Dynamics of a Soccer Ball

Brian Fredericks and Mark Landreneau

Professor: Dr. Tamas Kalmar-Nagy

Aero 211

Due April 29, 2008

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Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Procedure
  4. Experimental Results
  5. Summary
  6. Works Cited

Introduction

Roberto Carlos is perhaps most well known for his incredible free kick against France in 1997. Carlos took a free kick from about 30 yards out from goal, smashing what appeared to be a shot vary far off target. The ball then miraculously bent back towards the goal bouncing off the inside of the post and into the goal. The shot appears to curve so unexplainably that we decided to brake down the aerodynamic forces acting on the ball and discover the reason behind one of the most incredible free kicks in soccer history.

Problem Formulation

In order to explain the free kick by Roberto Carlos, we first created a simple model to explain the rotation of a sphere.

Ball as seen from above

If we were to take the velocity of two points on the perimeter of the ball, one at the inside (Va) and the otherat an outer point (Vb), the velocity at point (a) would have a greater magnitude than Vb with respect to the center of the ball. The velocity at point (a) is greater due to the rotation of the ball, which causes Va to move in the same direction as the air flow where as Vb is hindered by the airflow which moves in an opposite direction. Thus the airflow is added to the magnitude and direction of Va where as Vb has an opposite direction so airflow subtracts from its speed. Bernoulli’s equation proves that at a point of higher velocity there is a decrease in pressure and at a point of slower velocity there is an increase in pressure. Therefore the rotation causes a greater pressure at point (b) than at point (a).

The aerodynamic forces of the soccer ball hold many similarities to the forces on an airfoil. Just as a higher pressure under the airfoil causes a lift force perpendicular to the drag force, the difference of pressure on the soccer ball causes a lift force from high pressure to low pressure. This resultant lift force is also referred to as the “Magnus” force. The Magnus force is responsible for the curved flight of the ball.

Procedure

In order to begin this project we needed to first find some values and parameters of the free kick. Our primary goal is to create equations of motion for the curvature of the soccer ball’s flight path so we are simply looking at the ball from above and neglecting the ball’s upward trajectory. We know that the shot was thirty meters away from the goal and two meters from the center of the field. We also know that it took approximately one second to cover the distance. We have decided for the purpose of this project that there is no wind force acting on the soccer ball. After defining these initial conditions, we constructed a free body diagram of the soccer ball that left us with only two forces, the drag force and the lift force, or Magnus force. (Appendix 1) From the FBD we were able to derive a linear balance equation. (Appendix 2) With these two forces we had the basis for our equations of motion. The equations we necessary for the drag force and lift force were:

FD = CDpV2S/2

FL = CLpV2S/2

By defining a new coordinate system we could take components for the drag and lift forces. From there we broke the velocity into x and y components and began dotting the drag and lift forces with i and j. This gave us our two equations of motion with respect to the velocities and accelerations in the x and y direction. Using ode45 in MatLab we solved these two non-linear differential equations with respect to position and time.

Experimental Results

The following graph shows our experiment data using ode45 in MatLab.

The first graph shows the path in the left and right direction with respect to time for the turbulent phase of the kick. It should be noted that the plot is actually going in the same direction as the kick and the graph at first looks different because of the direction of the positive axis. Due to very low coefficients of drag and lift, the turbulent phase of the kick slows down little and has almost no lift force causing the path to almost be a straight line. Around 0.2 seconds in to the kick and just past the human wall, the kick goes into a laminar phase. Now the coefficients of drag and lift are high causing the ball to slow much faster and the ball to begin hooking back towards the goal. Our experimental data shows the ball going in the net close to the center not the post like in Roberto Carlos’ kick.

Summary

Our experimental data disagrees slightly with Roberto Carlos’ actual kick. The ball goes in the net closer to the center then to the post. This can be accounted for in our assumptions that we made at the beginning of the project. We assumed the coefficient of drag to be constant during each phase while in reality they are changing with the velocity of the ball at all times. We also modeled the ball as a perfect sphere not the more oblong shape the ball would take in flight. Finally the rotation on the ball as it is kicked and the dimensions of the kick itself were approximated by the geometry of the web video. Each assumption was very slight but there is still a considerable difference between the kick and our data. This just goes to show how difficult this kick really is to pull off and how little assumptions can really have an impact.

Works Cited

  1. "Soccer Ball Physics." 1 June 1998. 20 Mar. 2008 <
  2. Dr. Tamas Kalmar-Nagy
  3. Anderson, John D. Introduction to Flight. 5th ed. New York: McGraw-Hill, 2005.
  4. Dr. Kaibin Fu, TexasA&MUniversity, Mathematics.