal ball launch data for this golfer are dictated to a high degree. This is further illustrated by
Table 3 and Figure 8.
Table 3 - TrackMan™ recommended optimal Vb, VLA, ω and DL, for maximum carry across different combinations of
clubhead speed and attack angles. The numbers are calculated assuming sea level conditions, no wind, and a tour type pro
ball. This information is part of the TrackMan™ Live release 3.1 documentation.
Vc AA DLopt. Vb opt. VLA opt. ωopt. Carry opt.
40.2 m/s -5 deg 13.9 deg 57.7 m/s 11.1 deg 61.5 rps 175 m
40.2 m/s 5 deg 18.4 deg 59.0 m/s 16.4 deg 43.8 rps 196 m
53.6 m/s -5 deg 8.1 deg 78.7 m/s 6.1 deg 57.1 rps 257 m
53.6 m/s 5 deg 13.9 deg 80.0 m/s 12.6 deg 39.0 rps 284 m
In summary, to maximize the carry distance for a golfer, ensure the following:
1. Center impact with as positive attack angle as possible
2. Align club path and face angle (this means also that the spin axis will be horizontal)
3. Balance the vertical launch angle and spin rate carefully via club adjustments in order to
achieve the optimum dynamic loft for a given club head speed and attack angle.
7
Proper club fitting and instruction must address all 3 (above).
Figure 8 TrackMan™ carry distance chart for clubhead speed of 53.6 m/s. Depending on the attack angle of the golfer,
the optimal ball launch data and potential carry distance changes dramatically. The chart is taken from the documentation
delivered with TrackMan™ Live release 3.1, but converted into SI units
REFERENCES
Jorgensen T.P. (1999). The physics of golf, (Springer-Verlag, New York).
Beasley, D. and Camp, T., 2002, Effects of Dimple Design on the Aerodynamic Performance of a
Golf Ball, In Science and Golf IV: Proceedings of the World Scientific Congress of Golf,
pp.328-340.
Quintavalla, S.J., 2002, A Generally Applicable Model for the Aerodynamic Behavior of Golf
Balls, In Science and Golf IV: Proceedings of the World Scientific Congress of Golf,
pp.341-348.
Quintavalla, S.J., 2006, Experimental Determination of the Effects of Clubhead Speed on Driver
Launch Conditions and the Effects on Drive Distance, USGA Technical Report
RB/cor2006-01.
1
Optimization of Driving Distance –
Importance of Determining the Attack Angle
Fredrik Tuxen, ISG A/S, Vedbaek, DENMARK
ABSTRACT
The trajectory of a golf ball is determined by the following ball launch data: 1) Ball Speed, 2)
Vertical and horizontal launch angle, and 3) Spin rate and spin axis orientation. From this point,
the trajectory will evolve in a deterministic way influenced only by the golf ball aerodynamic
properties and the environmental properties - wind speed and direction, temperature, pressure, and
humidity. Many different combinations of golf swings and club builds can produce the same
launch data, but for only a few of these combinations are the obtained launch data optimal. While
there is consensus in the golf community that optimal driving distance is obtained from a straight
shot hit squarely in the center of the club face, there are, however, many variations in the
recommendations for optimal vertical launch angle and spin rate for a given club head speed. By
using the commercially available ball flight monitor TrackMan™ that measures the club delivery,
ball launch data, and ball flight trajectory, the parameters to obtain optimum carry can be studied.
As it turns out, the attack angle and speed of the club head at impact are essential in determining
the optimal ball launch data for a given golfer in both club fitting and instruction.
KEYWORDS
Club fitting, driver optimization, attack angle, TrackMan™.
INTRODUCTION
Regarding optimization of driving distance, some work has been produced by Quintavalla (2006).
Quintavalla studied the effect of variation of the club head speed for the different balls used on the
US PGA TOUR. Quintavalla found that 1 mph higher club head speed Vc results in 3.05 yards
(2.78 m) more carry/total for a Vc of 90 mph (40.2 m/s) and 2.3 yards (2.1 m) for a Vc of 120 mph
(53.6 m/s). The diminishing effect at higher club head speeds is caused by a slowly decreasing
COR for higher club head speeds. However, Quintavalla did not study the impact on launch data
and distance by changing the attack angle.
2
The Golf Shot Chain
It is convenient to divide up the golf shot in different stages. See the “Golf Shot Chain” (Figure 1).
Club and Ball collision
The collision between the club and the ball has been studied theoretically by Jorgensen (1994),
who introduced the D plane. Jorgensen taught that the initial direction of the ball Vb always will be
in the plane suspended by the club direction Vc and club orientation N, and that the spin axis is
oriented in the direction of the vector cross product Vc x N. The later is not true for off center hits
– in this case the spin vector is influenced by the gear effect spin vector.
Ball Flight Trajectory
Multiple people have studied the details that influence the ball flight trajectory. See Beasley (2002)
and Quintavalla (2002). However, all this work has been oriented toward providing a theoretical
model for estimating ball trajectories.
With the commercially available TrackMan™ system that accurately measures real time
trajectory and carry for every shot, the knowledge about which combinations of ball launch data
provide the longest carry distances has been widely spread.
Figure 1 The “Golf Shot Chain” is divided into 4 processes: (I) the swing, (II) the club/ball collision, (III) the ball flight
trajectory, and (IV) the bounce and roll of the ball. After each process is a “result”: the club delivery, the ball launch, the
carry landing and the final/total position of the ball
Figure 2 The club/ball collision: The COG of the club head moves at impact in the direction Vc, the vertical angle of Vc
relative to the horizon is the club head attack angle (AA). The club face orientation N is perpendicular to the club face at
the point of impact with the ball, the vertical angle relative to the horizon is the dynamic loft DL. The initial direction of
the ball Vb is inclined relative to the horizon by the vertical launch angle VLA.
3
Bounce and Roll
The details regarding bounce and roll will not be discussed in this paper. However, it is important
to note that turf conditions must be considered in order to maximize total distance.
METHODS
Two sets of experiments were carried out using a golf robot. In both cases, a TrackMan™ Pro
system was used to measure the club delivery, the ball launch, and the entire trajectory.
In Test 1, the attack angle was varied by having the tee position back, neutral, and forward.
When moving the tee, caution was taken to align the tee for a center hit. Two different drivers were
tested: a conventionally shaped driver (Driver 1) and a square shaped driver (Driver 2). To
indicate if there could be any ball specific issues, two different balls were used for each setup:
Titleist ProV1 (Ball A) and the R&A/USGA calibration ball (Ball B). Also, multiple vertical swing
planes vSP were tested (49, 54 and 59 degrees). The sample size was six shots for each setup.
Throughout this test, the club head speed was 48.8 m/s, which is slightly below the US PGA
TOUR average of 50.3 m/s (112.5 mph).
In Test 2, a driver with 3 different lofts (8, 10 and 13 degrees) was tested. Except for the loft
variable, all other club specifications were identical. Only ball B was used for this test. Throughout
this test, the club head speed was constant at 51.6 m/s (115.4 mph) and the attack angle was -3.0
deg. This attack angle setting is comparable to the 2007 season average from some of the world’s
top players, including Tiger Woods and Phil Mickelson (who both have a slightly higher club head
speeds than the test speed).
TrackMan™ Pro
The TrackMan™ Pro system is a Doppler radar based system that is placed 2-4 meters behind the
golfer and measures 3-dimensionally the club head position vector Xc(t) and velocity vector Vc(t)
during the bottom of the forward swing arc. TrackMan ™ also measures the entire ball flight
trajectory unit landing (position vector Xb(t) ) and velocity vector Vb(t). In addition, TrackMan™
measures the total spin rate ω(t) of the ball, independent of spin axis orientation.
From these fundamental measurements, several important measurement parameters can be
computed, including club head speed Vc, attack angle AA, and club path CP, which are determined
from the club head velocity vector Vc(t) at the point in time representing first contact with the ball.
Similarly, the initial ball speed Vb , vertical launch angle VLA, and the horizontal launch angle
HLA are derived from the ball velocity vector Vb(t) at the point in time when the ball leaves the
clubface. Also the launch spin rate ω is computed at point in time when the ball leaves the
clubface.
4
68
69
70
71
72
73
74
75
76
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
B
a
ll
S
p
ee
d
[m
/s
]
Ball A
Ball B
y = 0.75x + 8.61
0
2
4
6
8
10
12
14
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
V
e
r
t
ic
a
l
L
au
n
ch
A
ng
le
[d
eg
]
Ball A
Ball B
30
35
40
45
50
55
60
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
S
p
in
R
a
te
[
rp
s
]
Ball A
Ball B
170
180
190
200
210
220
230
240
250
260
-10 -8 -6 -4 -2 0 2 4
Attack Angle [deg]
C
a
r
ry
D
is
ta
n
ce
[m
]
Ball A
Ball B
RESULTS
Results for the attack angle test (Test 1) for Driver 1 and vertical swing plane of 49 degrees are
shown below in figure 3-6.
With attack angle the only variable and everything else regarding the club delivery constant, it is
expected that VLA to be the only ball launch parameter among the primary three launch data (Vb,
VLA and ω), to change significantly. This is because the angle between club direction Vc and club
orientation N remains unchanged. Consequently, it is expected that VLA would follow 1:1 with
changes in the attack angle.
Figure 3 and 5 shows indeed that Vb and ω are independent of the AA, and figure 4 shows that
VLA increases linearly with AA – but with a ratio less than 1:1.
When looking at the results for all the combinations tested, see Table 1, in the big picture it is
confirmed that Vb and ω are independent AA, and that VLA increases linearly with AA. When
studing the results more detailed there are deviations from the general picture. These deviations
are believed to be a result of non-ideal test situation. It should be realized that it is difficult to
conducts these experiments insuring that there will be no (or constant) influence of gear-effect on
Figure 3 - Ball Speed vs. Attack Angle
(Test 1, Dr. 1/49)
Figure 4 - Vertical Launch Angle vs. Attack Angle
(Test 1, Dr. 1/49)
Figure 5 - Spin Rate vs. Attack Angle
(Test 1, Dr. 1/49)
Figure 6 - Carry Distance vs. Attack Angle
(Test 1, Dr. 1/49)
5
6 7 8 9 10 11 12 13 14
Club Loft [deg]
Ball Speed
Vertical Launch Angle
Spin Rate
Carry Distance
80
79
78
77
76
75
74
73
72
71
70
B
a
l
l S
p
e
e
d
[m
/s
]
17
16
15
14
13
12
11
10
9
8
7
V
e
r
t
ic
a
l L
a
u
n
ch
A
ng
le
[d
eg
]
90
85
80
75
70
65
60
55
50
45
40
S
p
in
R
a
te
[r
p
s
]
270
265
260
255
250
245
240
235
230
225
220
C
a
rr
y
D
is
ta
n
ce
[m
]
spin rate and launch angle for the different attack angles. This is very difficult to insure even for
robot testing.
Table 1 - Average variations on Vb, VLA, ω, and carry distance when changing AA. The clubhead speed was 48.8 m/s.
∆AA ∆Vb ∆VLA ∆ω ∆Carry
49deg vSP, Driver 1, ball A 8.6 deg 0.0 m/s 6.5 deg -1.4 rps 40 m
49deg vSP, Driver 1, ball B 8.2 deg -0.1 m/s 6.3 deg -2.5 rps 47 m
54deg vSP, Driver 1, ball A 10.1 deg 0.2 m/s 7.3 deg -1.7 rps 39 m
54deg vSP, Driver 1, ball B 10.2 deg 0.4 m/s 7.4 deg -3.3 rps 43 m
49deg vSP, Driver 2, ball A 6.8 deg 0.6 m/s 6.5 deg -10.1 rps 14 m
49deg vSP, Driver 2, ball B 6.8 deg 0.5 m/s 6.6 deg -13.1 rps 16 m
54deg vSP, Driver 2, ball A 8.5 deg 1.3 m/s 6.3 deg -0.4 rps 15 m
54deg vSP, Driver 2, ball B 7.9 deg 1.8 m/s 6.2 deg 0.1 rps 16 m
59deg vSP, Driver 2, ball A 8.5 deg 1.8 m/s 7.4 deg -0.1 rps 19 m
59deg vSP, Driver 2, ball B 8.3 deg 2.3 m/s 7.8 deg 2.2 rps 21 m
Figure 6 demonstrates that the carry distance increases with an increase in attack angle, clearly
illustrating the advantage of a positive attack angle for driving distance optimization. While ball A
and B had identical launch conditions (Vb, VLA and ω), the carry distance is longer with ball A
than ball B. This illustrates the difference in aerodynamic properties of the two balls.
For the club loft test (Test 2), the results are shown in Figure 7.
Figure 7 Results from TrackMan™ for Test 2, where club loft was the only variable. This was accomplished by using
different lofted clubs of the exact same type (head design, shaft specifications, etc.).
6
From Figure 7, the following is observed:
• When increasing the club loft, a small reduction in ball speed and a more dramatic increase
in spin rate are observed.
• VLA increases very close to 1:1 with increasing club loft.
• The longest carry is obtained with a 10 deg lofted driver, which offered the best comprise
between high vertical launch angle and low spin rate for this club speed and attack angle
combination.
DISCUSSION AND APPLICATION
For a golfer, it is very difficult change one’s own club head speed. Also, the attack angle is
difficult for most golfers to change without intervention, such as instruction. So, the easiest thing
the golfer can do is change the club!
Table 2 - A generalization of the sensitivity of the primary launch conditions (Vb, VLA and ω) are listed as function of Vc,
AA and club loft. The data are averages from the experiments described in this paper and added with data from
Quintavalla (2006) regarding the club speed variation
Ball Speed, Vb Vertical Launch Angle, VLA Spin Rate, ω
Club Speed, Vc +1m/s 1.4 m/s -0.04 deg 0.69 rps
Attack Angle, AA +1deg 0.1 m/s 0.82 deg -0.42 rps
Club Loft +1 deg -0.25 m/s 0.68 deg 2.8 rps
From Table 2, some clear conclusions can be made. The club head speed and attack angle of a
golfer will offset the achievable combinations of his/her vertical launch angle and spin rate.
This means that depending on the club head speed and the attack angle of a golfer, the set of
optimal ball launch data for this golfer are dictated to a high degree. This is further illustrated by
Table 3 and Figure 8.
Table 3 - TrackMan™ recommended optimal Vb, VLA, ω and DL, for maximum carry across different combinations of
clubhead speed and attack angles. The numbers are calculated assuming sea level conditions, no wind, and a tour type pro
ball. This information is part of the TrackMan™ Live release 3.1 documentation.
Vc AA DLopt. Vb opt. VLA opt. ωopt. Carry opt.
40.2 m/s -5 deg 13.9 deg 57.7 m/s 11.1 deg 61.5 rps 175 m
40.2 m/s 5 deg 18.4 deg 59.0 m/s 16.4 deg 43.8 rps 196 m
53.6 m/s -5 deg 8.1 deg 78.7 m/s 6.1 deg 57.1 rps 257 m
53.6 m/s 5 deg 13.9 deg 80.0 m/s 12.6 deg 39.0 rps 284 m
In summary, to maximize the carry distance for a golfer, ensure the following:
1. Center impact with as positive attack angle as possible
2. Align club path and face angle (this means also that the spin axis will be horizontal)
3. Balance the vertical launch angle and spin rate carefully via club adjustments in order to
achieve the optimum dynamic loft for a given club head speed and attack angle.
7
Proper club fitting and instruction must address all 3 (above).
Figure 8 TrackMan™ carry distance chart for clubhead speed of 53.6 m/s. Depending on the attack angle of the golfer,
the optimal ball launch data and potential carry distance changes dramatically. The chart is taken from the documentation
delivered with TrackMan™ Live release 3.1, but converted into SI units
REFERENCES
Jorgensen T.P. (1999). The physics of golf, (Springer-Verlag, New York).
Beasley, D. and Camp, T., 2002, Effects of Dimple Design on the Aerodynamic Performance of a
Golf Ball, In Science and Golf IV: Proceedings of the World Scientific Congress of Golf,
pp.328-340.
Quintavalla, S.J., 2002, A Generally Applicable Model for the Aerodynamic Behavior of Golf
Balls, In Science and Golf IV: Proceedings of the World Scientific Congress of Golf,
pp.341-348.
Quintavalla, S.J., 2006, Experimental Determination of the Effects of Clubhead Speed on Driver
Launch Conditions and the Effects on Drive Distance, USGA Technical Report
RB/cor2006-01.
....
