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 well-known 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 t2

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 real-world 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 add-on 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 three-dimensional 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 x-axis.
  • 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 one-dimensional 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 two-dimensional 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 single-number 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 nose-to-tail 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 nose-to-tail, and the combination is the sum of the vectors.

If you have more vectors, take them in any order and lay them nose-to-tail. 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 A-B 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 A-B.

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 nose-to-tail 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 par-4, 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 latitude-longitude coordinates of the green and the coordinates of where you are standing with it. So all it can measure is the straight-line 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] two-dimensional 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 x-axis. Let's consider this for a moment. If I shine a light down, with all its beams perpendicular to the x-axis, the original vector casts a shadow on the x-axis. The shadow is exactly as long as the x-component of the vector. You can think of it as the vector projecting its image on the x-axis, 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.
  1. The first is the world view, horizontal-vertical, x-y 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.
  2. 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