**Nicole Imhof**

**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.****
**

**Introduction**

** Process**

**Assumption**
**Bottom
Corner**
**Top
Corner**
**What
is Alpha???**
**What
is Theta???**

**Bottom
Corner Results**

**Top
Corner Results**
**Conclusions**

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

**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).**

**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.**

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**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.**

**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.**

**Conclusions**
**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!!!
Smile! :)**