Newton and the Divot:
Followup
Dave
Tutelman 
January 28, 2014
This article stirred up much more comment than most I write. I
got a lot of email with very interesting questions and comments. And
Fredrik Tuxen of Trackman told me that I had misinterpreted Trackman's
definition of Attack Angle as stated on their web site. Actually, I
used it completely literally. The definition needs a little work
to state exactly what Trackman measures and reports.
So it seems appropriate to add a page of followup and correspondence,
because a lot of the questions and comments require more than just a
small tweak on the existing article. (I did that where I could, but it
left over a lot worth discussing.) So here it is.
Trackman's
definition of Attack Angle
The most controversy, and the biggest changes needed to my article,
dealt with the way Trackman defines what they report as Angle of
Attack. At the time I original wrote this article, the definition on
the site was:
Attack
Angle  The vertical direction of the club head’s center of gravity
movement at maximum compression of the golf ball
Today in 2018, it has been slightly updated to:
The up or down movement of the club head at the time of maximum
compression. Attack angle is measured relative to the horizon.
It is a different wording, but still says the same thing.
Fredrik Tuxen, the CTO of TrackMan, responded to the article in email
to me and a post to the Facebook group for Professional Golf
Instructors. The latter post said:
TrackMan's angle of attack is
taken at point
of time of max compression
of the ball. However, ONLY by using preimpact data. Meaning the attack
angle from TrackMan is very close to first contact with the ball but
shifted app. 0.3 deg due to the 1/4000 second time difference between
first contact and maximum compression of the ball.
Tutelman has unfortunately misunderstood the definition of TrackMan's
angle of attack, but he is excused it was not sharp enough.
TrackMan will update the definitions of attack angle and club path to
avoid future misunderstandings.
Tutelman's 'Newton divot' is an insightful description in the
consequences of conservation of linear momentum for the oblique impact
between club and ball.
He followed up with a video Skype session with me, covering the
details of how they measure and report Angle of Attack. Here is my
understanding of it, explained in the context of a diagram. Fredrik
says this is a correct interpretation.
TrackMan
picks
up the clubhead well before impact, and tracks it to the ball, getting
as close as possible to the ball while maintaining accuracy. In the
diagram, it tracks through positions 1, 2, and 3. (Actually, it notes a
lot more than three positions in this interval.) Those positions follow
an arc, the red dotted line, which is mathematically fitted to the
clubhead's measured positions as it approaches the ball.
TrackMan is capable of measuring the time and position of the
beginning of impact very accurately. It extends the clubhead along that
same arc (the red dotted line), to where it would be without any clubhead
deflection in the middle of
impact at the point of maximum ball compression, shown as position 4 in
the diagram. The slope
of the arc at
position 4 is the Attack Angle reported to the user. Note
that:
 While the clubhead is indeed deflected downward
during
impact, that deflection is not considered part of TrackMan's AoA
measurement.
 Since the clubhead had been traveling a curved arc,
the
reported AoA is a bit more upward than the preimpact AoA. TrackMan has
found this difference to be 0.30.4 degrees.
I went back and recalculated the tables. The calculations are much
easier using this definition. I was able to dispose of half the green
math section of the article. The results are now on the first page of
the article.
Thanks, Fredrik.
While the short definition of Attack
Angle on the TrackMan web site has not changed appreciably, those
wanting to dig deeper can find the correct details in their new article
on "Club Data Definitions".
This article acknowledges momentum transfer's significant contribution
to clubhead movement during impact, and explicitly excludes the
influence of the ball from the reported measurement.

Correspondence
There were quite a few emails and forum posts with worthwhile questions
and comments. Let me address them.
In what follows, I
don't
identify the commenter. I present the
comment that was made, and respond to it without attribution. If you
recognize your own comment and want to be credited, let me know and
I'll be glad to do it.

I
have not
reread your article
but my first run through doesn't leave me with any more information
about why you wrote it and what it is suppose to accomplish. Nice
charts and physics discussion but isn't what you presented taking the
phrase "angle of attack" and nit picking it?
I enjoy discussions like you presented but I cannot take that to the
golf course as the guys I play with would look at me and proceed to
throw me into one of nearest ponds.
I write articles for different audiences, and very few articles are
aimed at all of them. From your comment, I suspect you are a golfer
with a technical interest, but not a clubfitter, club or component
manufacturer, instrument designer, professional golf instructor, or
researcher. This article is probably of marginal interest to you, but
I've gotten a lot of very interesting discussions from instructors and
clubfitters, and instrument designers.
When I go out on the course with my own friends, there are precious few
that I could mention this to without the chance of a dunking, so I
understand what you say.
Even so, you might take this away from your read:
If you hit a ball from the ground with
any club at all, you better leave at least a scuff mark on
the
ground after the ball  and a real divot with a middle to short iron.
If your club never touches the ground, that doesn't mean you picked it
clean; far more likely, you hit it thin.

Your
interpretation of AoA is measured just before impact. So what is that
"just before"? We need two points so we need a time frame, but how
large a time frame? It would seem to me that the value here is
dependent on the sample rate or fps of the camera used!
Absolutely right! In fact, even more points at a high frame rate give
us a more accurate estimate. And that is what TrackMan does. Their
sampling rate (20,000 samples per second) is comparable to the fastest
cameras. They use this to fit an arc to the clubhead path. If you
factor in the ability to estimate the time of impact very closely, you
can extrapolate that arc to either the moment of impact or the moment
of maximum compression. They choose the latter.
But
even more
important, does it really matter to the ball and the ball flight? I
would say it doesn't. In fact it's the real movement of the head during
the impact period (till max compression) that is important.
Depends what you are trying to do with the data. In my case, I was
doing momentum transfer calculations. Momentum must be computed before
impact begins and after impact ends. Anything that happens during
impact is just going to confuse the calculations. Let's take a closer
look at what goes into a momentum transfer.

I
had always
assumed that the duration of impact was so short that there wasn't much
change in clubhead presentation during the impact interval but that
appears not to be the case.
Definitely not the case! Momentum transfer is a wonderful thing. 1/2000
of a second is not very long, but 2000 pounds is a lot of force. To get
a feel for why this matters, we need a concept called "impulse", which I
haven't mentioned
yet.
Let's look at how momentum
comes about. One of Newton's three famous laws is F=ma.
We should
examine that a little more closely. Acceleration is the change of
velocity over time. So we can rewrite F=ma
as
Where the delta represents a change or interval. We rewrite this as:
F
Δt = m Δv
We recognize the right side as the change in momentum. What the whole
equation says
is, "The change in momentum is equal to the force acting on a mass
times the time
interval the force is exerted." The left side, the force times the time
interval, is called "impulse", and we see that momentum change of a
free body is equal to the impulse on that body. So impulse =
change in
momentum. (For a more complete explanation, try this.)
Now think about momentum in terms of impulse. Two thousand pounds of
force applied for 1/2000 sec gives the same momentum transfer as if it
were one pound for one
second. Much easier to think about  and not negligible at all.

The
pros hit
the center of the club face, and the tables in the article are based on
the pros.
Would you see less effect for the average
golfer?
That's
a great question! The effect is based on momentum transfer, and the
pros transfer momentum very efficiently. So perhaps they will see more
of a deflection than you or I would. Let's look at some of the factors
involved in the momentum transfer:
 Momentum is MV,
mass times velocity. Everybody plays with the same ball mass and the
clubhead mass doesn't have a lot of variation. But the pros generate
more clubhead speed and especially
more ball speed. (Their center strikes have a better smash factor.) So
that suggests more deflection for the pros.
 The downward deflection of the clubhead is based on
upward
motion of the ball. The pros hit the ball with a flat or bowed wrist,
resulting in a fairly low trajectory. I see a lot of somewhat cupped
wrists among my playing companions, which will result in a higher
launch angle. This means a proportionally larger vertical component of
momentum, which will give the duffers more downward deflection.
 True, the average golfer is not a centerface
striking
machine. But it's worse than that. Often enough, they won't get it on
the face at all  something you almost never see from the pros. A thin
shot that is at least partially off face
(partially bladed or worse) will not generate much downward force on
the clubhead. And you can see it as a momentum transfer thing by
watching the wormburner scoot along. You hardly ever see that from a
pro.
So I see three factors: one where the pro would experience more
deflection,
another where the average golfer might see more deflection, and a third
where an average golfer would sometimes encounter a situation of almost
no deflection.

Does
Flightscope have the same
problem? What is its definition of Angle of Attack?
Henri Johnson, CEO of Flightscope, told me through a mutual friend that
their definition is:
"The
angle between
the velocity vector and level ground, just prior to impact, is AOA."
That is the definition that I understand, and the one I use here. I
hope it is in fact the one they use in their measurement.
The reason for my uncertainty is, you won't find the definition if you
check the Flightscope web site. Instead, you will find a video on the subject. The
definition implied in
the video
has a final sample just before separation (not just before
impact), which
would include the downward deflection of the clubhead. That is not in
accordance with their definition.

Concerning
the TM
tour average numbers, are you aware that TrackMan reports AoA and
clubpath
from the geometric center of the clubhead and not the clubface? I have
a sense that this could change some of the calculations you have
preformed.
The truly relevant position on the clubhead would be the center of
gravity (GC), not the geometric center and not the clubface. So neither
TrackMan nor I get it exactly right. But I think we both do pretty
well. Here's why:
The center of gravity is usually pretty close to the geometric center
of the clubhead. Not exactly the same, but close. So TrackMan will not
be far off.
I did pick my points near the face, so your comment is on target. But:
 For the irons, the CG is
not far from the
face at all. Any error due to
picking a point on the face is quite small.
 For drivers the problem is
potentially much
larger, because the CG is
well back from the face. (I mentioned it, in fact, as the last bullet
point in the section "Even
the
driver?") For that reason, my case study
for a driver involved center impact, where the clubhead did not rotate.
Any up/down motion of the face in
that particular case corresponds to
an up/down motion of the CG.
So you raised a valid concern. I had the same concern, and
believe I have it covered.

Momentum
is
not conserved in the
club/ball collision. Some energy is turned to heat or sound, while
compressing the ball and clubface. You should include COR in your
calculations, or they will be called into question.
You are confusing conservation of momentum with conservation of energy.
(I ran across the same mistake last week in a book on golf physics by
John Zumerchik, so you are not alone.)
When you set up the equations for a collision, you need both
conservation of momentum and conservation of energy. Both! The
conservation of energy equation involves COR and energy losses.
Momentum, however, is conserved without loss.
In the calculations here, only momentum needs to be considered. We are given the clubhead
velocity before
impact and the ball velocity after impact. Therefore, the COR has
already been
accounted for; the ball's velocity reflects the energy losses. The
problem to be solved here is just conservation of momentum.
While the objection has been answered, let's assure ourselves that we
can indeed conserve momentum perfectly and still reflect energy loss.
Here's an example. Consider two possible twoball collisions, one
lossless and one very lossy.

Lossless
collision
COR = 1.0
Balls bounce perfectly

Lossy collision
COR = 0
Balls stick together


Before
Collision

After
Collision

Before
Collision 
After
Collision 
Picture





Momentum

MV

MV

MV

2M×½V
= MV

Energy

½MV^{2} 
½MV^{2} 
½MV^{2} 
½×2M×V^{2}/4
= ¼MV^{2} 
The important thing to notice is: in the lossless case, both momentum
and energy are
conserved. That is, they are the same both before and after. But in the
very lossy case (COR=0),
momentum is still conserved but half the kinetic energy in the system
is lost. So it is OK to ignore losses if the problem is simply
conservation of momentum. Losses must be accounted for when you do
conservation of energy.


Given
that
the clubhead slows down during impact and the shaft is bent forward
just prior, I was wondering how much of the lowering (and delofting)
may be due to just unbending of the shaft. Or do you think that in this
short period of time the clubhead really acts more like a free body
disconnected from the shaft?
The latter, definitely! I address this in other
articles,
and in great detail in my article
on gear
effect. The influence of the shaft on the clubhead during impact is
almost nothing,
relative to the other forces acting on the clubhead. Consider: it takes
an inch of deflection of the shaft to produce a twopound spring force,
but the force between ball and clubhead is of the order of two thousand
pounds. So the
inertial forces are three orders of magnitude greater than the forces
the shaft is exerting.

In
one iron
shot in particular,
the ball was hit towards the toe. The face opens during and after
momentum transfer, which causes a motion that has a circular pattern
for the reference point on the club. The reference point on the club
hence is moving downward, causing the apparent perception of a
"downward movement".
If you mean the measurements from the iron in the animated GIF, then
you are correct. There is in fact plenty of actual downward
movement, it is not limited to just apparent movement. But the motion
of the toe  a little
downward and and even more "backward" motion  exaggerates the angle
of deflection. Probably for that reason, the deflection angle measured
from the
video frames is 12º, about twice the value calculated for Tour strikes.
But make no mistake, there is plenty of real downward
movement in the video
 just not as much as it looks like.
It is worth saying more about this. In order to take accurate
measurements from frames of a video, you can't just use random
videos. Mea culpa! I used random videos, even knowing the problems. I
was looking to demonstrate an effect, which I would quantify by
calculation. I don't have a highspeed video camera, and had to make do
with what I could find on the Internet. If you looked closely, you
noticed the
measurements from the videos were inconsistent with the calculated
values. The iron video measured at twice the calculated value (12º vs
6º), and the driver video measured at half the calculated value (3.2º
vs 6º).
In order to get an accurate measurement, here are a few things you
should be careful about before you even start to film:
 The camera should be pointed perpendicular to the
target
line.
 The camera should be far enough from the clubhead so
perspective distortion is minimized. (That is, motion closer to the
camera should measure almost the same as motion farther from the
camera. If the camera is within a few feet, closer will measure
substantially more than farther.)
 The clubhead should be marked with cameravisible
dots,
preferably dots with sharp corners, so clubhead position can be
determined
to within a pixel.
 The position of the impact on the clubface should be
controlled  or enough clips taken so the right position can be
selected  to avoid the valid criticism applied to my measurement.

One
point can
not be used to measure the 3dimensional movement of the clubhead. If
you take a club and rotate it around its CG, then that point would be
moving but the CG would not! Two points would be better, as you could
use the distance between them as an indication of movement towards the
camera. To get the correct 6DOF data from any object in one video a
minimum of 3 points is needed and those point have to be arranged in a
correct way.
All very true, but not a big issue here. 6DOF (jargon for "six degrees
of freedom") includes three degrees of rotation and three of linear
motion. I maintain we only need two degrees of linear motion for this
exercise, so a single point is sufficient. Looking at my the previous
response:
 Rotation of the clubhead at impact does not need to
really be measured,
just avoided in the frame we use. That will depend on the position of
impact on the clubface  as noted above. We need to pick neutral
videos  nearcenter impact  to do our measurements.
 Multiple dots serve two purposes: rotational degrees
of
freedom (which for our purposes only need to be detected, not measured;
multiple dots not needed) and motion toward or away from the camera.
 If the camera is perpendicular to the target line,
then
motion toward or away from the camera is not an issue. And the camera
has the other two dimensions covered.

Last modified  Aug 8, 2018
