Article: Spin Doctors

Dr Ken Bray and Professor David Kerwin, at the University of Bath, have been developing a mathematical model of the trajectory of a swerving football, using Mathcad to calculate solutions to multiple differential equations covering the three-dimensional flight path. Their astoundingly accurate model may have significant implications both ball design and the tactics adopted by defenders facing the likes of David Beckham.

On 6 October 2001 England’s hopes for automatic qualification for the 2002 World Cup hung by a thread. The Greek side had led for most of the game but conceded a free kick in the dying seconds. David Beckham placed the ball, struck it with perfect control and less than a second later it was nestling in the Greek goal. England had qualified!

Supreme technical skill may have delivered the killer blow but the outcome owed as much to physics and aerodynamics as Beckham’s artistry.

The Department of Sport and Exercise Science, opened in 1999, is a new and growing department of the University of Bath. Within the Faculty of Science, it brings together biomechanics, physiology and psychology in the study of sporting activity.

Dr Ken Bray, a visiting Industrial Fellow, and Professor David Kerwin, head of the department, have been developing a mathematical model of the trajectory of a swerving football. Using Mathcad to calculate solutions to multiple differential equations covering the three-dimensional flight path, they have fitted the results to real data to establish the aerodynamic constants governing the flight of the ball.

With this approach, they have created an astoundingly accurate model, which may have significant implications for both ball design …and the tactics teams adopt for a free kick with a defensive wall.

Dr Bray explained: “There was an incredible amount of calculation involved, with three empirically determined unknowns and three second order differential equations to solve. There would have been no way of modelling this in either a biomechanical or even mathematical sense without the power of Mathcad”.

Professional and semi-professional footballers studying at Bath play for the University’s elite soccer squad, Team Bath. Within the controlled environment of an indoor sports hall, the flight of curving free kicks taken by a Team Bath player were captured using digital video cameras. The cameras recorded the flight from two angles, giving a record of the three dimensional coordinates of the ball.
A set of differential equations describing the flight was established from theoretical considerations. These incorporated terms representing the aerodynamic spin and drag coefficients, together with the implied spin axis of the ball.

Initial estimates of these variables, together with the ball’s initial velocity and direction, were fed into Mathcad to calculate the resultant flight path, which was matched with the real data. The coefficients could then be adjusted iteratively to give the best fit.

“There are many contributed differential equation solvers within Mathcad”, explained Dr Bray. “We used a Runge-Kutta to do the calculations. Not only could we develop an excellent fit between the raw data and the theoretical equations, but we were able to show that the equations deliver the correct results for certain special conditions (such as zero spin) as further evidence of their robustness.” The team is confident that they have reduced the errors to around 0.1% of the field of view – and can position the football to within 2.5cm of its actual position in a flight of some 20m. The work has provided confident determinations of the drag and spin coefficients for standard footballs, data for which has been very sparse in the open scientific literature.

Meanwhile, it is apparent that the art of set piece kicking is extraordinarily sensitive. “The best free kick takers such as David Beckham create a very high degree of controlled spin, with the axis of spin close to or even beyond the vertical. Using the model in conjunction with a correctly placed defensive wall, we can see that for a direct free kick around 20 m from goal, there is very little margin for error.
“Just a couple of degrees variation in the direction of the kick, a slight tilt in the axis of the spin, or a marginal variation in speed, would put the kick into the wall or over the bar.”

Although the results so far have been very revealing, Kerwin and Bray feel that they are only just getting to grips with the subject. They plan to extend the work into an analysis of the strategic options open to both attackers and defenders for various placements of the wall in relation to the free kick. More experimental and supporting theoretical work is on the cards, probably involving the use of a ball launching machine as well as human subjects.

Why a machine especially? “We believe this will give us much greater control over the input variables such as speed, elevation and spin. It’s very demanding of a live subject to produce repeatable conditions, kick after kick. And unfortunately, our budgets won’t stretch to David Beckham!”