The Penalty Kick Equation

By on 01/07/2017

Naked Maths is a brand new project from Naked Scientist Tom Crawford that looks at the maths that’s all around us…

With the ongoing semi-finals of the Confederations Cup and the Under-21 European Championship both being decided by penalty shootouts, I thought I would help out all the players who managed to miss their kicks by explaining the maths behind the perfect penalty. The maths equation which describes the movement of the ball is given by:

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The term on the left-hand side, D, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.

So, what are the inputs on the right-hand side? The symbol pi just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that pi will pop up in the equations – it’s sort of its job. The ball itself is represented by R which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by m. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by r, which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.

Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by v. The distance the ball has moved in the direction it is kicked is given by x, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is w – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so w will be small, but for Beckham – aka the king of curl- w will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…

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So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…

A version of this article originally appeared on tomrocksmaths.com where you can find all of the material by the Naked Mathematician Tom Crawford. You can also follow him on Facebook, Twitter, Youtube and Instagram @tomrocksmaths.

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