Golf and Projectile Motion

Sian Langham

Why do we study Maths? A question I’m sure most of us have asked ourselves at some point – maybe it was when studying long division in school or when trying to help your child with their maths homework for the umpteenth time… What’s the purpose of it all? Well, let’s start simple. We need to use arithmetic when adding up the cost of our groceries in a supermarket, or percentages when working out the sale price of a discounted shirt. Delving a little deeper you will in fact discover that maths can be found in the most unexpected places, and in this series of articles I plan to explore some of these in detail to give you the perfect answer next time someone asks “why do we study maths?”.

Article 1: Binary Code and Storing Music on Computers

Next up, let’s talk about golf…

There is lots of maths behind a simple game of golf the average person doesn’t consider. Mathematicians can use equations to predict the flight path of the ball and how it is affected by factors such as air temperature, the speed of the swing and the quality of the golf ball.

A golf ball in flight is an example of a projectile because it follows a curved path called a parabola. An example of a parabola is shown below. The shape of the curve is affected by two main forces, gravity and air resistance. Let’s assume air resistance as negligible and just consider gravity, as this is the easiest to analyse.

Firstly, we will split the projectile motion of the ball into horizontal and vertical components. In the absence of air resistance, the horizontal velocity of the golf ball stays constant throughout the flight as there are no external forces acting on it. The vertical velocity however, is affected by gravity, so it changes over time. Acceleration due to gravity g is equal to 9.8m/s2. This acts towards earth so the velocity of any object decreases by 9.8 m/s every second when the object is travelling upwards and increases when the object is travelling downwards. For a golf ball, let’s say it’s initially hit at a vertical velocity of 49m/s. This will initially decrease as the ball is travelling upwards, and will reach zero after 5 seconds (9.8 * 5 = 49). When the vertical velocity equals zero, the ball is at its peak height. This is because after this point, the ball will begin to travel downwards as g causes the velocity to increase back towards the ground. This means the greater the initial vertical velocity is, the higher the ball travels.

Due to differences in the two components – horizontal and vertical – of the velocity, the angle at which the ball is hit will affect the shape of the path. Let’s suppose the golf ball is struck at speed U at an angle θ to the horizontal direction. This can be resolved into horizontal and vertical components using trigonometry. As you might expect, if the angle is small, the golf ball will have a large horizontal velocity and a small vertical velocity. This results in a path with a long range but a small height. If the angle is large the opposite happens, and the path has a large height and small range. The golfer must decide which situation they need depending on the terrain and the type of shot they are performing. The angle that gives the maximum height and range simultaneously is 45 degrees, which is often ideal. The type of golf club used usually determines the angle at which the ball travels. The figure below shows the flight paths of balls being hit at different angles.

Now, how do we actually measure the height and velocity of the ball? The vertical velocity can be quite accurately measured using a set of equations referred to as the equations of motion or the SUVAT equations. They are derived from a velocity time graph but are fairly easy to apply.

V = U + AT;            V2 = U2 + 2AS;        S = UT + AT2/2;              S = (U + V)*T/2

where S = displacement; U = initial velocity of the object; V = final velocity of the object; A = acceleration (usually g); T = time of flight. These equations hold true for both the vertical and horizontal components of the velocity.

When using the SUVAT equations in practice, you select an equation with one unknown and substitute in the other known values to solve for that unknown. Let’s go through an example to make it clearer. Suppose a ball is hit at an initial velocity of 40m/s at an angle of 30 degrees from the horizontal. We want to find its maximum height, range and time of flight.

First, let’s find the initial horizontal and vertical velocity. The horizontal velocity using trigonometry is cos(30)*40 = 34.6 m/s. The vertical velocity is sin(30)*40 = 20 m/s.

For the height and time we need to use the SUVAT equations. We know the initial vertical velocity U is 20 m/s; we know that at its peak height the final velocity V must be 0 as it changes direction from going up to going down; we know the acceleration A is equal to -9.8 m/s2 because it acts in the opposite direction to the motion of the ball; and we want to find the displacement S. So, the equation we should use is V2 = U2 + 2AS. If we plug in our values, 0 = 202 + 2*(-9.8)*S  ⇒  19.6*S = 400  ⇒  S = 400/19.6 = 20.4 m. So the peak height of the ball in this situation is 20.4 m.

Now we use the SUVAT equations again to find the time of flight. This time we want to use the equation V = U + AT since T is the only unknown. Substituting in our values we have 0 = 20 – 9.8*T   ⇒   T = 20/9.8 = 2.0 seconds. However, this is the time it takes to reach its peak so the total time of flight is 4.0 seconds. Finally we can use this time to work out the range using the equation speed = distance/time so for the horizontal velocity we have distance = speed x time = 34.6 x 4.0 = 138.4 m.

Of course, this is just the bare bones of the situation, but it is a great place to start when analysing projectiles. Air resistance will impact both the horizontal and vertical velocities, and in fact is the reason dimples were added to golf balls in 1905. When a golf ball travels through the air it creates an area of high pressure at the front of the ball and an area of low pressure at the back, causing drag. Adding dimples to the ball causes the air to cling to the balls surface for longer, thus reducing the size of the area of low pressure behind the ball, which in turn reduces the drag allowing the ball to travel faster. The image below shows the difference in airflow around a smooth golf ball and a golf ball with dimples. You can see that when it has dimples, the air sticks to the ball for longer, reducing the drag. 

The ideas of projectile motion discussed here can be applied to other situations such as the launch of a rocket into space and an arrow being fired from a bow. In fact most sports include a projectile of some form, and can therefore be analysed using the SUVAT equations. Think about this article next time you kick a football, throw a ball for your dog or play tennis. Projectile motion is everywhere and maths is our tool for understanding it.

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