Two Dimensional Collisions

## Question:

Hi Mr. Jones.
My partner and I were looking at your notes on the web and
found some interesting information about momentum.
Coincidentally, my partner and I are working on a project
involving billiard balls. We would just like to confirm that we
are interpreting your notes and equations correctly.

For the 2D equation case, we just need to use the two
equations:

v1 = v1i * (m1-m2)/(m1+m2) + v2i * (2*m2)/(m1+m2),
and

v2 = v1i * (2*m1)/(m1+m2) + v2i *
(m2-m1)/(m1+m2)

where v1,v2 represent the change in line of sight velocity
vectors. v1i and v2i can be vectors (x,y,z) formed by the initial
velocities projected onto the line of sight. v1 and v2 are then
added to the corresponding initial velocities.

I'd like to make sure our interpretation is correct and
that we can indeed use "vectors" rather than scalars
for v1i and v2i.

Thanks!

(My physics are a bit rusty =). It's been 3.5 years since
I had a course in physics.)

## Answer:

Your interpretation of the equations and concepts involved in two
dimensional collisions is essentially correct. Your statement
"use 'vectors' rather than scalars for v1i and
v2i" made me think a bit. The projection of a vector onto a
particular line is not much different than a scalar. In fact the
magnitudes of the projected vectors, which are scalars, are the
quantities used in the equations.
You might want to check out an alternative way to model
multi-dimensional collisions. In our MechLab program we create
virtual objects with spherical symmetry by a representation of
the potential energy resulting from their interaction and
actually let them collide in model space to see what happens.
This allows us to handle up to three objects in three dimensions,
a situation where the geometric approach is way too messy.

This information is brought to you by M. Casco Associates, a
company dedicated to helping humankind reach the stars through
understanding how the universe works. My name is James D. Jones.
If I can be of more help, please let me know.

JDJ