Energy conservation in elastic multi-dimenstional collisions

[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) traveling 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 high school, the total kinetic energy is conserved within an isolated system. Thus, I conserve both angular kinetic energy and translational kinetic energy:

$\frac{1}{2}$ω_1i²m₁d²+$\frac{1}{2}$m₁v_1i² + $\frac{1}{2}$ω_2i²m₂d²+$\frac{1}{2}$m₂v_2i² = $\frac{1}{2}$ω_1f²m₁d²+$\frac{1}{2}$m₁v_1f² + $\frac{1}{2}$ω_2f²m₂d²+$\frac{1}{2}$m₂v_2f²
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 = E_kin + E_pot = constant
E_pot = $\frac{kx^{2}}{2}$
-$\frac{\delta E}{\delta x}$ = -kx = F
And this is what I'm interested in, F_x and F_y 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
F_1x = -$\frac{\delta v}{\delta x}$ = -$\frac{\delta}{\delta x} [ \frac{k(d_{12}-d)^{2}}{2} ]$ = 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).

 

[randunel]

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

 

[xts]

Thus, I conserve both angular kinetic energy and translational kinetic energy:
$\frac{1}{2}$ω_1i²m₁d²+$\frac{1}{2}$m₁v_1i² + $\frac{1}{2}$ω_2i²m₂d²+$\frac{1}{2}$m₂v_2i² = $\frac{1}{2}$ω_1f²m₁d²+$\frac{1}{2}$m₁v_1f² + $\frac{1}{2}$ω_2f²m₂d²+$\frac{1}{2}$m₂v_2f²
I don't know where to go from here, since i have a single equation with 4 unknown variables.

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.

 

[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 high school physics :D).