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).

 

[PJC01]

Hello! I can definitely help you with this problem. First of all, you are correct in stating that the total kinetic energy is conserved in an isolated system. This means that the sum of the kinetic energies of the two discs before the collision must equal the sum of the kinetic energies after the collision. However, as you pointed out, there are multiple unknown variables in this equation, so we need to approach the problem from a different angle.

Your idea of conserving the total energy is a good one. However, it is important to note that the potential energy term in this problem is not relevant, as there are no external forces acting on the system. Therefore, we can focus solely on the kinetic energy.

To solve for the final velocities of the discs, we can use the conservation of momentum in both the x and y directions. This means that the sum of the momenta of the discs before the collision must equal the sum of the momenta after the collision. Since we have two unknown variables (final velocities), we will need to set up two equations to solve for them.

In the x direction, we have:

m1v1i + m2v2i = m1v1f + m2v2f

In the y direction, we have:

m1v1i + m2v2i = m1v1f + m2v2f

From these two equations, we can solve for the final velocities v1f and v2f. Once we have these values, we can use the conservation of energy equation you provided to solve for the final angular velocities.

I would also recommend checking your equations for the forces in the x and y directions. It seems like you may have made a mistake in your derivation, as the force should be proportional to the distance between the discs (d12), not the difference between the distance and the disc diameter.

I hope this helps and good luck with your problem!