Read the original article in the Daily Mirror here and the original article from Sian here.
Next time you go for a game of golf, maybe don’t go with a mathematician, because you’re probably going to lose.
It turns out that there’s plenty of hidden mathematics behind a simple game of golf that many of us don’t realise.
In a Tom Rocks Maths blog, maths whizz Sian Langham talks about how much mathematics is involved in the popular sport, and it’s pretty mind blowing.
Mathematicians are able to use equations to predict the exact flight path of the ball, and how certain factors may affect it.
Factors include the speed of the swing, air temperature, and even the quality of the golf ball itself.
In the blog, Sian said: “A golf ball in flight is an example of a projectile because it follows a curved path called a parabola.”
“The shape of the curve is affected by two main forces – gravity and air resistance.”
Now it’s time for the complicated part.
With the absence of air resistance, the horizontal velocity of the golf ball stays constant through flight as there are no ‘external forces’.
She continued: “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).”
If you want the ball to travel high, you need to ensure the ‘vertical velocity’ equals zero.
This is when the ball is at its peak height, and after this point, the ball will travel downwards towards the ground.
Basically, the angle in which the ball is hit will affect the shape of the path.
Sian explained: “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.”
You need to decide which situation you need, depending on the terrain and the type of shot.
In terms of which angle, the maximum height and range is usually 45 degrees, so be sure to jot that one down.
For those who want to get really mathematical, Sian explains how to work out the height and velocity of the ball.
She explained: “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
To simplify: “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.”
“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.”
Granted, this is the simplistic scenario, but is a great way of analysing projectiles and taking into account air resistance impact.
In fact, that was why dimples were added to golf balls 1905.