Physical principles for the golf swing
Introductory concepts
Dave Tutelman
 February 26, 2020
Kinematics vs kinetics
Let's start with something simple  the definitions of two important
terms in biomechanics: kinematics
and kinetics.
Kinematics
is the study of motion. Just motion! Not what causes the motion, just
what the motion is.
As such kinematics concerns itself with position, velocity, and
acceleration, both linear and rotational. It does not concern itself
with the forces or torques that create the motion, nor the energy or
momentum represented by the motion. It's just motion, all by itself.
Kinetics
is the study of the physical principles behind the motion. The forces
and torques. The energy and momentum. What causes
the motion.
Let's look at a couple of examples, specifically two that are central
to everything in biomechanics studies:
 Newton's wellknown second law, F=ma, is a kinetic
statement. It tells exactly how force causes acceleration.
 The relationship between acceleration and velocity, a = dv/dt, is a kinematic
statement. It tells how to compute one kind of motion from
another, without ever worrying about forces, torques, energies, nor
momentum.
Here are a few basic and important kinematic facts you will need to
know when reading these notes or, for that matter, lots of biomechanics
papers. The authors (including me) expect that you will just recognize
this off the top of your head. So try to wrap your head around this  understand and memorize.
If you have a constant acceleration a (linear or
angular/rotational) for a time t, then
Velocity v = a t
Distance x = ½ a t^{2}
This is true for linear distance (meters, feet, miles) and angular distance (degrees, radians).
It is true for linear velocities (mph, ft/sec, m/s) and angular velocities (rpm, deg/sec).
The graph shows the shape of these two curves.
 For constant acceleration, the velocity increases at a constant
rate. (Hey; acceleration is the rate at which velocity increases, by
definition.)
 Because the velocity is increasing, the rate at which distance is
covered keeps increasing. (Again, this is pretty much by definition.)
So the distance curve's slope is constantly increasing.
We will revisit this with examples when we talk about Newton's laws of
motion, but I will feel free to count on your recognizing these two
relationships at any point in the lessons.
Units of measurement
A word of advice
before we start. Much of the effort in doing realworld physics
problems is assuring that the data and the answers are in the proper
units of measurement. Inches? Feet? Miles? Centimeters? Meters?
Kilometers? These are all units of length, but the formulas of physics
don't work with all of them. Or at least any given calculation will
only work with one of them at most,
once we are given the units for the other quantities. For example, the
vitally important and very simple F=ma
talks about forces, masses, and velocities. If we know that we are
given the mass in kilograms and the acceleration in meters per second
squared, then the only unit of force
that will work as an answer is Newtons. Not pounds. Not tons.
Not grams. Just Newtons.
Conversely, if the mass is in slugs and the velocity in feet per
second, then we know the answer is in pounds.
Fortunately, there is a system of units that can be consistently
applied to all our Newtonian physics. In fact, there are several such
systems.
 The MKS
system is based on:
 The meter as
the unit of length or distance.
 The kilogram
as the unit of mass.
 The second as
the unit of time.
 While not part of the basic system, the unit of force is the Newton.
This system forms the original underpinning of the
International System of Units. Any serious technical publication is
likely to demand measurements and calculations be in this system and
its extended additional units.
 An American
system, with a selection of units most of which we Americans can
relate to.
 The foot is
the unit of length or distance.
 The slug is
the unit of mass. You probably are not familiar with it. It had to be
invented to make F=ma work, and only students of physics really get it.
 The second is
the unit of time.
 Again, the unit of force is not part of the system, but the
addon unit is the pound.
 The CGS
system,
which we won't use in our examples, is based on the centimeter, the
gram, and the second. But the formulas for Newtonian physics work with
this combination as well.
The bottom line is that we need to use a consistent set of units when
we do calculations, and there are two such systems you will see used in
these notes. Which brings us to the question, "If I don't have my
inputs in consistent units, or if I want the answer in a different set
of units, how do I get there?" The answer is unit conversion. There are
whole books (well, fat pamphlets at least) of factors for unit
conversion. But they are becoming obsolete, as unit conversion
calculators are becoming
available. You can search for any pair of units for the same quantity,
for instance units for length, and find an online calculator to
do the converting. And if you can use some form of Windows on your
computer or device, there is a free program for download called Calc98
that is a favorite of mine to do unit conversion. I used it for all the
examples in these notes.
But one of the goals of this tutorial is to allow you to read and
understand published research in golf biomechanics. In order for you to
follow along with the papers, you need to develop at least a little
intuition for the units. So let's get some approximations
for the more
important units. Here's a list of rough
conversions worth committing to
memory:
 Length: there are 2½ centimeters in an inch.
 Length: there are 40 inches in a meter.
 Length: a meter is a little more than a yard.
 Mass/Force: there are 2.2 pounds in a kilogram.* (See footnote at
the bottom of this list.)
 Force: there are 5 Newtons in a pound.
 Velocity: there are 2.2 miles per hour in a meter per second.
 Velocity: there are 22/15 feet per second in a mile per hour.
That's about 1½, if you prefer to think about it that way.
* (footnote)
Americans tend to describe both mass and force in pounds. The rest of
the world tends to describe both mass and force in grams or kilograms.
Neither is correct, strictly speaking. But really, only scientists talk
about slugs (for mass) or Newtons (for force). Still, you do
need to know this if you intend to read what biomechanists write about
golf.
So, for reference purposes:
 A long iron is about a meter long. A driver is more than a meter,
a wedge is less.
 A middle iron weighs about 400g or about a pound (1#). A driver
weighs less, a wedge more, and a modern putter a lot more. Wait! A
driver weighs less?
Yup! The driver is the lightest club in the bag. You probably know this
already, but too many people are surprised by this fact. The driver is
long and the head is big. It looks
heavy, but it needs to be light to do its job.
 A 100mph golf swing is about 44m/s.
 A 6'2" tall golfer is less than 2m tall (close to 1.9m).
 A 180# golfer weighs 82kg.
In this last list, I transitioned from the units written out as words
to their usual symbols in technical writing.
There won't be a quiz on this, at least not from me. It will come the
first time you read a technical paper and try to make sense of it.
Vectors
Simply put, a vector is a quantity (a number) with a
direction. This is in contrast to a scalar, which is just a number.
Here's how to tell the difference.
In
the picture, the scalar is the number 142. It doesn't have to be an
integer. It could have been 142.2, or 142.32549, or even the square
root of 142. But it is an amount, a quantity.
In the picture, the vector has the magnitude
(the quantity) 142, but it also has a direction. Since this is a
threedimensional picture, we have a 3D vector. So we have to specify
the direction with two different angles:
 In the horizontal direction, it is right 5 degrees from the
xaxis.
 In the vertical direction, it is up 15 degrees.
An
interesting side note: to completely specify a vector, you need as many
numbers as the dimensionality of the space. For a 3D vector, you need
three numbers; in our example, that would be {142, +5°, +15°}. For a 2D
vector, you need only two numbers. And a scalar might be considered a
onedimensional vector, where negative numbers are left and positive
numbers are right. (But that is overkill in thinking; you can do things
mathematically with a scalar that you could not do with a vector.) 
Another interesting side note: the three numbers for our 3D vector need
not be a magnitude and two angles. A very useful vector
representation
is three magnitudes, one for the "component" in each dimension. We'll
learn more about component vectors when we talk about resolving
vectors to their components.

Vectors in golf
Why would we ever need a vector? What is it
good for? After all, we can say a clubhead is traveling at 105mph, and
everybody knows what we mean. That's a scalar, a single number. So why
buy the added complication from a vector.
It turns out that is wrong!
To characterize clubhead velocty as just clubhead speed (a scalar), is
inadequate for analysis purposes. Clubhead velocity is a vector. It
consists of:
 Clubhead speed (yes, that is a scalar).
 Angle of attack (a vertical direction).
 Clubhead path (a horizontal direction).
Suppose
you want to do serious analysis, like plotting the trajectory of the
golf ball. For that, you need the full vector representation of
clubhead velocity, not just the clubhead speed. The directions will
have a lot of effect on things like launch angle and spin, which in
turn have a huge effect on the trajectory.
In golf physics, here are most of the quantities we need to remember
are vectors, not just scalars:
 Force
 Torque
 Velocity
 Acceleration
 Momentum
 Position or distance
That doesn't mean that everything is a vector. There are some things
that are inherently scalar,
for instance:
 Mass. (There is no direction involved; mass is the same
whatever direction you look at it.)
 Energy.
 Volume.
But
some things you think might be scalars turn out not to be if you
need good answers. Let's take spin, for instance. You can assign spin a
number, say 2700rpm. But that is only part of the story. You don't know
if that is pure backspin or if it includes a hook or slice component,
until you give its spin axis  the angle of tilt of the spin. So a
specification of the spin is a twodimensional vector: both a magnitude
(2700rpm) and a direction (axis tilt).
In fact, like many vectors,
there are multiple useful ways to specify it:
 Magnitude and direction, as we are used to. These are the
blue quantities in the diagram.
 Backspin and sidespin, which is just a resolution
of the spin vector into vertical and horizontal components. These are
the green and red quantities, respectively, in the diagram.
I'm not going to get into an argument about which one is correct,
in some purist sense. The point is: spin
is a vector, so choose the vector representation
that best deals with the problem you are solving. Sometimes that will
be spin and tilt, and other times it will be backspin and sidespin. And
remember, it is easy to convert back and forth between the two.
We've mentioned several times the resolution
of a vector into components.
We'll look at that soon, but first we have to know how to add vectors
 because resolution will be defined as finding the vectors
(components) we add together in order to get the original vector. 
Adding vectors
Vectors
can be added together. The result is another vector. But you can't just
add the magnitudes of the vectors. Adding vectors works differently.
Example
 Two position vectors: 10 miles north and 10 miles east. Add them! We
know they are 2D vectors because each has a magnitude
and a singlenumber direction. North and east are typically expressed
as degrees from north, so the directions would be 0° and +90°
respectively. If you want a word problem that turns into these vectors:
"I am 10 miles north of my homebase. I walk 10 miles east. What is my
new distance and direction from the homebase?"
How do we add the
two vectors. We don't just add magnitudes. The resultant vector is not
20 miles in some direction. The diagram shows how it is done. Lay the
vectors to be added in a nosetotail fashion. Then the vector from the
tail of the first to the nose of the last is the sum, called the
"resultant" vector. In this case, it is 14.1 miles directly northeast,
that is, 45°. 
General
 What
you just saw is general. It is the way
to add vectors. If you have two vectors, you can lay them
nosetotail, and the combination is the sum of the vectors.
If
you have more vectors, take them in any order and lay them
nosetotail. Again, the tail of the first to the nose of the last is
the sum of all the vectors. It will come out the same no matter what
order you take them.
The sum of a collection of vectors is
called the "resultant" vector. I'll be using that word and assuming you
know what it means. 
Subtracting
 You can also subtract vectors. "A  B" is the
same as A
plus B.
That
brings up the question: what is the negative of a vector. The answer is
simple and intuitive. It is the vector pointing in the opposite
direction.
Here are the same two vectors we used before, but now we want AB instead of A+B. We set it
up the same way, but we turn vector B to point the
opposite way to make it negative. Now we have the difference vector AB.
This
same approach works for adding and subtracting in 3D, or in fact any
number of dimensions. Just take all the 3D vectors to be added, and lay
them nosetotail in any order. (It turns out that any ordering will
give the same resultant. It is fairly easy to prove, but we won't in
this article.) 
Golf example
 Here's an example of vector addition I hope you can relate to. It
shows rather graphically the difference between a vector sum and just
adding the magnitudes of all the vectors.
You are on the tee of a dogleg par4, and you and your friends are
puzzled. "The scorecard says this is 404 yards, but my GPS says it's
only 337." You are at the back of the tee box, so that can't be the
problem. What is wrong?
The problem is that the scorecard distance shows the way the hole was
meant to be played. But the GPS doesn't know anything about that! All
it knows is the latitudelongitude coordinates of the green and the
coordinates of where you are standing with it. So all it can measure is
the straightline distance to the hole.
Let's look at the distances as vectors. They are
vectors, because they have both a magnitude and a direction. The red
distance vectors are the way the hole has to be played. (Unless you can
hit a high fade over tall trees, carry 330yd, and land softly enough to
stay on the green.)
 The red vectors are the intended shots.
 The arithmetic sum of the magnitudes of the red vectors is the distance on the scorecard.
 The vector sum
of the red vectors is the green vector. And that is the number your GPS
is showing you  because that is all it has the information to
calculate.

Resolving vectors
I mentioned resolving
vectors earlier. It is a very important topic, so let's cover it now.
First a definition...
A
vector is "resolved" within a set of axes by finding a vector on each
axis such that they add up to the original vector. ("On each axis" can
be interpreted as "parallel to each axis.")
In the diagram, we have an [x,y]
twodimensional axis system. Resolving the dark blue vector means
finding the x
and y
component vectors such that they add up to the dark blue vector. The
two light blue vectors are the components we seek. We know they add up
to the original vector because, laid nose to tail, they touch the nose
and tail of the original vector.
The example is the standard set of 2D axes, x and y.
The axes are perpendicular to one another, and that is something we
will require when we go to resolve a vector.
The component in the x direction (or
any axis direction) is sometimes called the "projection" of the
original vector onto the xaxis. Let's
consider this for a moment. If I shine a light down, with all its beams
perpendicular to the xaxis, the
original vector casts a shadow on the xaxis. The
shadow is exactly as long as the xcomponent of
the vector. You can think of it as the vector projecting its image on
the xaxis,
hence the term "projection". That is often a good, intuitive way to
think about resolving components. 
Similarly, we can resolve a vector in three dimensions. Here we are
using the standard x,
y,
and z
axes, the usual "world view" 3D coordinates, the default axes for many
physics problems. We can immediately see the components of the dark
blue vector:
 The red vector is the x component.
 The green vector is the y component.
 The light blue vector is the z component.
If
this isn't immediately obvious to you by looking at it, then look at it
some more. It is important that you feel comfortable with resolving
vectors in order to understand much of what follows.
(Note:
If you had trouble understanding this, and some other way of looking at
it unlocked the secret for you, please let me know. I will probably
want to add that insight to this article.)

The
usual "world view" axes have the x and y axes
horizontal and perpendicular to one another, and z as the
vertical axis. But these are not the only axes you might encounter.
Here
is an example where there are two interesting sets of axes (called
"coordinate systems") on which you might want to resolve a vector.
The
vector to be resolved, the dark blue vector, is the force of the
clubface as it strikes the golf ball. The diagram shows two coordinate
systems, and both of them have uses.
 The first is the world view, horizontalvertical, xy
set of axes. You would probably use these axes if you were making a
trajectory program that was plotting the height and the distance along
the target line of the ball's travel.
 The second is a pair of
axes that are parallel and perpendicular to the clubface itself. The
force the clubface exerts on the ball includes a component
perpendicular to the face, that gives the ball most of its speed, and a
component parallel to the face. The latter, which is due to friction,
imparts spin to the ball, and also moves the launch angle away from the
clubface direction and toward the clubhead path. You might want to use
these coordinates for a study of impact to create launch conditions.
That
is all we have to say about vectors as vectors. But vectors will keep
coming up in our discussion of forces, torques, and the resulting
motion. In fact, forces, torques, and motion (position, velocity, and
acceleration) are vectors.
If you don't feel comfortable with vectors, the rest might prove
difficult. 
Time now to look at the specific vectors for kinetics: force and torque.
Last
modified  Aug 14, 2024
