Why do tall golfers hit it farther?
Dave Tutelman
 June 5, 2015
"Help
me scientifically understand  because the stats are there  why 15
of the
top 20 players on tour are 6'0"+ and are the longest drivers on tour?
How does height seem to equal longer distance with these facts?"
So
asked Jonathan David Muccio on June 1, 2015, in the Golf Sports
Science
discussion group on Facebook. A long and interesting discussion
followed. Some was very traditional (levers, longer arc) and
some
was very scientific (torque, mathematical models). As you might expect,
I come down on the scientific side. Here is my take on the question.
First... Is it true?
Do tall golfers actually hit it farther?
As the discussion got under way, Liam
Friedman pointed out, "This is only significant if less than 3 of 4 PGA
tour players is over 6 feet tall." Spot
on! If the 25^{th}
percentile of height on the PGA Tour is six feet or more, you would
expect  just from chance  that 15 of the top 20 players would
be six feet or taller. Same for 15 of the longest 20 drivers. That
would just be an accident of the numbers.
So what is the truth
here? Nobody wanted to pick up the question, so I looked around the
Internet and found confirmation that this is indeed an interesting
question. Interesting in two ways: it is not a statistical anomaly, and
there is
indeed a statistical trend that taller golfers hit it farther.
Not
a statistical anomaly
I
checked by looking up PGA Tour player height statistics,
and importing them into Excel. The stats available were from a much
larger set than the fullyexempt PGA Tour. With the numbers there, it
must have included anybody with any standing at all in the "big show",
the senior tour, and the Web.com tour.
But it is the data available, and I used it.
They had a huge number of players there, a page for each letter of the
alphabet. I didn't want to deal with that, so I made the blatant
assumption that there was no bias based on initial of last name. Then I
loaded pages until I had over 400 names. Not all the data was available
for
all golfers; the 444 Excel rows I had contained heights for 326 of the
golfers. I sorted those 326 by height and looked for the median and the
quartile
points. Here's what the data showed:
Percentile 
Height 
Comment 
25 
5' 10" 
closer to 5'9" than 5'11" 
50 (median) 
6' 0" 
closer to 5'11" than 6'1" 
75 
6' 1" 

So the 25^{th} percentile would appear to be
between 5'9" and 5'10",
definitely below 6 feet. Jonathan's question of "why?" is therefore
interesting, not just what we should expect statistically.
There
is indeed a trend
Continuing
my Internet research, I found a page on Golf Magazine's site
that had an interactive graph plotting driving distance vs height
for PGA Tour players during the 2013'14 Tour season. Here is
a
noninteractive picture of it. If you go to the site itself, you can
identify each dot with a player.
The
article says the slanted red line is the average driving distance for
each height.
I doubt it; I'm sure it is a bestfit straight line for driving
distance vs height. But
this is even better than
just averages!
The slope of the straight line shows how much an inch of height is
worth in driving distance. Some quick measurement shows an inch of
height
is worth
about 1.3 yards of driving distance.
But...
There is a lot of scatter in the data. In statistical terms, the
"Rsquared" of the bestfit line is way below 1.0. In fact, you'd be
hardpressed to draw the line by eyeball; I certainly couldn't. What
this says is: Height matters, but other factors matter a lot more.
Having verified that the question is interesting and meaningful, let's
get back to the original question, which was...
Why???
As
I said, the speculation in the discussion group ranged from traditional
buzzwords to mechanical analyses. As one of the contributors of an
analysis, let me present it here and argue for it.
I
used a mathematical model of the swing, and changed parameters one by one to reflect
differences that height might mean. The model was the
double pendulum of
Jorgensen. (There was some discussion of that model choice. I defend it
below, but for now let's just accept it.) I have a computer program that
simulates the double pendulum model (SwingPerfect
by Max Dupilka) that is quite
adjustable in terms of the parameters you
can set.
I
set up for a "nominal" golfer 5'10" tall, then varied things one at a
time to
compare the 5'10" golfer with a 6'2" golfer. Some details for the
nominal
golfer:
 24"
arm on a 5'10", 176 pound golfer. Shoulder
torque during downswing
is a constant 58 footpounds; wrist torque is zero; we're
freewheeling.
 45"
driver of typical shaft and head weights.
 Clubhead
speed is 109mph. (That just worked out, given the rest of the
parameters.)
We
will grow this golfer from 5'10" to 6'2". That's a 5.7% increase
linearly. We will grow one thing at a time, and see what happens 
what kind of difference each change makes.
Longer
arms
Let's
start out by checking the most traditional of the explanations, longer
arms, which is usually expressed as "increased leverage" or
"bigger arc". We'll lengthen the arms 5.7%, in proportion to the taller
golfer. The arms are now 25.4" long. The
clubhead speed drops to 107mph, a 2mph loss. So much for that theory!
We'll look more closely in the discussion below at why
"leverage"
and swing arc don't seem to help; in fact, they hurt a little bit.
Summary:
longer arms reduce clubhead speed 2mph.

Longer
club
Geometry
says that the taller golfer should be able to handle a
proportionally longer club. (I know, I know! I do clubfitting too, and
it
often
doesn't work that way. But, averaged over all possibilities, it does.)
We also know that a
longer driver should give a higher clubhead speed. So, if we
can control the longer club well enough to give good impact, we should
get more distance.
I tried scaling up the club length proportionally to the height. Our
5.7% longer driver, barely legal at 47.6", gives back exactly
the 2mph we lost with longer arms. That's
sort of interesting. It says scaling the linear dimensions of the
golfer and the club leaves clubhead speed just about the same. Sounds
counterintuitive, but I'm pretty sure it has to do with the fact that
we have the same torque driving a longer arm lever, so less force at
the hands is
applied to accelerating the club. You need to increase club length, or
the lower handpull force will give decidedly lower clubhead speed.
Summary:
a longer club gives back the 2mph we lost from longer arms.

Golfer
weight
A
taller golfer in the same proportions  with the same "build"  as
the shorter golfer will weigh more. Let's account for that.
Weight should scale with volume, if the golfers have
roughly similar builds  and volume scales as the cube of length. We
increased the length 5.7%. So we must
cube 1.057; we get 1.181, or a volume increase of 18.1%. The 176pound
golfer is now a 208pound golfer. Do this
and
the clubhead speed drops off again. Not all the way down to 107mph, but
most of the
way (107.4).
Why?
Changing
the weight increased the golfer's moment of inertia (MOI). It now takes more torque to
rotate the torso and the extended arms. That's why the clubhead speed
drops.
Summary:
increasing the golfer's weight drops clubhead speed 1.5mph, almost as
much as arm length increase did.

Shoulder
torque
Merely
changing the golfer's weight does not convince the
computer to increase the torque applied. I checked, and changing the
weight had no effect on torque. But
it should! More muscle
mass should mean more muscular force. All other things being equal,
force should go up with the
crosssectional area of the muscle, which would be proportional to the
square of the
height.
Not only that, but remember that the torque produced at any joint is
the muscle strength times the moment arm from the joint's pivot to the
muscle attachment. The larger golfer probably has
proportionally larger joints. The distance from muscle attachment to
pivot is longer. So the torque (force times lever arm) probably scales
with
the cube of the height. Why cubed? It is height squared for muscle
strength, times height again for lever arm.
For
the reasons above, I increased the torque by 18.1% to 68
footpounds. That did it! The clubhead speed went up to 115mph.
Summary:
increasing the torque to reflect the golfers increased size gives a
marked advantage in clubhead speed. This is the only place in the whole
study where the tall golfer's clubhead speed actually increased,
compared with a shorter golfer.

Shoulder torque revisited
Mike Duffey
pointed out to me, "Muscle strength scales to body weght reasonably
well across an athletic population, but to the 2/3 power. I don't have
access to how weight scales to height right away (especially in
golfers), but that information might help adjust the predicted increase
in force output. Here is a good read on the topic in
athletes."
Let's
incorporate this new information. We will back off from a cube scaling
on height to a 2/3power scaling on weight. In the absence of
information on how weight scales with height statistically on
the PGA Tour, we are left with scaling geometrically,
as we did explicitly with the golfer's weight above. So scaling to the
2/3 power of weight would be the same as scaling to the square of
height.
I did that. The 5.7% height scaling becomes an 11.7%
strength scaling, or a shoulder torque of 65 footpounds. Plugging this
into the computer model gives a clubhead speed of 113mph. Not as much
as before, but still markedly more than we got from arm length, club
length, and weight put together. We will use this number in the
discussion that follows. And we will amend it as new information
becomes available  like the statistical scaling of weight to height
for golfers.
I even checked to see what a 1/3 power scaling for strength would do.
(That is, strength proportional to height rather than body weight.)
Even that gives a gain of clubhead speed, albeit a small one, to
110mph.

My
conclusions
The
reason taller golfers hit it farther is mostly because
their size provides a frame for more muscle and larger lever
arms at the joints. The rest of it  larger arc, more massive body  reduces
clubhead speed, and a proportionally longer club doesn't get enough back to
make up for it. But the bigger golfer is also stronger (or at least has
the potential to be stronger, given proper conditioning and nutrition).
If he/she realizes that potential, the result is higher clubhead speed.
Now
don't go throwing Jamie Sadlowski at me... nor, for that matter, Nick
Faldo. Technique matters, of course. But the question was asked in
terms of trends, statistical norms. What does height do, all other things being
equal? I'm answering in that spirit.
More
discussion
Facebook discussions have an interesting dynamic, even if they are in
technical discussion groups like this. If they draw enough interest and
comments, they often break into subthreads on related topics. I'd
like to comment on some of the subthreads that were still related
to the original question.
Longer arc, levers
Before we get into
this topic, we need to clarify the terminology. This is especially
important because the golf instruction world uses these terms as
buzzwords. We need to be very specific if we are going to draw correct
conclusions; vaguelydefined buzzwords will not do, not even if
"everyone knows what they mean."
A "longer arc" or "bigger arc"
refers to the arc traveled by the hands. It can occur two ways in the
golf swing, a bigger radius (longer arms) or a bigger angle (more
shoulder turn).
In
the diagram, the green arc is longer than the red arc in both cases. So
the term "bigger arc" is ambiguous. Let's resolve that ambiguity. When
discussing a longer arc in this article, we
mean a bigger radius!
There's a good reason for that. A scaledup taller golfer will have a
bigger radius because of longer arms. But there is no reason to expect
taller golfers to make a bigger turn. Some will, some won't, but it
won't have anything to do with their height.
Now that we are explicit about what it means, here are some quotes
purporting to answer the original question. Some are from
right at the beginning of the discussion, and others close to the end.
"Longer
arms, bigger arc!"
"Bigger
wheels go faster i.e wider arc"
"Let's
don't try to turn this into atomic science. A smaller man, like
Ben Hogan, has to have a longer swing arc and swing the club faster to
match a 6'6" golfer with similar technique and abilities and a much
larger swing arc. Small wheel vs large wheel. It can be done though."
"Levers,
levers, levers!!!!!"
"Length
of Arc, swing radius, leverage..."
... And the same thought, but with math instead of buzzwords:
"Tangential
velocity v=rω.
In this instance,
v
= club head velocity, r
= radius of
the lever from the point of rotation (arm and club), and ω
= rotational
velocity. Therefore if a taller golfer creates the same rotational
velocity ω,
they will hit the ball farther base as v
will increase."
Simply
not true! The last quote there is a good mathematical
justification for the argument, but it assumes the same angular
velocity
for shorter and longer arms. This usual golfinstruction
assumption about "leverage" is purely kinematic (just motion), and
ignores kinetics (the forces that produce the motion). In fact, a
longer arm loses angular velocity, unless you put in enough extra
driving torque to make up the lost rotation rate. I
have an article on
my site proving that this scenario
does
not give any increased speed.
Why doesn't
the speed
increase proportionally to the length? The article cited above
discusses the answer in detail. But here's the short form of the answer.
The
leverage argument has a tacit assumption that it takes no more effort
to accelerate a payload, a mass, on a long arm than on a short arm. But
we know
otherwise. The rotational equivalent of Newton's famous F=ma
is
τ = I α
...or
torque equals moment of inertia times angular acceleration. So we get
the same angular acceleration from the same torque only if the moment
of inertia does not increase.
But
moment of inertia is proportional to the square of the radius (at
least; it could be even more under many circumstances). Thus I
is increasing even faster than the radius. Unless we can find more
torque somewhere, "bigger wheels" or "bigger arc" will go no faster,
and perhaps slower. And our computer modeling said the answer is slower
for longer arms if the action is a golf swing. This means that, in
order to maintain angular velocity and increase clubhead speed, you
need a bigger torque τ.
So much for the longer lever theory.
Longer
club
Michael
Deiters asked, "Why
does a long club hit it farther than a shorter club?"
It
doesn't, necessarily. Yes, the clubhead speed is greater. The ball
speed
might not be greater if you can't control impact on the sweet spot with the
longer club. So the question might be better phrased, "Why does a long
club give more clubhead speed?" I'm not just splitting hairs; I have
data that this happens with some golfers  including myself, so I'm
personally sensitive to the problem.
I've
done a study that goes into gory detail on the question of longer driving clubs.
The short, simplified answer is this. If you can lighten the components
to keep the club MOI the same, then the same swing should give the same
angular velocities. And the same inplane angular velocity means that
the tangential component of clubhead speed (angular velocity times club
length) will obviously go up with club length. Since angular velocity
contributes about 80% of the total clubhead speed, that will boost the
total
clubhead speed.
Mike Duffey posted a response very similar to this.
But the longer club often has more speed even if we don't take care to
maintain the moment of inertia. The reason is most of
that clubhead speed is due to inertial release, not a torque driving
the club from the hands. The torque driving inertial release is
dependent on the length from pivot to balance point and the weight of
the club. So what? Well, those are the same things that increase the
MOI. As the club gets longer, both the MOI and the torque
grow. So, unlike a hubtorquedriven member, the club does not lose
much angular acceleration as it gets longer. True, the MOI grows faster
than the torque, but that is still a lot better than the situation with
the arms, which are driven by shoulder torque. So τ
= I α is a more valid argument for the club than
the arms.
Why so long?
Was the increase too
much? Consider... The revisited gain of 4mph of
clubhead speed will give us 12
more yards for the fourinch increase
in height,
assuming we don't lose
smash factor.
That is 3 yards per inch of height. The increase in
distance of the bestfit line from the Golf Magazine
article was only 1.3 yards per inch of height. We now have slightly
more than 2 times the gain the Tour statistics show.
Why
so much? I have a few guesses, which taken together might explain the
discrepancy. Most of my guesses involve the net torque increase not
being the full heightcubed increase.
 My modeling did not
account for the torquevelocity curve, a phenomenon wellknown to
biomechanics study. It says that you can't exert as much force (we are
interested in torque, but there is a similar forcevelocity curve) on
an object that is moving away from you. The faster it is moving away,
the lower the force you can exert. Consequence: A faster swing due to
faster body rotation reduces the torque that the body can apply; the
taller golfer is operating on a lower portion of the torquevelocity
curve.
 Another biomechanical effect is that torque does not
appear instantly; it builds up on an exponential basis. (Sasho
MacKenzie's models use a time constant of 20msec, about 7% of the
entire downswing.) My modeling does not have fine enough control of
torque vs time to reflect torque rampup. In reality the maximum torque
is not operating the whole time, but it is for the model. Why is this a
height issue? Because the modeled tall golfer's faster swing takes less
time  283msec vs 300 msec. So the rampup takes a greater percentage
of the tall golfer's swing than the shorter golfer's.
 A taller, heavier golfer may need more effort to
stay in
balance. Balance often implies opposing muscles exerting force (e.g.,
tricep as well as bicep in the example shown), thus reducing the
advantage obtained by the bigger muscles and longer lever arm.
There
may also be psychological or social effects at work. (For instance, the
taller golfer doesn't have to work as hard for an advantage, and so
doesn't work as hard).
Or perhaps my assumptions are incorrect
leading to the cube law for effective torque vs height. But a square
law or even a proportional law would be enough to give the taller
golfer a distance advantage.
The
double pendulum model
I
took a certain amount of flak for the model I used and
how I used it. Let me deal with those criticsms here. But
I'm going to try to stick to points that deal with the question we are
trying to answer,
not
just gratuitous criticisms of one model or another. Let's face it. No
model is exactly the golf swing. You want to choose the model to deal
with aspects of the swing that are relevant to the question being asked.
I am using
the double pendulum model, a 2dimensional model. It is old,
and has been
superseded in detail by 3D models. But there are a lot of questions
where the 3D models just add detail to the basically correct
information we learned from the double
pendulum. And sometimes it adds no new information; the 2D model is
accurate, as far as the particular question is concerned.
So let's look at the question we are asked: Why do taller golfers
tend to driver farther than shorter golfers? If the detail
added by
a
more complex model does
not distinguish taller from shorter golfers, then it does not matter to
this discussion.
For instance, Brian Manzella posted, "Wrist
torque is never zero." Let me respond...
I
have seen no model advocate a lot of inplane active wrist torque.
Those qualifiers  inplane
and active
 are important.
 The
2D double pendulum model considers only
inplane torque. If the 3D models show lots of beta and gamma torque
but not much
alpha torque, then they are not contradicting the 2D model. They are
just
adding
information that any 2D model would not address.
 There
are three sources of inplane wrist torque: (1) Active muscular
contraction. (2) Passive hardtissue interaction, mostly the limited
range of
motion for wrist cock. (3) Inertial torque applied by the club to the
wrist hinge, which is unopposed by either #1 or #2. I set #1 to zero,
and let #2 and #3 just happen; the model does account for them.
I
haven't seen any model of a good swing that has large amounts of wrist
torque that is both inplane and due to active muscle contraction. And
I will argue
that, for purposes of answering the question, "Why do tall players tend
to hit the ball farther?" other torques are not very interesting.
 Beta
torque will happen; it is needed to keep the club onplane, no matter
how tall the
golfer. Small variations in this will not affect inplane clubhead
speed.
 Gamma
torque will happen; it is needed to square the clubface at impact, no
matter how tall the golfer. Small variations in this will not affect
inplane clubhead speed.
So,
to a first approximation, I don't see a problem with the model I used.
I'll be glad to back off from that if shown a better, more detailed
model that tells a different story when
answering the question at hand.
So far, I haven't seen that, and I am reluctant to respond further
until I do.
Allometric
scaling and ground force moments
Let's
revisit  yet again  how strength scales with height. My first cut
used geometric scaling;
it assumed everything grows in proportion to size, and size grows
proportionally in all dimensions. Mike Duffey pointed out research on allometric scaling;
it quantifies experimentally how things like strength and weight grow
with the actual
size of the organism. (It is often applied to crossspecies
comparisons, involving size differences of more than an order of
magnitude. The question on
the table here involves a size difference of less than 10%.)
Why do I want
to examine this question yet a third time? Because in the group
discussion, Don Parsons asked a really good question:
"Since we are looking at tall
golfers, it seems your model
should include lower body as an option for developing speed. Longer
legs allow the taller golfer to create a larger moment
arm. How is the lower body accounted for in a double pedullum
model?
"I spent this
weekend in YoungHoo Kwon's biomechanics class.
One of the more interesting slides to me was this one. It seems that a
taller golfer has more capacity to shift COP away from COM and get
closer to M."
There is a simpler diagram in Dr. Kwon's talk than the one Don posted
make this point; I have adapted the simpler diagram here. Also, I'm not
sure Don got the terminology quite right; I believe that he was trying
to say, "It seems that a taller golfer has more capacity to shift force
F away from
COM and get a larger M."
And indeed it would seem so. If the positions in the swing scale up
with height, then the perpendicular distance d
would scale proportionally. So that would increase the turning moment M
proportionally to height, which I'm sure is what Don is saying.
With my original assumption about a cubelaw scaling, this just falls
out
in the wash. (We'll see why below.) But with any other scaling, it has
to be dealt with explicitly. So let's look again at the scaling, and
how to deal with Don's observation.
I went back to the references suggested by Mike Duffey, as well as
searching for other papers on the subject. Here is what I found:
 The reference scaling strength to weight (Jacobson, et al) concludes that
the scaling
is a 2/3 power relationship. This is strength in terms of force, not torque.
It still doesn't completely answer the question; we still need a
weight to height
scaling.
 The reference scaling weight to volume is not really
what I want for the question at
hand. I would still need to scale volume to height. The obvious scaling
based on geometry is a cube law.
And the paper (even when written way back in 1966) suggested that there
were more
direct answers around to the weight to height question. The obvious
solution, in the absence of measured data, is that weight scales to
height cubed.
 Looking over the web, there
are some measured studies around, almost all of which are unavailable
to me.
They are in journals that are archived behind paywalls. As a
nonprofessional hobbyist, I'm not about
to buy a subscription.
 There is an interesting
article on allometric scaling at Wikipedia.
Not a primary reference,
but at least I can access it. They point out that the challenges occur
if you are trying to scale across orders of magnitude. (E.g. mammals
have a mass range of more than
1000 times, or three orders of magnitude. That would be one order of
magnitude in height, assuming geometric scaling.) We are looking at a
problem
where the length range is less than 10%.
 There are many articles
around that relate muscle strength to its crosssectional area. (E.g. "Strength and crosssectional area of human
skeletal muscle" and
Crosssectional
area and muscular strength: a brief review)
They find different relations based on things like gender and
conditioning, but they seem to accept that within a sufficiently
uniform population strength should be proportional to cross sectional
area. It seems reasonable to assume that PGA Tour golfers constitute
such a population.
In the absence of documented studies that show otherwise, I'm
going to assume simple physical scaling within the population of
interest. The population is adult male athletes, specifically golfers.
I'm not including couch potatoes, nor sumo wrestlers, nor marathon
runners. I would expect the elite golfers to have similar training and
nutritional habits, at least similar enough to expect a reasonably
restricted range of body
compositions and proportions. So the physical scaling I assume is
a geometric one:
 Lever arms proportional to
height.
 Volume and mass proportional
to cube of height.
 Cross sectional muscle area
proportional to square of height.
Interestingly, this gives a 2/3 law for muscle strength vs weight.
 Strength →
crosssectional area → square of height.
 Weight → volume
→ cube of height.
So far, the strength we talk about is pure muscle force. Jacobson et al
reports that the 2/3 law also applies for output
force. So let's try to deal with ouput, and not just the
sheer force exerted by a muscle.
All the joints I can think of turn muscle tension into torque. But the
tasks in the Jacobson paper  and any task where the measurement is
force exerted by the body  requires turning this torque back into a
force. That requires a lever, a moment arm. It seems very reasonable to
assume this moment arm is proportional to height as well. (That is
certainly less of a stretch than was the cube law for weight.)
In fact, that is the explicit assumption underlying Don's
observation.
Let's see how this scaling affects output force.
 Muscle strength scales as
square of height.
 The moment arm of the joint
in question scales as the height.
 Therefore the torque at the
joint scales as the cube of height.
 The force exerted at the end
of the lever  say, force exerted by the foot due to torque at the
knee joint  is torque divided by lever length.
 Therefore the force exerted
at the end of the lever scales as the square of height. (That's a
cubelaw torque divided by a proportional lever length.)
 With my assumed scaling,
output force varies as 2/3 power of weight. (That's square of
height, with weight assumed the cube of height.)
All this is completely consistent with Jacobson, et al! (Though it
still depends on
the assumption that weight vs height scales geometrically.)
I don't see inconsistencies with any of the reference material I have
seen.
Let's continue with this line of thinking, and address Don's question
about whether a taller golfer can make better use of the ground forces.
The computer program for the swing model is driven by the torque that
rotates the left arm. It is a derivative of a lot of torques generated
in different places in the body. Originally, I was looking at torques
in the joints. Don pointed out that there are largerscale torques
involved, that have a major effect on clubhead speed  specifically
the ability of the golfer to generate torque (a moment M)
by moving the ground force F
away from the golfer's center of mass.
My original simulation assumed that the "shoulder torque" input to the
computer model scales as the cube of height. That is because it is an
accumulation of torques at the joints, each of which would be
proportional to the cube of height. (Forces follow a 2/3 law, torques a
cube law.) Let's extend this to the groundforce moment in Dr. Kwon's
diagram.
 The moment M
is the product of perpendicular distance d
times ground force F.
 It is reasonable to assume that d
grows in proportion to the golfer's height. That is, with a swing of
the same shape, a taller golfer generates a proportionally larger d.
 Forces vary as the 2/3 power of weight. With our
assumed geometric scaling, that would be the square of height.
So the moment M
scales as d
(proportional to height) time F
(proportional to height squared). In other words, M
is proportional to height cubed.
That is completely consistent with the scaling of torque at the joints.
So the ground force moment contributes to shoulder torque in the same
way as any individual joint torque would contribute. It is covered by
the generalization that shoulder torque grows as the cube of height,
all other factors being equal.
Don and Mike, thanks for making me think a little harder about it. I'm
sure I learned something.

Extreme case
When I reposted the article with the new section on allometric
scaling, the discussion started again. Most of it was people who didn't
notice the first time, but still clung to the "longer lever, bigger
wheel" way of thinking. In support of that agenda, Rick Marcy asked, "Would you not agree that a 10
foot tall skinny man would have a faster swing if done correctly than
DJ for example?" Very interesting question! Sometimes
analyzing an extreme case can teach us a lot. So let's work this one.
We have to start by tightening the definition of "10 foot tall skinny
man". Here is what we will use for the analysis:
 DJ (Dustin Johnson, a tall, bighitting PGA Tour player and
#3 golfer in the world at the time of this writing) is 6'4" tall and
weighs 190 pounds. Based on scaled norms, I'm going to use 26" for the
arm length in the calculations.
 Our tall comparison golfer, Skinny, is 10 feet tall. The height ratio we need for our
calculations is 10'/6'4" = 1.58. Skinny is 1.58 times as tall as DJ.
 We will further assume that Skinny's limbs are all 1.58
times the length of DJ's, so he is in proportion lengthwise.
 But he is
skinny! Let's make his crosssection at any section (arm,
leg, torso) the same as DJ's. That has consequences for weight and
strength:
 His weight is not proportional to the cube of his height,
but rather directly proportional. That's because volume is proportional
to height times crosssection. The height ratio is 1.58, but the cross
section does not change.
 The force exerted by his muscles is the same as DJ's,
because the crosssectional area of the muscle is the same.
Let's go ahead and see the effect of one change at a time, as we plug
changes into the computer model. We start with DJ, and plug in new arm
length, club length, weight, and strength to get to Skinny.
What changed? 
New
clubhead
speed 
Comments 
Start with DJ 
121 mph 
Tweaked the torque until, at 80 ftlbs,
we got a clubhead speed over 120mph 
Lengthen arm to 41" 
101 mph 
Lengthen proportional to height. We lose
a lot of clubhead speed. Wingspan alone doesn't work. 
Lengthen club to 48" 
104 mph 
It helps, but not much. OK, to hell with
conformance to the rules.
Let's lengthen the club proportionally to height. 
Lengthen club to 71" 
120 mph 
Got us back almost to DJ. 
Weight to 300 lb 
114 mph 
Big weight gain, but it's only
proportional; he's skinny. Lost speed again. 
Torque gain averages
somewhere between
the present 80 ftlb
and 126 ftlb 
Between
114 mph
and
138 mph 
Big gain possible! See *** below for why
the torque increases, and by how much. 
Again, we see that almost everything we do to account for height loses
clubhead speed  until we look at the implication of increased torque
driving the longer arm lever. Only then does clubhead speed show a
potential increase. It is given as a range, and the bottom of the range
is lower than DJ's clubhead speed. But Skinny's clubhead speed is
probably higher than the bottom of the range, and the range goes well
above DJ's.
So yes, Rick, 10foottall Skinny will likely develop more clubhead
speed than DJ. But it isn't the wingspan that does it. Without a torque
boost above what DJ's muscles and joints generate, wingspan gives a
speed loss, not a gain. But larger size allows Skinny to develop more
torque to swing the longer lever; that and only that
develops the increased speed.
*** Here's why the
torque increases, and why it's hard to compute exactly how much.
(Notice that the torque increase is given as a range, not a single
number.)
The torque at each joint is the force the muscle can generate times the
moment arm. (Reviewing, moment
arm is the distance between the centerpivot of the joint
and the line along which the force acts.) The force itself is the same
as it was for DJ, because Skinny has the same crosssection as DJ for
each limb. As for how we find the moment arm, look at the diagram:
 On the right, the moment arm D
is parallel to the length of the forearm. The length of the limb is
proportional to the height, so D
will be 1.58 times what it was for DJ. Since the forces are the same,
the torque will be 1.58 times what DJ generates. If all the joints were
like this, then the shoulder torque for the computer model would be
80*1.58=126 footpounds.
 On the left, the moment arm d
is perpendicular to the length of the forearm. Skinny's crosssection
details are identical to DJ's, so Skinny's d
is the same as DJ's. If all the joints were like this, then the
shoulder torque would be the same 80 footpounds as DJ.
Some joints are lengthwise, some are crosswise, and most are in between
for most of the swing. Don't forget; joints move during the
swing; that's their job. And as they move, we can expect changes in
whether they are lengthwise or crosswise or in between. So, netting out
over all the joints and the whole swing movement, the effective
shoulder torque is somewhere between 80 and 126 footpounds. And that
means the clubhead speed is somewhere between 114mph (what we got with
80 footpounds and Skinny's dimensions) and 138mph (the result with 126
footpounds).
So now you might well ask: why did we have single number, not a range,
with the original study? That was because we didn't assume the taller
golfer was skinny; we scaled him up proportionally. So it didn't matter
whether the moment arm was lengthwise or crosswise; all the dimensions
were scaled up by the same amount. 
Additional topics
There
was some discussion way outside my area of expertise, and I hesitate to
incorporate it here. It included things like adding muscle mass vs
stronger (or perhaps more effective) muscles at the same mass, and also
the role of fascia in producing clubhead speed. If you are interested
in such things and have access to the discussion, you can read those
topics yourself.
Bottom
line
 Strength
matters! If you assume the taller golfer is scaled up in all
dimensions  weight, muscles, size of joints, etc  there is an
implied strength advantage that accounts
for all of the
extra driving distance.
 Length
of arc is detrimental, if it comes from a bigger radius. That's
because it increases the need for strength to get the same rotation
rate. (I have done other studies that show length of arc helps
if it
comes from a bigger angle. But
that's not a tallvsshort thing. You can have the extra
shoulder turn whether you are tall or short.)
 "Taller
implies farther" is not a statistical accident. Taller golfers are
statistically likely
to hit a golf ball farther. But the statistical correlation is far from
the whole story; other factors are more important for driving distance.
Last
modified  Aug 30, 2022
