Two particles of same mass in a 2D frame collide with known initial (i) velocities. I would like to know the final (f) velocities of them after the collision.

As in any other collision, momentum is conserved after the collision. Writing in components:

$$ v_{x,1}^{i} + v_{x,2}^{i} = v_{x,1}^{f} + v_{x,2}^{f} $$ $$ v_{y,1}^{i} + v_{y,2}^{i} = v_{y,1}^{f} + v_{y,2}^{f} $$

The total energy (not the mechanical) is also conserved. K accounts for the thermal energy. $$ v_{x,1}^{2,i} + v_{y,1}^{2,i} + v_{x,2}^{2,i} + v_{y,2}^{2,i}= v_{x,1}^{2,f} + v_{y,1}^{2,f} + v_{x,2}^{2,f} + v_{y,2}^{2,f} + 2 \cdot K/m $$ I obtain with this 3 equations for 4 unknown quantities, the x and y components of the velocities of particle 1 and particle 2. How could this be solved?. What information should I add? I can only think of modelling the collision with a potential of some kind.

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  • $\begingroup$ You need more information. You have to know how inelastic the collision is. You have the extremes of perfectly elastic and perfectly inelastic, and a continuum of possibilities in between. For example, you have to know the fraction of energy lost, or how much heat was generated in the collision. $\endgroup$ – garyp Sep 26 '16 at 19:06
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    $\begingroup$ Write the equations in the center of mass frame. $\endgroup$ – Floris Sep 26 '16 at 19:11
  • $\begingroup$ My question was made by assuming that the thermal coefficient is known in advance. I know it is not realistic but I wanted to focus on the obtainton of the final velocities provided the rest is known. $\endgroup$ – Fisiquin Sep 26 '16 at 19:16
  • $\begingroup$ point masses are ideal objects, there can be no other collision between them except absolutely perpendicular ones; at any other angle, the collision will fail; point masses cannot collide in plastic manner (e.g., there can't take any sort of deformation). they can't take friction; their internal temperature is absolute zero and it's a constant. Meaning, it can't change! Meaning, there is no possibility for "inelastic collision" to occur between two point masses. $\endgroup$ – Bekim Bacaj Jul 13 '17 at 13:27

As garyp says, you need more information. Even knowing what % of KE is lost in the collision, you still don't have enough equations, because this only modifies the conservation of energy equation. You have 4 unknowns but only 3 equations. You need more equations.

The difficulty with point masses is that the direction of the force between them is ambiguous. One way you can get round this ambiguity and get more equations is to make the point masses into circles. Then there is a definite point of contact relative to the centres of mass, and the direction of the forces between the particles is along the line joining the centres. This allows you to resolve the 2D collision into one 1D collision along this line.

For further information see :

Physics of simple collisions
How to get the new direction of 2 disks colliding?
Is it possible to determine the outcome of any impact knowing only the ratio of masses?
Elastic collision in two dimensions

Circle-Circle Collision Tutorial (Eric Leong)
Collisions in 2 Dimensions (Farside Physics)

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