# Some Elementary Physics

## CLUB HEAD SPEED AND SHOT DISTANCE

This should not come as a surprise. Club speed is related directly to the distance of a golf shot. The faster the club moves, the farther the ball will go.

The graph below relates distance to club head speed. The table shows a direct relationship between club speed and distance. For every 1 mile per hour increase in club speed, total distance increases by about 2.57 yards. Keep this in mind as a point of reference.

## DISTANCE TRAVELLED BY THE CLUB HEAD

There are two elements to club head speed: distance travelled and time. Speed is the distance travelled divided by the time taken to travel the distance.

We are all familiar with straight line distance. We measure straight line distance with familiar measuring devices such as our feet, our paces, rulers, yard sticks, tape measures, and range finders.

Distance around a circle can also be measured using measuring devices, but it is usually easier to calculate this distance. To do so, we need to know the circumference of the circle, and the portion of the circle traversed.

- The circumference of the circle in turn can be calculated from the radius, using the formula where the circumference is 2 times the radius of the circle (straight line distance from the centre of circle to the circumference) times the constant pi (approximately 3.1416).
- The portion of the circle traversed can be calculated from the degrees of rotation in the downswing, divided by number of degrees in the circle (360).

Almost all movements in the golf swing, taken individually, cause the club head to move around the circumference of a circle

Determining the radius of the circle traversed by the club head through particular movements (taken in isolation) is more complicated, and involves two notions: circle radius, and distance from the circle centre to the ball. In some cases, the circle radius is the distance from the circle centre to the ball. In other cases, it is different.

To illustrate the difference, picture a stick hanging down from a shaft that can rotate vertically to the ground. If the shaft rotates very slowly, the stick appears to not move but continues to hang vertically to the ground. The distance from the circle centre to the tip of the stick hanging from the shaft is the length of the stick, but the radius of the circle traced out by the hanging end is zero.

If the shaft rotates sufficiently quickly, the stick will move in a circle perpendicular to the shaft and horizontal to the ground. The radius of the circle traced out by the loose end of the stick will be the length of the stick, which is also the distance from the circle centre to the circumference.

One can imagine the situation where the shaft rotates at a rate that causes the loose end of the stick to trace a circle with a radius greater than zero but not the full length of the stock. The distance from the circle centre to the circumference of the circle traced by the loose end of the stick is the length of the stick, but the radius of the circle being traced is less than the length of the stick.

The picture below depicts what is happening. As the angle between the axis of the shaft and the stick increases from 90 degrees to 180 degrees, the circle being traced out by the loose end gets smaller. Note the difference in size between the circle with the dotted line compared with the one with the solid line. In both cases, the distance from the circle centre to the circumference is the same.

Call outs in the picture emphasize key concepts to be applied to the movements in the golf swing: the rotation centre; the axis of rotation; the distance from the rotation centre to the circle circumference traced out by the golf club and including the golf ball; and the angle between the axis of rotation and the line from the rotation centre to the golf ball at the circumference of the circle.

The picture shows that the distance to the circumference from the centre of rotation is not always the radius of the club head path/circle. The critical issue is the angle between the axis of rotation and the line from the centre of the circle to the circumference.

The sine of this angle is used to calculate the radius of the club head path/circle when the angle is from 90 to 180. The table provides the sine of various angles between 90 and 180 degrees. Note how the sine of the angle starts to drop quickly after 130 degrees.

When the angle is 90 degrees, the radius of the circle traced out by the club is the distance to the circumference from the centre of rotation. When the angle is 130 degrees, the radius of the circle traced by the club head is the distance from the circumference of the circle from the rotation centre times .766.

## CALCULATING DISTANCE TRAVELLED

A key aspect of physics is measurement, and here are some basics we will apply for the golf swing in total and individual movements:

- Lengths of arms, legs, spine and distance between body parts such as shoulders and hip joints. A tape measure works fine.
- Angles such as between the club and lower arms, the upper and lower legs, the spine relative to the vertical, etc. Pictures of the golfer and a protractor are excellent tools.
- Amounts of rotation through the various movements such as cocking and uncocking the wrists. A protractor and mirror are useful tools.
- Trigonometry can assist in calculating distances which are difficult to measure directly. For example, if one knows three of six elements (lengths or angles) of a triangle, trigonometry can calculate the remaining lengths or angles. There are triangle calculators on the internet.

## MEASURING TIME

Measuring the time required for the golf swing and its components is essential for calculating club head speed. Accurate measurements require sophisticated equipment, but a simple technique can help.

High speed video cameras typically take pictures at rates defined in terms of frames per second. Counts of the number of pictures in the downswing will let you determine the seconds required for the downswing.

An analysis of all the frames during the downswing can generate a great deal of time related information, including duration of the downswing, the duration of particular movements within the downswing, the number of degrees of rotation of the club during the frame and the relationship between degrees of club rotation and time in the downswing.

The table below illustrates the results of a detailed analysis of the golf swing pictures from a typical high definition camera. The results have been incorporated into the golf swing model.

Observations from the table include:

- The downswing has taken 11 frames and the camera takes 27 frames per second, so the elapsed time is (11/27 =) 0.407 seconds.
- The club begins the downswing horizontal to the ground, and ends vertical. From a 2 dimensional perspective, it has rotated 270 degrees.
- The number of degrees of rotation increases throughout the downswing.
- Not all movements start at once. The hips start rotating first, followed by the spine, the arms and shoulder sockets, and then the forearms and wrists.
- The rapid increase in degrees of rotation occurs because different movements are kicking into action during the downswing, but may also occur because each movement imparts some acceleration to the club head during the course of the movement.

## AVERAGE SPEED AND IMPACT SPEED

In all movements, we will be calculating the distance travelled through the movement divided by the time required for it. The result is the average speed during the movement.

More relevant than the average speed for the movement is the speed at the moment of impact.

We are not in a position to provide a definitive relationship between average speed and impact speed. However, here are some ideas:

- From the previous section, it is clear that the club head speeds up during the golf downswing. However, the previous section provides one explanation, namely that the acceleration during the downswing occurs because the various movements kick in a different times.
- One would expect that each movement in an efficient golf swing, taken in isolation, would see an increase in speed (acceleration) during the downswing, with the maximum speed occurring at impact. Specifically, for each movement in the downswing taken in isolation, the club head will begin with a speed of zero at the start of the movement, then accelerate throughout the downswing (i.e. increase its speed), leading to a peak speed at impact. After impact, the club will decelerate (i.e. decrease its speed) and end up at speed zero at the end of the movement. The graph below shows this process, and illustrates that with such a pattern, the speed at impact will be greater than the average speed. The blue curve illustrates the relationship between distance travelled by the club head and time during the downswing. At the start of the downswing, there is relatively little distance travelled per unit time. However, as time passes, the club head travels larger distances and the slope of the curve gets steeper. At impact, the distance travelled per unit time is at its maximum, and the slope of the curve is its steepest. After impact, the club head slows down, with less distance travelled per unit time. At the end of the swing, there is no further movement. The red dotted line illustrates the slope of the curve at its steepest point, namely the point of impact. The red dashed line passes through the start position (time and distance travelled = 0) and the impact point. Its slope illustrates the average club speed during the downswing. The dotted line is steeper than the dashed line. Impact speed is greater than average speed.

- One reason for acceleration in the downswing is inertia. An object (the club) will stay at rest unless a force is applied. Initially, the force works to overcome inertia. However, in the golf swing, the club is relatively light, some muscles are much stronger (legs and ankles) than others (wrists), and some movements occur in shorter time periods (uncocking wrists) than others (hip rotation).
- Attention must be paid to how muscles work. They work through signals to contract from the brain to fibers within muscles. The messages probably arrive relatively simultaneously. In other words, muscle contraction is more akin to an instantaneous process leading to rapid transition to full speed, rather than sequenced, gradually accelerating process.
- That said, the brain probably takes into consideration the risk of injury when rapid acceleration occurs at the end of a range of movement. This could lead to slow rates of contraction at the extreme end of the range of movement, followed by fast rates when the injury risks are lower.
- In addition, muscles do not generally contract at the end of their range of movement in day to day living. As a consequence, they may not be as strong or conditioned to contract rapidly .

Each movement will likely see acceleration during the downswing because of the need to overcome inertia, muscle weakness at the end of range of movement, lack of muscle conditioning at the end of the range of movement, and the sequencing of contraction instructions from the brain at the end of the range of movement. However, the acceleration may not be rapid because the club head is light relative to the strength in most muscle groups and the fact that for the most part brain messages are instantaneous.

This Guide refers to the “scalability factor”, which is impact speed divided by average speed. In essence, this Guide calculates average speed in the downswing, and recognizes that these average speeds will need to be scaled up to determine impact speed. Because of the uncertainty around scalability factor, this Guide assumes a factor of 1 for all movements and recognizes that impact speeds will likely be greater than speeds estimated herein. With a scalability factor of 1, all speed estimates are minimums.

As an observation, average speeds calculated using the golf model in this Guide are about 15 percent less than overall impact speeds calculated through golf club speed measuring devices.

## THE BASIC ROTATION EQUATION

In applying the above to the movements in the golf swing, for each movement we need to know:

- the rotation centre;
- the distance from the rotation centre to the ball;
- the angle between the line from the rotation centre to the ball and the axis of rotation;
- the amount of rotation in the downswing relative to a full circle ;
- the duration of the movement within the downswing; and
- a factor converting the average downswing speed of the club head from the movement to speed at impact.

In the following chapter, we shall explain how to estimate the club head speed from the individual movements in the swing.

The Bottom Line

- We can calculate the contribution of each movement in the swing when we know the foregoing elements.