Hello and thanks for coming to this page!
This page is about a mathematical modeling project for my senior capstone class. The reason I chose this project is because I have been playing soccer most of my life. Sadly I still can't juggle as well as the little girl above, but I have improved. I simplified my project down to penalty kicks because I have always assumed that goalkeepers do not really have a chance of saving the goal. So, this project was basically designed to determine if my assumption is correct.
What is Alpha???
What is Theta???
Bottom Corner Results
Top Corner Results
Do you remember seeing this picture in the newspapers or on magazine covers?
This picture is of
Brandi Chastain celebrating after she scored the winning penalty kick against
China in the 1999 Women's World Cup. Penalty kicks don't only occur
at the end of a game if it is tied, they also occur during the game if
there is a foul in the penalty box. back
to top of page
The way I went about doing this is I took what people consider to be the best penalty kicks, which is the top or bottom corner, and found the time for the ball to go into the goal. I varied the velocity of kicks from 12 to 22 meters per second, which then varied the times. After that I had a goalkeeper act as though she was saving top or bottom corner penalty shots. With a stop watch, I timed how long it took her to get to the corners. In the end, I just compared these times. back to top of page
These are a little bit silly of assumptions because it makes the problem quite unrealistic, but it just gives things for people to work on in the future. First, I have that when the ball is on the ground, there is no friction with the ball and the grass. Next, I am assuming that when the penalty kick is taken, it is a perfect day. The sun is shining and there is no wind blowing... kind of like the movies, I guess. Also I assumed that if I would tell someone the exact measurements and degrees of how to kick the soccer ball, they could do it. back to top of page
I started out by finding the distance to the corner. Next, I picked a speed. Then, by using the formula,
The top corner is a bit more complicated. I have three equations, one for moving the ball right or left of the center (x), another for getting the ball from the penalty spot to the goal (y), and the third for getting the ball that high (z).
As shown in the picture below, alpha is the angle from the ground to the flight of the balls path.
line is the side-view of the goal.
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As shown in the diagram below, theta is the angle to the right or the left of the penalty spot.
The graph below shows my results. The blue line is the time it took the ball to go into the goal from the penalty spot. The pink and yellow lines are the times for the goalkeeper to get to the corners. Each goalkeeper has a good side, the side they are good at diving to, and a bad side. The pink is the good side while the yellow is the time for the bad side.
Below is shown the results from a penalty kick shot to the top corner of the goal. Here there is an extra line compared to the bottom corner. That line is the blue line which represents alpha, the angle off the ground that the ball is kicked at. So, there are more requirements when kicking it in the top corner. As the velocity changes, time (the pink line) and alpha change also. But the time for the goalkeeper to get to the corners on her good side (yellow) and her bad side (light blue) remains constant as the velocity changes.
Overall I have found it to be that penalty kicks are unfair to goalkeepers. Goalkeepers really have no chance of saving the kicks.
Thank you for visiting
my website. If you have any further questions or comments please
email me. Have a great day!!!