# Golf – Its just A Level Maths

And so the superb summer of sport rolls on. We’ve had the women’s success in the football World Cup, watched some sublime rallies at Wimbledon and thumped the Aussies to secure the First Test in the Ashes series. Now our attention swings northwards to St. Andrews, Scotland, for the Open Golf Championship.

Golf, despite what they’d have you believe with their birdies, bogeys, niblicks, yips and eagles, is really a such a simple, straight forward sport, well within the capacity of anyone doing A Level Maths.

Newton’s First Law tells us:

An object either remains at rest or continues at a constant velocity unless acted upon by an external force.

So picture the scene, the crisp white form of an unblemished golf ball highlighted against the perfectly manicured green grass of the first tee. The only thing that is going to make that ball move – the external force of Newton’s first – is the head of the golf club as it strikes the ball, and this contact is so momentary, certainly less than a second in length, but it is this moment that is the difference between success and failure.

Lets look at that “moment” (of contact) in a little more detail.  In the Mechanics 1 (M1) module, we learn that:

Impulse = Force x Time

and also:

Impulse = Change in momentum

So to increase the impulse (force x time) we can either increase the force or the time (that the club is in contact with the ball), or both.  Why do we want to increase the impulse? The impulse of the club on the ball is the same as the impulse of ball on club. The initial velocity of the ball is zero, so the initial momentum of the ball is zero, so the impulse is equal to the final momentum of the ball, and momentum is mass x velocity. The mass of the ball is constant, so the bigger the impulse, the higher the velocity of the ball as it leaves the tee, the higher the velocity the further it goes.

Still with me? Good!

So to increase the force the golfer has to swing the club faster as force = mass x acceleration

To increase the time the golfer has to keep the club in contact with the ball for as long as possible – this is what follow through is all about, keeping ball and club together for as long as possible, to increase the impulse on the ball.

And once the ball leaves the club, the only external force acting upon it is gravity, and the problem becomes a straight forward SUVAT question (we can work out its initial velocity as it leaves the club as we know the impulse, hence the momentum, and momentum equals mass x velocity.)

But what about when the ball fades to the right or draws to the left? To change direction some external force must be applied. This can only occur if the ball is rotating as it flies through the air, creating a pressure differential on either side of the ball. I admit – we are now into Bernoulli’s Principle and some undergraduate level aerodynamics. Perhaps that’s a topic for another post, another day.

The only other thing the golfer can control is the direction of the ball which will be determined by the direction of the clubhead when the ball ceases contact with it.

So, its all pretty simple, basic A Level maths. There are only three variables that dictate where the ball goes:

1. The speed of the club as it strikes the ball
2. The length of time the club is in contact with the ball
3. The direction of travel of the club as it is in contact with the ball

Three little things that take place in the blink of an eye. But it does raise one very important, challenging question:

If its so simple, how come the ball never goes where I want it to when I play the blommin’ game?

Posted 16/07/2015 at 1:09 pm | Permalink

A great article Jeff.

I think that this proves just how much a game of MILLIMETRES golf really is, and that a strike on the ball just 1 or 2 degrees ‘out’ (not square/perpendicular to the ball) can have crazy consequences.

• A Maths Teacher
Posted 16/07/2015 at 8:00 pm | Permalink

Hi Marc

You are right – a game of millimetres and a very unforgiving game, a game that amplifies and magnifies your errors – a millimetre or degree out when you strike the ball can result in many metres difference to the outcome.

And this is verging on the (mathematical) meaning of chaos – small changes in input have a big effect on the output:

Chaos Theory – the branch of mathematics that deals with complex systems whose behaviour is highly sensitive to slight changes in conditions, so that small alterations can give rise to strikingly great consequences.

I’ve never thought of golf as being chaotic, although my game is best described as random!