A Perfect PenToss:
How to Survive the MCM!
By: Ann Monville

Welcome to my Math Modeling project page!  This project
was done for my math modeling class at California Lutheran
University, Spring 2001!

How My Project Was Born

I came up with this project while competeing in the MCM.  MCM stands for....
Myself along with two of my classmates were in a room for four continous days working
on our problem for the competition.  While in this room we once again tossed pens onto
the whiteboard tray to talk breaks away from our problem.  Tossing pens has been a
ritual for math majors/minors before exams.  Therefore, I wanted to put a mathematical
approach to one of our favorite pastinmes.

Assumptions for My Model

~ The pen is thrown two meters from the board ( X=2 meters )
~ The pen is released at the same height as the whiteboard tray ( Y=0 meters )
~ There is no air resistance
~ All pens have the same bounce

My Model

I started out using projectile motion equations:
X=Vo cos (q) t
Y=. 5 g t2+Vo sin (q) t
Where:
Vo is the velocity
q is the angle in which the pen is thrown
g is the gravity which is -9.8 m/s
X=2, Y=0
t is the time
Vo cos (q) is the velocity in X direction
Vo sin (q) is the velocity in Y direction

From these equations I got the Vo/theta relationship which means at a certain velocity
what two angles the pen can be thrown at.
Then I got the max height of the pen to make sure the height is appropriate for a
standard room.  Then these equations told me where the pen hits: the tray, the board, or
the ground.  I did not work with the ground because that means that the pen cannot land
on the tray because it did not make it as far at the tray.

Pen Hitting the Tray First

I experimented with throwing pens and seeing how much they boundce.  I found that
a good approximation would be 10% of the starting velocity, so that is what I used as my
bounce factor.
Using geometry, I found that theta is equal to theta used in the first set of equations ( as
long as the tray is the same height as the release of the pen ).
So the bounce factor equations are:
Xb(t)= Vo (.9)cos (q) t +X final
Yb(t)= -. 5 g t2+Vo (.9) sin (q) t + Y final
Where X final is .02 and Y final is 0.  This means that the pen hits the tray at .02 away
from the board and Y final is zero because the pen is the same height as it was when it
was thrown.
From these equations, I was abel to determine if the pen would land back on the tray
using different velocities.  As the velocities increase you need to throw the pen at a
smaller angle for the pen to land back on the tray.

Pen Hitting the Board First

I used the same approximation of 10% of the starting velocity as my bounce factor
again.  This time the angle coming in is different then the angle thrown (theta).  The angle
coming in is alpha (a) and the angle after hitting the board is b.  I solved for b, using the
slop which is -1/(tan b).  Then using geometry, I found these bounce factor equations to
be:
Xb(t)= Vo (.9)sin (b) t +.02
Yb(t)= -. 5 g t2+Vo (.9) cos (b) t
Notice that in the X direction sin (b) is being used and in the Y direction cos (b) is being
used.  That is because of the angle being flipped around from the x direction to the y
direction.
From these equations, I was able to determin if the pen would land onto the tray at
different velocities.  As the velocity increases, the pen needs to be thrown at a higher
angle for the pen to land onto the tray.

Results

The angle you should use for a certain velocity depends on if you are going to hit the
tray or the board first.  If you are going to hit the tray first, you should use a smaller angle
the if you hit the board first given a certain velocity.

The following graph is hitting the tray first:

This graph shows that at a certain velocity the pen will land back on the tray only if you throw the pen at a                       smaller angle which is theta.

This following graph is hitting the board first:

This graph shows that at a certain velocity the pen will land onto the tray only if you throw the pen at
a higher  angle which is theta.

Special Thanks to:

Dr. Wyels
Dr. Stanley
Classmates