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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.

Fisiquin
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1 Answers1

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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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sammy gerbil
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