Last Updated on February 13, 2015 by Jimson Lee
This article is guest blogged by University lecturer Dr. Vassilios McInnes Spathopoulos, author of An Introduction to the Physics of Sports.
He also wrote The Effects of Shooting a Basketball and The Effect of Wind on Curve Running.
To read all his articles on this Blog, click here.
Scoring Goals from Free Kicks in the FIFA World Cup
In a few hours from now, a sporting event will begin that probably unites more people around the world than any other human spectacle. The football (or soccer, depending where you live) World Cup will excite populations from London to Sydney and from the biggest cities to the most remote villages. Just think of one fact that really encapsulates the global reach of football: there are more country members of the International Federation of Association Football (FIFA), than the United Nations!
Although football remains as popular as ever, in recent years there has been a lot of criticism on several aspects of the game. A common complaint is that the modern version is based more on brute force and good physical condition rather than skill and accuracy. It is of course evident, even just by watching clips from competitions of a few decades ago, that footballers are nowadays stronger and faster. Does this mean though that accuracy is still not as important for success?
Some of the most spectacular goals scored are those from a direct free kick. When players like Ronaldo step up to take one from just outside the box it is almost expected that they will at least hit the target. Commentators will say that it was a “bad” free kick if it hits the defensive wall or goes over the bar. Is it really that simple though to hit the target? A few years ago it was this question that actually got me interested in a field you could call “the physics of sports”. I knew that with the help of science and the calculating power of a simple home computer I could at least obtain a rough idea of the shooting accuracy of those big stars.
If we wish to create a simulation of a free kick, we must fully solve a set of complex equations derived from the application of physical laws. Putting it in simple terms, when a ball is in the air it is acted upon by a number of forces, such as its weight and air resistance. There are two things that will determine the actual path of the ball. The first one is the forces themselves and there is nothing the player can do about it, it is the laws of the universe that take over. The second one is what scientists call “initial conditions” i.e. the set of initial parameters with which the ball leaves the foot of the player, for example the initial speed, spin and inclination to the ground. This is where the skills of Ronaldo, literally, come into play.
Taking all the above into account and by using suitable software (for this case see www.mathworks.com), it is possible to simulate the flight of a ball. The technique that is evoked is similar to that employed by flight simulators and describing it in detail would go beyond the scope of this article 1. It is the actual results that we are more interested in.
Let us for example simulate a free kick just outside the penalty box, 20 yards (18.28m) from the goal line. If the initial speed of the ball is 25m/s, the spin rate is 7Hz (7 revolutions per second) and the inclination of the ball trajectory is 17 degrees, we see from figure 1 that the ball ends up in the top left corner of the goal.
Figure 1: Free kick with 17 degrees inclination
For the next kick, we just increase the inclination by one degree, to 18 degrees. If we do this, the ball hits the bar.
Figure 2: Free kick with 18 degrees inclination
Finally, if we reduce the inclination to 16 degrees, the ball hits the defenders’ wall (positioned at a distance of 9.15m as determined by the rules).
Figure 3: Free kick with 16 degrees inclination
It is very interesting that only a slight difference in the initial inclination given by the player modifies the final result by so much. Free kicks must be taken with an accuracy of 1 degree! In fact, by use of appropriate mathematical calculations, it can be shown that for the free kick simulated here, an error of 1 degree will produce a difference of 32cm to the height the ball will have when it crosses the goal line.
So if you see any goal scored from a direct free kick over the next few weeks, just remember what physics has revealed: only 1 degree of error would have been enough to completely change the outcome! The team that eventually wins the world cup will be strong and fast but it will also still possess the skill and talent necessary to score goals!
 For a more detailed analysis see Chapter 2 of my book “An Introduction to the Physics of Sports”.
About the Author
Dr. Vassilios McInnes Spathopoulos graduated from the University of Glasgow (UK), with a joint honours degree in Aerospace and Electronic Engineering, in 1995. The following year he completed a MSc course in Flight Dynamics at Cranfield University (UK). In 2001 he obtained his PhD from the University of Glasgow, conducting research on the validation of a rotorcraft mathematical model by means of flight testing a gyroplane. He teaches undergraduate subjects at the Department of Aircraft Technology, at the Technological Education Institute (TEI) of Chalkis, Greece. His research interests include the aerodynamics of sports balls and improving engineering education.
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