# Energy conservation in elastic multi-dimenstional collisions

1. Aug 7, 2011

### randunel

Hi everyone. I believe that I'm in the right place for this topic. If not, please excuse me for the misplacement :)

I am a programmer, not a physicist, so please excuse my lack of knowledge. I currently need help with an energy conservation problem.

I am currently working on this problem: i have two discs (perfectly round, equal diameter d and equal mass distribution, but different masses m1 and m2) travelling at different translational velocities v, with different angular speeds ω. These two discs collide with each other at a random point in time t1, with a random Θ angle between their velocity vectors.

From what I remember since highschool, the total kinetic energy is conserved within an isolated system. Thus, I conserve both angular kinetic energy and translational kinetic energy:

$\frac{1}{2}$ω1i2m1d2+$\frac{1}{2}$m1v1i2 + $\frac{1}{2}$ω2i2m2d2+$\frac{1}{2}$m2v2i2 = $\frac{1}{2}$ω1f2m1d2+$\frac{1}{2}$m1v1f2 + $\frac{1}{2}$ω2f2m2d2+$\frac{1}{2}$m2v2f2
I don't know where to go from here, since i have a single equation with 4 unknown variables.

But, looking at the problem from another angle, I was trying to conserve the total energy:
E = Ekin + Epot = constant
Epot = $\frac{kx^{2}}{2}$
-$\frac{\delta E}{\delta x}$ = -kx = F
And this is what i'm interested in, Fx and Fy on each disc.

Having the velocity of disc1
V(r) = $\vdots \stackrel{0,d_{12}>d}{\frac{k(d_{12}-d)^{2}}{2},d_{12}\leq d}$
where d12 = distance between centers of discs 1 and 2
We can derivate
F1x = -$\frac{\delta v}{\delta x}$ = -$\frac{\delta}{\delta x} [ \frac{k(d_{12}-d)^{2}}{2} ]$ = k(r12-d) * $\frac{\delta}{\delta x} ( d_{12}-d )$

And so on, typing formulas in here is really time-consuming.

Am I going in the right place with this? Or am I too off-course, away from my target?
I just want the final velocities, translation and rotation (or accelerations of some sort).

Last edited: Aug 7, 2011
2. Aug 8, 2011

### randunel

Is there anyone you'd recommend me to go to and maybe send a private message to?

3. Aug 8, 2011

### xts

1. You should rather say: I conserve sum of angular kinetic energy and translational kinetic energy: Anyway in such collision kinetic energy is not conserved - some must be dissipated on friction
2. Formulas you used for angular kinetic energy are wrong, but (as 1. was wrong) it doesn't matter
3. Thus you need 4 (not 3) more equations:
3.a. two of momentum conservation (only two, as the problem is two-dimensional)
3.b. Constraint that at the point of collision velocities of points at edges of both discs must be the same
3.c. Total angular momentum must be conserved.

4. Aug 8, 2011

### randunel

Thank you for the reply, I will try to work with a,b,c from here on, I'll post back with the results (if any, since i can hardly remember the highschool physics :D).