# If you play golf with a mathematician, you will probably lose

There’s a lot of math hidden behind a simple game of golf that we don’t consider – from swing to air temperature

Next time you go golfing, maybe don’t go with a mathematician because you’re probably going to lose.

It turns out there’s a lot of math hidden behind a simple game of golf that many of us don’t realize.

In a Tom Rocks Math Blog, Math whiz Sian Langham talks about the importance of math in popular sports, and it’s pretty mind-blowing.

Mathematicians are able to use equations to predict the ball’s exact flight path and how certain factors may affect it.

Factors include swing speed, 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 is the time for the complicated part.

With no air resistance, the horizontal velocity of the golf ball remains constant during flight because there are no “external forces”.

She continued: “Vertical speed is affected by gravity though, so it changes over time. The acceleration due to gravity g is equal to 9.8 m/s2.

“It acts towards the earth so that the speed of any object decreases by 9.8 m/s every second as the object moves up and increases as the object moves down.

“For a golf ball, let’s say it is initially hit at a vertical speed of 49 m/s. This will initially decrease as the ball travels upwards and will reach zero after 5 seconds (9.8 * 5 = 49).

If you want the ball to travel high, you need to make sure the “vertical velocity” is zero.

This is when the ball is at its maximum height, and after this point the ball will descend towards the ground.

Basically, the angle at which the ball is hit will affect the shape of the path.

Sian explained, “Suppose the golf ball is hit with velocity U at an angle θ from the horizontal direction.

“This can be resolved into horizontal and vertical components using trigonometry.

“As you would expect, if the angle is small, the golf ball will have high horizontal velocity and low vertical velocity.

“This translates into a path with a long reach but a small height. If the angle is large, the opposite happens, and the path has a large height and a small span.

You have to decide what situation you need, depending on the terrain and the type of shot.

In terms of angle, the maximum height and reach is usually 45 degrees, so be sure to note that one.

For those who want to get really mathematical, Sian explains how to determine the height and speed of the ball.

She explained: “Vertical velocity can be measured quite accurately using a set of equations called equations of motion or SUVAT equations.

“They are derived from a velocity time chart but are quite 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 speed of the object; V = final speed of the object; A = acceleration (usually g); T = flight time.

“These equations are true for the vertical and horizontal components of velocity.

“Suppose a ball is hit at an initial speed of 40 m/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 speed is sin(30)*40 = 20 m/s.

Granted, that’s the simplistic scenario, but it’s a great way to analyze projectiles and factor in the impact of air resistance.

In fact, that’s why dimples were added to golf balls in 1905.