Soccer Kicks to Understand the Physics Behind Them

Analyzing the Use of the Magnus Effect in Soccer Goals Evamvid Sharma, Ifedayo Ajasa Summer Ventures in Math and Science 2015 Visual and Image Processing Dr. Rahman Tashakkori, David Kale, Sina Tashakkori, Grayson Fenwick Appalachian State University

Abstract – The goal of this research was to analyze some of the most famous soccer kicks to understand the physics behind them. The research examined in particular each player’s use of the Magnus Effect to create opportunities that may not have been possible without utilization of the Effect.

1.0 Introduction

This research studies one of the 2015 goals from the FIFA Women’s World Cup, as well as several other famous goals. All goals relate to the Magnus Effect, and several other research studies that explains its purpose. The first step was to review pre-existing literature on the subject. A paper that specifically relates to any of the goals we selected was Depeux 2010[2], which asserted that the path of a curving soccer ball was a spiral rather than a circle, and used the goal by Roberto Carlos from the Tournoi de France to demonstrate this. Other related literature includes several papers on the Magnus effect, some of those describing it in sports, and two analyses of drag coefficients for soccer balls [3, 4, and 6]. Although it is simple to view these goals on the most basic level, there was plenty more going on in the background that was not so easy to understand but worth studying. For each goal, we looked at the factors that made the shot into a goal. Most had in common human error and the Magnus Effect. After reading several articles discussing the effects of a spinning ball, we came across a theorem that explains the dramatic movement of soccer balls – The Magnus Effect. We wanted to examine the physics behind each goal in-depth. For the goals we chose, the common thread was the Magnus Effect.

2.0 Methodology

Many Americans watched the final of the 2015 FIFA Women’s World Cup. The American women were set to face the Japanese national team, in a rematch of the 2011 World Cup Final. The teams were widely considered to be equals, so it was a pleasant surprise for the American fans when Carli Lloyd scored two goals in the first five minutes. Then, in the 11th minute, her teammate Lauren Holliday scored to put the USA up 3-0. When Lloyd scored again, in the 15th minute, she did so in spectacular fashion by shooting from the midfield line. She took the shot from almost directly on the midfield line during the final match of the FIFA Women’s World Cup. The Japanese goalkeeper, Ayumi Kaihori, allowed the ball into the goal after being caught looking into the sun as she moved back from the 18-yard goal box. She tripped on her own foot, but still managed to tip the ball to the side as she fell. However, it wasn’t enough to knock it wide, and the ball went in and hit the side netting of the goal.

Figure 2-1: Carli Lloyd's Half Field Goal

Roberto Carlos is well known in the soccer community for his skill on set pieces. We examined two of his most famous goals. The first goal by Roberto Carlos that we examined is commonly known as the “banana kick”. A “banana kick” can be any kick that curves dramatically, but this goal in particular, scored in the opening match of the 1997 Tournoi de France, has acquired the name. This goal was scored off of a free kick where Carlos’ shot bent several yards around the “wall” (a group of players standing between the ball and the goal with the object of stopping the shot) and back to bounce off the inside of the post. The curve of the shot was so dramatic that the keeper, Fabian Barthez, didn’t even make an attempt for it, as he thought the ball would go wide.

Figure 2-2: Roberto Carlos’ Banana Kick Goal

The second goal by Roberto Carlos, colloquially known as the “Impossible Goal”, was for Real Madrid against Tenerife on the 21st of February 1998. Carlos took the shot from the corner just before the ball would have gone out of bounds across the goal line. Carlos kicked the ball with an anticlockwise spin that tricked the goalkeeper, Bengt Andersson, into stepping off the line, leaving him unable to reach the ball when it subsequently curved back towards the goal. The easiest way for Carlos to kick it would have put a clockwise spin that would have curved away from the goal. However, by putting an anticlockwise spin on the ball, Carlos caused it to spin towards the goal. The shot was even more deceptive for the keeper because the initial force of the ball moved back towards the goal. This goal was not subject to the analysis that the others were, but presented as an example of some of the principles that were discussed.

Figure 2-3: Roberto Carlos’ Impossible Goal The fourth goal that was analyzed in this research was David Beckham’s famous 2001 free kick against Greece that sealed England’s trip to the 2002 World Cup. 2 minutes and 40 seconds into stoppage time, England was awarded a free kick from just over 10 yards outside the goal box. Beckham’s kick curved within a couple of feet of the head of the closest member of the wall before swerving outwards to the top left corner of the Greek goal. This leveled Greece and England at 2 - 2. This tie and the result of the concurrent Germany-Finland game, also a tie, meant that England won the Qualification Group and advanced to the World Cup proper.

Figure 2-4: Beckham’s Free Kick The fifth goal we worked with was by Papiss Demba Cissé for Newcastle, against Chelsea on the 2nd of May, 2012. A teammate brought down a long throw in off of his chest near the top left corner of the goal box. It bounced towards Cissé, who struck it mid-bounce, imparting a spin that caused it to go towards the goal before flattening to go into the right side netting of the goal, over the head and outstretched arm of the Chelsea keeper, Peter Cech.

Figure 2-5: Papiss Dempa Cissé’s shot In order to clearly understand these topics we needed to study several theories that proved the physics behind Carli Lloyd’s, Roberto Carlos’, and several other goals - Reynolds Number, Drag, and Magnus Effect. Reynolds Number is a dimensionless number governed by wind speed, air density, and the shape of the object. The coefficient of drag for any object in a given environment

varies with Reynolds Number. The drag and surface friction of the object create a so-called boundary layer across the surface, which consists of a layer of fluid being pulled by the surface of the ball in the direction of its spin. Drag is created as an object flows through the air, creating a viscous wake from behind. The larger the wake the larger the drag. The size of the boundary layer, which is itself affected by Reynolds’ Number can affect the size of the wake. Overall, drag increases with faster speed, a large Reynold Number, and a thick wake. The Magnus Effect is a principle that specifically explains the curve of a soccer ball. As a ball with continual rotation moves through the air, surface friction causes a boundary layer of air that is dragged in the direction of the spin to form. On one side, the boundary layer follows the same direction as the air passing. This interaction results in no collision, meaning the air flows faster by the ball and creating a low-pressure. On the other side, the boundary layer of air flows against the air passing by. This collision reduces the speed of the air around the ball, creating a high-pressure zone. Just like the concentration differential in diffusion – high-concentration to low-concentration, the high-pressure zone forces itself onto the low-pressure zone, forcing the object in the direction of the low pressure zone [1, 5]. 2.1 Lloyd Goal Physics

To get data about the projectile motion variables of Carli Lloyd’s half field strike, we used ImageJ to measure some attributes of the shot, including Lloyd’s distance from the midfield line at the time of the shot, the height of the goalkeepers hand at the time of the tip, and the distance of the ball from the 6-yard goal box at the time of the tip. Using this data, we used some standard physics equations to find distance and time data for an idealized version of the shot.

2.2 Magnus Demonstration

After seeing a video of a person utilizing the Magnus Effect by dropping basketballs off a dam with backspin and watching them be pushed away from a 650 ft. dam by the Magnus Force, we were inspired to do our own demonstration. We used the staircase in the Anne Belk Hall of Appalachian State University to perform several drop tests with a miniature foam soccer ball spinning in the air. We used two camera angles to properly show the movement of the spinning ball as it fell through the air. One of these cameras generated high framerate footage, at 124 frames per second, that was converted to slow motion to show in greater detail the flight of the ball. Once we had enough footage, we used Windows Live Movie Maker to create a video presentation to show the bend of the ball. Though it wasn’t nearly as dramatic as the video that we took our inspiration from, the effect of the Magnus Force is still perceptible.

Figure 2-6: Test drop for Magnus Effect Figure 2-7: Bottom view of Magnus Effect Demonstration Demonstration

2.3 Python Projectile Motion Simulation

To generate a graph of an idealized projectile motion path that the Carli Lloyd shot would have taken without the effects of the Magnus force and air resistance, we used Python to create a simulation that calculated the x and y positions of a projectile given initial x and y velocities. We then used ImageJ to and Windows Live Movie Maker to calculate the flight time and velocity of the real-life kick. The simulation output y position data, and graphed and animated the path of the ball through the air. To mimic the Lloyd goal, we used the idealized physics equations to find the approximate initial x and y velocities. Watching the simulation and comparing it to the real flight path of the ball allowed us to visualize the effects of air resistance and the Magnus Force on the kick itself.

2.4 Angle Measurements

In order to quantify the effect of the Magnus Force on Beckham’s and Cissé’s goals, we used Windows Live Movie Maker and OpenShot to pull frames from the videos of the goals. Afterwards, we used ImageJ to measure the angle of the ball’s trajectory to the field lines at the beginning and at the end of its flight. The field lines were used as a reference to the vertical and horizontal positions. We then were able to calculate the approximate change in trajectory using these measurements.

Figure 2-8: Screengrab of Cissé's Strike

Figure 2-9: Measuring the angle of the initial trajectory of Cissé’s shot to the horizontal.

2.5 Angular Size Methodology

Using ImageJ, we were able to measure the distance from the ball to each post for several of the examined goals. We set up a triangle whose sides were the distances from the ball to each post and the distance between the two posts. We were then able to use the Law of Cosines to solve for the angular size/vision for each player in each scenario. It should be noted that some of the goals were not analyzed in this way because of a lack of suitable camera angles. It should also be noted that we believe there to be a significant amount of error with ImageJ because of the perspective of

the images, so all results for angular size are inconclusive. However, they are valuable for giving a general idea of the angular size of the goal to each shooter and how they compare to each other.

Fig 2-5: Angular Size Triangle Image courtesy WIRED/Conde Nast Entertainment 3.0 Results

The results shown below explain how various attributes of the Magnus Effect and projectile motion control the movement and placement of the soccer ball.

3.1 Carli Lloyd

Lloyd’s goal was an impressively well struck shot. The biggest obstacle to the goal was the Japanese goalkeeper, Ayumi Kaihori. Kaihori was caught out of position because of the rapidity of the Lloyd counterattack. She was in position while her team was attacking, at the top of the other team’s half. A bad Japanese pass allowed Carli Lloyd to steal the ball, get to the midfield line, and shoot in a span approximately 2.8 seconds. Kaihori didn’t get back quickly enough, but the situation would still have been salvageable had she not been looking into the sun as she ran backwards. Seemingly unable to see the ball, she stumbled and fell eventually letting the ball in. However, to even get to the point of goalkeeper error allowing the goal, the ball first had to be struck almost perfectly. The ball was actually kicked about 0.35 meters away from the midfield line. After Carli Lloyd kicked it, the ball spent about 2.56 seconds in the air before it was tipped by the falling Kaihori. It was almost another half second before the ball actually crossed the line, meaning the ball spent approximately 3 seconds in the air between being kicked and crossing the goal line. The ball traveled 49.4 meters before being tipped by Kaihori, and approximately another 3.7 meters before it actually crossed the line. Furthermore, Lloyd utilized the Magnus Effect. By leaning back as she struck the ball, she imparted more backspin on the ball, meaning the ball was rotating towards her as it moved away from her. This created a Magnus Force pushing up on the ball, giving it a little extra height and keeping it in the air longer. In this way, Lloyd strategically used the Magnus Effect to help get the ball in the back of the net.

3.1 Carlos Banana Kick

Carlos’ kick was one of the most dramatic uses of ball spin and the Magnus Effect ever in soccer. We couldn’t in this case calculate the exact angular size because of a dearth of camera angles of the shot that included Carlos, the ball, and the goal all in one frame, but we can draw some conclusions based on the camera angles we do have and the information from Dupeux 2010. We know that the angular size was relatively large, because Carlos was almost directly in front of the goal; he was only offset by a little bit. However, a significant portion of the goal was blocked by the wall, decreasing the effective angular size. To score this goal, Carlos took advantage of the Magnus Effect to create an opportunity that would have been impossible with a linear kick. He curved it so far around the wall that the goalkeeper did not make an attempt for the ball and a ball boy standing behind and to the side of the goal ducked his head to avoid the ball, before it bent back and went in off of the right side post of the goal. 3.2 Beckham

Beckham’s goal was one of the hardest in terms of sheer pressure on the shooter. Because it was so late in the game, missing the shot would have all but doomed England’s chances of going to the World Cup. However, in terms of the technical difficulty of the shot, it wasn’t the opportunity given or the shot itself that made this goal hard, it was the way that David Beckham chose to execute the shot. To score the goal, he had to get the ball close enough to the wall that the goalie shifted his weight in the other direction, but far enough from the wall that the man on the end of the wall couldn’t make contact with the ball and get it away from the goal area. Then, the ball had to curve out far enough that the goalkeeper wouldn’t be able to reach it should he make an attempt for it, but not so far that it missed the corner. All this meant that he had to have the perfect amount of spin on the ball in order to score, which he did. Human error also played a large part in this goal. Normal defensive strategy for free kicks calls for the wall to cover one part of the goal and for the goalkeeper to cover the other half. However, most likely because of Beckham’s reputation for bending the ball, the goalkeeper didn’t trust the wall and stayed in the middle of the goal. Most likely, he thought that Beckham might be able to curve the ball around the inside of the goal. In fact, when Beckham kicked the ball, the goal keeper was expecting the ball to go to the outside of the wall. This is apparent because he had to visibly shift his bodyweight to the other side when the ball went to the opposite corner of the goal. So, he created an opportunity for Beckham by being out of position.

Figure 3-1: Greek Goalkeeper Out Of Position 3.4 Angular Size Results After measuring “The Impossible Goal”, we found that the distance from the ball to the far post to be approximately 24 yards. The base length of the triangle otherwise known as the goal box was 8 yards, and the last side was 18.3 yards. Solving the triangle, the angular size for “The Impossible Goal was 15.4 degrees.

Figure 3-2: Angular Vision for Roberto Carlos

After measuring Papiss Cissé’s goal vs. Chelsea, we found that the distance from Cisse to the far post was approximately 20.8 yards. The base length was 8 yards, and the last side was 21.07 yards. The angular size for Cissé’s goal was 23 degrees.

Figure 3-3: Angular Vision for Papiss Cissé’s shot Because there was a wall in front of David Beckham as he was taking the free kick, we needed to take two measurements in order to solve for the angular vision. First, we needed to measure the maximum angular vision of the wall, and then find the angular vision from the goal. Once those measurements are recorded, we can subtract both, giving us the overall angular vision of the shot. The measurements of the wall showed a hypotenuse of 5.6 yards, a base of 2.5 yards, and a leg of 4.9 yards. The maximum angular vision of David Beckham’s shot with the wall was 26.5 degrees. The measurements without the wall showed a hypotenuse of 21.69 yards, a base of 8 yards, and a leg of 21.1 yards. The maximum angular vision of David Beckham’s shot without the wall was 30 degrees. The maximum angular vision of the entire shot was 3.5 degrees. Because the wall was so much closer to Beckham than the goal, the wall covered the vast majority of the goal.

Fig 3-4: Angular Size Without The Wall Fig 3-5: Angular Size of the Wall

4.0 Conclusion

Using tools such as ImageJ, OpenShot, Windows Live Movie Maker and Python, we were able to examine goals scored by Carli Lloyd, Roberto Carlos, David Beckham and Papiss Cissé. Despite having in some cases, very small angles in which to score, they all took advantage of the Magnus Effect to help them increase their chances of scoring. We reached the conclusion that the Magnus Effect is a very important strategic tool in soccer. We were also able to simulate the motion of an idealized projectile, not accounting for drag or the Magnus Effect, in Python. By doing this, we were able to highlight the difference between the reality of the Lloyd goal and the idealized projectile motion. This showed the effect of air resistance, but also the Magnus Effect in the vertical plane. Through our analysis and simulations, we were able to make somewhat clearer the strategic role of the Magnus Effect in soccer, even in situations where it would appear not to play a role. References

[1] What is the Magnus effect and how to calculate it, http://ffden- 2.phys.uaf.edu/211_fall2010.web.dir/Patrick_Brandon/what_is_the_magnus_effect.html [2] Guillaume Dupeux, “Football Curves”, Journal of Fluids and Structures, Elsevier, 2010 [3] F. Alam, “Aerodynamic Drag of Contemporary Soccer Balls”, 19th Australasian Fluid Mechanics Conference, RMIT University, Melbourne, Australia, 2014 [4] Takeshi Asai and Kazuya Seo, “Aerodynamic Drag of Modern Soccer Balls”, SpringerPlus, SpringerOpen, 2013 [5] H.M Barkla and L. J. Auchterlonie, “The Magnus or Robins Effect on Rotating Spheres”, Journal of Fluid Mechanics, Cambridge University Press, 1971 [6]John Eric Goff, “A Review of Recent Research Into Aerodynamics of Sport Projectiles”, Invited Paper, International Sports Engineering Association, 2013