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

#Assumptions for My Model

#My Model

#Pen Hitting the Tray First

#Pen Hitting the Board First

#Results

#Special Thanks

#CLU Math Links

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.

~ 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

I started out using projectile motion equations:

X=Vo cos (q)
t

Y=. 5 g t^{2}+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.

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:

X_{b}(t)= Vo (.9)cos (q)
t +X final

Y_{b}(t)= -. 5 g t^{2}+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.

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:

X_{b}(t)= Vo (.9)sin (b)
t +.02

Y_{b}(t)= -. 5 g t^{2}+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.

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.

**Math Modeling page link is:**

http://www.clunet.edu/Academic_Programs/Departments/Mathematics/Math/Frm_Students.html

http://www.clunet.edu/Academic_Programs/Departments/Mathematics/Math/CLU.html