# Soccer Penalty Kicks... Are They Unfair to Goalkeepers?

Nicole Imhof

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.

Introduction:
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

Process:
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

Assumption:
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

Bottom Corner:  (2-dimensional)
I started out by finding the distance to the corner.  Next, I picked a speed.  Then, by using the formula,

distance = speed * time
I solved for the time it takes the ball to get from the penalty spot to the goal.  I did this situation many of times with different speeds each time.back to top of page

Top Corner:  (3-dimensional)
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).

By using these three simultaneously, I found the smallest time (t), which relates to a specific alpha ().   The time for all three equations is the same because in order to get a perfect kick, the ball has a reach the top corner for all equations at once.  I again did this for many velocities. back to top of page

What is Alpha???
As shown in the picture below, alpha is the angle from the ground to the flight of the balls path.

This line is the side-view of the goal.

What is Theta???
As shown in the diagram below, theta is the angle to the right or the left of the penalty spot.

Bottom Corner Results
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.

From the results above, it is shown that the goalkeeper does not have the chance of saving the goal.  This is true with any velocity that the shooter kicks the ball at from 12 to 22 meters per second.  Therefore, it has been shown that penalty kicks taken in the bottom corner of the goal are unfair to goalkeepers.   back to top of page

Top Corner Results
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.

As shown above, the goalkeeper only has a chance of saving the kick if it is shot really slow (12 or 13 m/s).  Other than that, the goalkeeper has no real chance of saving the goal.  Therefore, penalty kicks shot in the top corner are also unfair for goalkeepers.   back to the top

Conclusions
Overall I have found it to be that penalty kicks are unfair to goalkeepers.  Goalkeepers really have no chance of saving the kicks.

This picture above is the usual picture after a penalty shot... the kicker is cheering because she scored while the goalkeeper is upset she did not save the goal. A lot more research can be done in this area.  These are only preliminary results and should be tested many more times before a real-real conclusion is made.  Also looking at different sorts of shots can be done, especially since people usually do not do perfect penalty kicks when they are under that much pressure.  back to the top

Thank you for visiting my website.  If you have any further questions or comments please email me.  Have a great day!!!  Smile! :)