Understanding 2D and 3D Elastic Collisions: Solving Analytical Problems

AI Thread Summary
The discussion focuses on the analytical challenges of solving 2D and 3D elastic collision problems, highlighting the discrepancy between the number of variables and equations. In 2D collisions, there are four variables (velocity components of two bodies) and three equations (two for momentum conservation and one for energy conservation), with a fourth equation derived from the geometry of the collision. For 3D collisions, the situation is similar, with six variables and four equations, where tangential velocities remain unchanged due to the absence of impact forces in that direction. The confusion arises from the mathematical treatment of angles in the collision equations, particularly in the 2D case. Clarification on the mathematical concepts and the application of conservation laws is sought to better understand the problem.
Omri
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Hello,

I have recently been interested in the problem of 2- and 3-dimensional elastic collisions. I just don't understand how to solve these problems analytically: in the 2D case we have 4 variables (x,y components of the velocity times 2 bodies) and only 3 equations (2 conservation of momentum, 1 conservation of energy); in the 3D case (similarly) we have 6 variables and only 4 equtions.
I ran across this page: http://www.plasmaphysics.org.uk/collision2d.htm
but I stopped understanding when they started talking about theta as the sum of two other angles.
I would be happy if somebody could explain it to me.

Thanks a lot! :smile:
 
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In the 2D case, the "fourth equation" comes from the fact that you know the direction of the momentum change, from the geometry of the collision.

Resolve the velocities along the line of impact and tangential to it. The two tangential velocity components don't change, because there is no impact force in the tangential direction.

Apply conservation of momentum and energy along the line of impact: that gives two equations to find the other two velocity components.

In 3D there there are no velocity changes in the plane tangent to the impact, so 4 components of velocity don't change. Again, the two conservation equations give the two velocities along the line of impact.
 
I more or less get the idea, but the equations in the webpage (again, starting from the weird angles equations) sort of confused me.
Could you please explain what happened there mathematically (I'm referring to the 2D case)?
 

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