Mike's questions  Part 1:
Simple Formula for Driving Distance
Dave Tutelman  December 27, 2014
This work was instigated by a couple
of
questions asked by Mike Stachura. He wanted to know how many yards of
driving distance you get for every mile per hour of clubhead speed, and
also how many extra yards of driving distance could a golfer pick up
without increasing clubhead speed.
Reiterating Mike's two questions:
 Is there a simple multiplier to relate driving distance to
clubhead speed, something like "distance
= 2.5 * speed"?
 How much driving distance does the average golfer "leave on
the table", assuming his/her clubhead speed is what it is and won't
increase?
These are complementary questions, related by a paradox. Think about
it; if there is a
hard and fast answer to #1, then no
distance is left on the table.
If you know the golfer's clubhead speed, then you know the
distance.
So it is
obviously more complicated than a simple formula that relates clubhead
speed to distance. Here are some of the
complications, and how we will deal with them to answer the questions:
 Ball
flight is not linear, and there are no easy formulas to give an exact
answer. To answer question #1, we are going to use the
TrajectoWare Drive computer program, which deals
explicitly with the
complexity of
ball flight. (If you want to see what makes the problem complex and how
a computer deals with it, I have a tutorial
on the subject.) We will see if we can make any easy
generalizations from
the computed results. Turns out we can.
 Because
ball flight is complex, distance depends on a lot more than clubhead
speed. To answer question #1, we will study the effects of
clubhead speed, using the most reasonable or simple assumptions we can
about the other things that affect distance. Then, when we get to
question #2, we will hold the clubhead speed constant and open up the
other assumptions, to see where we can do
better.
 Mike
wanted total driving distance, not just carry distance.
This is a much harder problem to solve. I don't have a quantitative
handle on bounce and roll after the ball lands ("runout"). But we do
come up with some answers, based on research that others have done.
Let's dive into question #1: Is
there a simple multiplier to relate driving distance to
clubhead speed?
Assumptions and
approach
Mike was looking for a formula for driving distance based on clubhead
speed. He wanted a simple formula of the type:
Distance =
ClubheadSpeed * someMultiplier
He started with the assumption that such a formula is possible. Here is
his email to me, with the results of his research to that point.
At 11:01 AM
12/11/2014, Mike Stachura wrote:
I've studied a
handful of trajectory programs out there, seen fitting charts companies
have handed out to their fitters and tried to come up with a simple
multiplier formula based on club head speed. Messing around with the
numbers I came up with a series of stepped multipliers.
Less
than 70 mph: 2.4
7180: 2.45
8190: 2.55
91100: 2.65
More than 100: 2.7
But I'm afraid these
yield distances that are too high for human golfers with real drivers.
Maybe not?
I'll get back to the last conjecture  about human golfers with real
drivers  but let's deal with these numbers first and what they ought
to be. When you look at a simple fitting chart or a trajectory
program, the author or programmer often assumed away much of the
complexity.
Let's see some of the typical assumptions:
 Clubhead speed is a given, it is fixed for this golfer.
 Center contact  this implies two things:
 No gear effect.
 Maximum smash factor, 1.48 for low lofts.
 The 1.48 smash factor is
itself based on another assumption: a modern ball and club with the
maximum allowed coefficient of
restitution of 0.83.
 Square clubface at impact; no sidespin.
 "Flat" attack, angle of attack=0.
 The loft at impact is that built into the clubhead. (Other
things that can affect loft at impact are shaft bend and wrist cup.)
 Plain vanilla ambient conditions: temperature 80ºF, sea
level, no wind, flat fairway, no elevation of tee box, etc.
 A properly fit driver. This is an important assumption!
With all the other assumptions, the only fitting variable we have left
is loft. The charts (and our study) will best fit the loft to the
clubhead speed, consistent with all the other assumptions.
Let's make all these assumptions, plug them into the computer, and see
what it gives us. Then we will try to find a simple formula that fits
what is coming out of TrajectoWare Drive.
Finding the formula
Mike's
multipliers
Before we turn to
a program that really computes the ball flight, let's start with the
multipliers Mike suggested. Five different
multipliers for different clubhead speeds certainly isn't what you
want; it's almost easier to remember the distances for each of those
clubhead speeds. But let's look at the implications if that is the
right "formula".
Here is a graph of the distance implied by the varying multipliers. As
we should expect, there is a discontinuity, or jump, every time the
multiplier changes. The jumps range from 6 to 12 yards. So, for
instance, the difference in distance between 89 and 90mph is 2.5 yards,
but the difference between 90 and 91mph is 12 yards. This is not a
helpful way to look at distance vs speed.
So let's look to the actual, computed numbers and see what we get.

Curves and lines
Let's use the TrajectoWare Drive
computer
program to run the real ball flight, and plot the carry distance. We
will do this for clubhead speeds from zero to 120mph. For each clubhead
speed, we will find the driver loft that gives the best carry distance
for that clubhead speed. (TrajectoWare drive makes this easy; you can
use the scroll wheel on the mouse to vary loft, and watch the carry
distance number for a maximum.)
Here is the graph I got.
. 
Let's
take this plot, and try for multiplier's like Mike's
for 60mph, 70mph, 80mph, 90mph, and 100mph. We get:
Clubhead
speed 
Carry
distance 
Multiplier 
60 
103 
1.71 
70 
136 
1.94 
80 
169 
2.11 
90 
202 
2.24 
100 
233 
2.33 
A
few observations:
 The most obvious is that the multipliers
are lower than the ones Mike suggested. Upon further questioning, I
found out he is looking at total
distance rather than carry distance. So you would expect his
multipliers to give longer distance. We will revisit this below.
 Apart from that, the trend
is similar to Mike's multipliers. We don't have one single multiplier.
Instead, we have
different multipliers for different clubhead speeds, and they increase
with clubhead speed. So they are no more useful than Mike's multipliers.
One way to view the multipliers is shown on this graph, derived from
the graph above. I have drawn straight lines from the origin [0,0] to
the actual distance at three speeds: 60mph, 80mph, and 100mph. The
slope of each straight line is the multiplier for that distance.
Looking at the graph, we can see why there is a different multiplier
for each clubhead speed; the graph (the red curve) is not a straight
line. In order for there to be a simple formula of the form distance=speed*multiplier,
the graph would have to be a
straight line through the origin!
It isn't. We can tell just by looking. Does that mean a simple formula
is a hopeless dream? No!
Looking at the graph, we can see something very interesting that will
help us simplify the problem.

There is indeed a section
of graph that looks like a straight line. It is the part from 60mph to
120mph.
I use Excel for plotting the graphs, and I can ask it to do curve
fitting to the function I plot. I asked for the bestfit straight line
to the straightlooking section of the red curve (the actual
distances). Excel gave me the green line
on the graph, and told me that the equation for that line is:
Distance =
3.16*speed  85.2
Let's look at this line and see what the differences are between this
and what Mike  and most people who look for a simple formula 
assume:
 It is not just a multiplier; there is a
multiplication and an addition. (Well, a subtraction, but that is
addition of a negative number.)
 The line does not go through the origin. At zero mph
of clubhead speed, the ball goes 85 yards backwards. That's
crazy.
Well, it would be crazy if we were crazy enough to try to use it at
0mph clubhead speed. But remember where this line comes from. It is a
best fit to the curve over
the range from 60mph to 120mph. We really
should only use it within this range. (Actually, it is just as good at
50mph, then it ceases to be a good estimator.)
How good is the match between the straight line and the red distance
curve? Let's look at two different ways to measure it.
 When Excel plots a bestfit line, it gives a measure
of fit called "Rsquared". When Rsquared is 1.0, that is a perfect fit
to the data. When Rsquared is zero, there is no correlation at all
between the data and the line we have claimed fits it. In the case of
our carry distance curve, Rsquared is 0.9991. "Three nines" means the
line is an
extremely good fit to the data.
 I have compared the data to the line at 10mph
intervals. I found agreement within two yards from 50mph to 110mph. It
was up to a threeyard difference at 120mph, still quite good.

Explanations and variations
What is going
on, and why?
So we have a very effective, simple formula, but not what Mike
expected. Why not? Because it isn't based on just a multiplier, a
proportion between clubhead speed and distance. But we have found a
linear relationship
between carry distance and clubhead speed, that stays linear over the
range of interest. 60mph120mph covers the vast majority of all golfers.
If we insist on a single multiplier to do the job, we will be
disappointed.
But we learned in high school math that the formula for a straight line
is:
y = mx + b
where:
m
is the multiplier (the slope of the line),
b
is the y
value where x=0,
the place where the line intersects the yaxis, what is called the
"yintercept".
Insisting on a singlemultiplier solution would add the restriction
that b=0,
and that prevents the solution from being very good. By allowing a
nonzero yintercept, we
get a very good linear approximation,
which is also a very simple formula for distance. True, you have to
remember two numbers, a multiplier and an intercept. But that is better
than needing to remember five different multipliers, along with the
speed range to which each applies. And more accurate, I might add.
Ball speed
While we're getting a
simple linear relationship for clubhead speed and
distance, what about a similar relationship for ball speed and
distance. That way, we don't have to assume perfect contact, a maximum
smash factor. We can compute ball speed using whatever smash factor the
golfer is able to generate, then go from ball speed to distance.
To
do this, I recorded the ball speed when I did the Excel graphs for
clubhead speed vs distance. Here is a plot of ball speed vs distance,
and its bestfit straight line. Points worth noting:
 It is hard to visually separate the red data curve
from the green bestfit line. That is supported by the Rsquared of
0.9994, threeandahalf nines.
 The slope is very close to 2.0. For all practical
purposes, we can say that every 1mph increase in ball speed causes a
2yd gain of carry distance.
 Again, there is a yintercept. The straight line does
not go through zero. Thus it will give nonsense if we try to use it for
very small ball speeds. But in the range of interest, ball speeds
produced by swings between 60 and 120mph, it gives very good estimates.

How do we use the formula?
Two ways, depending on what you are trying to accomplish:
 The obvious one: given a clubhead speed, what is a
reasonable distance you can expect to hit it? Here's how you do that:
 Multiply the clubhead speed by 3.16.
 Subtract 85.2 yards from the result.
 You have the carry distance, within one or two yards.
 You can do something that will gain a few MPH of clubhead
speed. (Clubfitters work problems like that a lot.) How many
yards is that worth?
 Multiply the gain of speed by 3.16. Actually 3 is close
enough for most such problems. Not many things you can do to gain more
than 6mph, which is the minimum before you can see much difference
between
3 and 3.16.
 You have the gain in carry distance.
If you are given ball speed instead of clubhead speed, the numbers are:
 Multiply by 2.04 and subtract 65.2 yards, for the first
case.
 Multiply by 2 for the second case.
Can we make it even simpler?
No we can't, at least not without increasing the error. But, if
you're OK with a 5yd error at the extremes, and less than a 3yd error
between
70mph and 108mph, you can round things off to:
CarryDistance
= 3 * ClubheadSpeed  71
That is really simple.
Total distance
Mike's question was really aimed at total distance. TrajectoWare Drive
gives carry distance, not total distance. Total distance is difficult
to compute in general. Making it harder is the fact that it depends on
the fairway conditions where the ball lands. Baked fairways prepared
for PGA Tour events give very different runout from the soft courses
that I am playing in New Jersey in the late fall. And also different
from the hardfrozen fairways (more resembling concrete) that friends
Bill, Jim, and I expect
to find tomorrow at our traditional New Years Eve round.
But there is some hope. There are people doing studies of total
distance, and apparently making progress. And we do have a little data
from those studies...
Revisit Mike's
multipliers
I
asked Mike if the multipliers he cited are for carry distance or
total distance. He cited his earlier email, and clarified it with
another email.
The USGA has done a lot of work in this area, I know. It's one reason
they developed a testing device to measure the firmness of the
fairways. Your research does have me wondering unfortunately whether
there's not that much more yardage out there. I've studied a handful of
trajectory
programs out there, seen fitting charts companies have handed out to
their
fitters and tried to come up with a simple multiplier formula based on
club
head speed. Messing around with the numbers I came up with a series of
stepped multipliers...
These are total distance, based on some USGA and Trackman projections,
factoring in fairly positive angle of attack numbers (as high as +5).
So
Mike has already digested a handful of studies of total
distance.
Let's
look again at that graph of the multipliers he came up with, but this
time we will ask Excel to show us the bestfit straight line. The line
is very interesting for a number of reasons:
 The
Rsquared value says it is quite a good match to the data, even if the
data is a little jagged. It is visibly not as good as the carry
distance match, but "two nines" is not bad at all.
 The slope
is the same as the slope of the carry distance line. This one
is
3.1604, and the carry distance is 3.1632. That is more exact than you
would ever expect. So...
Let's compare the simple equations for carry distance D_{c}
and total distance D_{t}.
D_{c}
= 3.16*speed
 85.2
D_{t}
= 3.16*speed  50.5

What this says is that the runout is 35 yards, no
matter what the clubhead speed.
Here's a graph to show what it looks like.
That
is not only interesting, it is suspect. Might be true, but it
defies intuition  at least it defies my intuition. I would
think
that a drive struck at 100mph will bounce and roll out farther than one
struck at 70mph. I need more data and a deeper
understanding before this makes sense.
Let's look for other data to confirm or deny this curious result. 
TrackMan
TrackMan has published
a Total Distance Driver Optimization^{[2]} chart that I have seen
numerous places on the Internet. It gives the impact and launch
conditions for maximum
total distance, for a variety of clubhead speeds between
75mph and 120mph. It also lists both carry and total distance. So we
have enough information to make a graph comparable to the previous one.
Like our other results, the carry and total distances are straightline
graphs (to three nines and better) over the range of interest. And,
like the graph we interpolated from Mikes multipliers, the runout seems
not to depend on clubhead speed. (Well, it does  a little. At lower
speeds the runout is about 48 yards; it climbs to 5051 at higher
speeds. Still, almost constant runout, regardless of clubhead speed.)
I conclude a few things from this exercise:
 The runout is approximately the same  about 50
yards  for all clubhead speeds, if the driver loft was optimized for
that speed.
 Justin Padjen of TrackMan tells me that the data for their
runout
model was gathered from PGA Tour events, where the fairways are dry and
hard. So there is more runout than the average golfer would experience.
We should probably keep our total distance formula based on the
composite model from Mike Stachura's data.
Something to keep in mind about the runout being independent of
clubhead speed. What we have seen so far is data for distanceoptimized drivers
with very specific constraints. If you are using a club not
distanceoptimized for the clubhead speed, or optimized for different
constraints (like nonzero attack angle) or you strike it badly, you
are likely to get a very different runout. So don't generalize this
beyond the constraints that apply here.
By the way, I did a "sanity test"  a consistency check  to be sure
that TrackMan and TrajectoWare Drive were generating comparable
answers. The TrackMan optimization charts had full launch conditions
for each clubhead speed: ball speed, launch angle, and spin. I plugged
these into TrajectoWare Drive to find the carry distance. In every
case, the carry distance for TrajectoWare Drive and TrackMan were less
than 3 yards apart. So the numbers we see are quite comparable. 
Cochran
& Stobbs  1968
I found further confirmation in an interesting and unexpected place.
Return
with us now to those thrilling days of yesteryear! In 1968, Alastair
Cochran and John Stobbs wrote what I think is still the best book
around for getting started in golf science. "The Search for the
Perfect Swing" is a genuine classic.
I'm
the sort of person who reads appendices, and Appendix I includes
equations for carry distance and total distance. And lo and behold,
they are linear equations of the form y=mx+b.
To get them into a form we can compare, I had to:
 Convert the speed units from feet/sec to mph. That's
a factor of 1.47.
 Convert
ball speed (the C&S equations were in ball speed) to clubhead
speed. But remember, this was 1968. No spring face drivers, just wooden
heads. No "rabbit ball". The COR cited in the book was only 0.67. That
gives a maximum smash factor of only 1.36.
The
converted equations, shown here in a graph, were:
D_{c}
= 3*speed
 103
D_{t}
= 2.5*speed  27
Much to my surprise, the runout actually decreased
with clubhead speed. So maybe a constant runout isn't farfetched after
all. If the C&S equations were correct for the day (and I have
to
believe they were), then the question to ask is, "What has changed
since 1968 to improve runout for big hitters more than for shorter
hitters?"
The thing that comes to mind immediately is the modern
tournament ball. In the past 1015 years, ball manufacturers have
managed to make balls with less tendency to spin as they compress more
on the clubface. Not that they spin less  higher clubhead speed will
always generate more spin  but they acquire spin at a lower rate as
they compress more. A lower spin means (a) less
runoutkilling
backspin on landing, and (b) a less steep angle of descent. (AoD is as
harmful as backspin to runout.)
So, to my surprise, it is
conceivable and even likely that today the runout does not vary with
clubhead speed. The additional velocity on landing is countered by
spinrelated problems: angle of descent and backspin on landing. The
data in these last two graphs suggests that it took big changes in
modern
ball design (think ProV1x and comparable balls) to get the
big
hitters' runout to even equal the mere mortals' with a driver. 
Conclusion
There
is a good answer to the first of Mike's questions, but it is not in the
form he assumed. It does not consist of a single multiplier, but rather
a straight line that does not pass through the origin. The formula is
Carry
distance = 3.16*speed
 85.2
But
Mike wanted total distance, not just carry distance, and his data is
probably the most applicable to the typcial golfer. The bestfit
straight line to the Mike's data is
Total
distance = 3.16*speed  50.5
Note that the carry distance formula above relates to optimizing
carry distance. The carry distance you realize when you optimize total
difference will be somewhat lower.
On to Mike's second question:
how much can typical golfer improve
his or her distance, without having to achieve more clubhead speed.
Last
updated  Jan 12, 2015
