Coefficient of Restitution for a Hypothetical Collision where masses are not equal after the collision compared to before the collision

[rdemyan]

Homework Statement

Determine the equation for the coefficient of restitution for a hypothetical collision where the masses after the collision are not equal to the masses before the collision

Relevant Equations

See post

I'm trying to determine the equation for the coefficient of restitution for the hypothetical case shown in the drawing. The big difference between this and the typical oblique collision problem is that the masses are not the same after the collision as before the collision. BUT, the sum of the masses is the same so,

$$m_1 + m_2 = m_f + m_b$$

If the masses after the collision were equal to the masses before the collision, then the coefficient of restitution would be defined as,

$$e = \frac{v_{2fy} -v_{1fy}}{u_{1y} - u_{2y}}$$

So my question is, how might I redefine "e" based on the masses differing after the collision, but the sum of those masses after the collision is still equal to the sum of the masses before the collision.

Also, the diagram shows that the velocity for both $m_b$ and $m_f$ are the same (as opposed to $v_{1f}$ and $v_{2f}$). That is probably an assumption I will make once I am able to start solving for the variables. But right now I need another equation which for an oblique collision is usually the coefficient of restitution equation. So I need to modify the typical equation for "e" (where masses remain the same) to the problem shown in the diagram.

 

[haruspex]

I would consider energy.

 

[rdemyan]

Assuming a perfectly elastic collision, all energy does for me is to calculate a final velocity, which I think I would have to assume applies to both $m_b$ and $m_f$. What I think I need is some direct relationship of the final component velocities to the initial component velocities

 

[haruspex]

Assuming a perfectly elastic collision, all energy does for me is to calculate a final velocity, which I think I would have to assume applies to both $m_b$ and $m_f$. What I think I need is some direct relationship of the final component velocities to the initial component velocities

Consider a simple head on collision. In the usual case, e is directly related to the fraction of KE remaining. What is that relationship?
I am suggesting that you could define e in this mass transfer case such that the same relationship holds.

 

[rdemyan]

Okay, the fraction of energy lost is equal to $1-e^2$, but I still don't see how this is going to help me calculate the scattering angles shown in the figure. I didn't state it in my original post, but my goal is to calculate the scattering angles. I think that in order to determine the scattering angles I have to have component velocities. This statement is based on videos I have been watching on how to solve an oblique collision problem. Of course these videos only consider the case where the masses remain unchanged. I already know what the velocity of the exiting stream in the forward direction (for $m_f$) is and I am going to assume that it is the same in the backward direction (for $m_b$). While I realize that my question didn't ask for the scattering angles, I just don't see how this will help in the calculation of $\alpha, \phi$.

 

[haruspex]

I didn't state it in my original post, but my goal is to calculate the scattering angles.

The forces that act between two bodies in contact are friction in the plane of contact and the "normal" force, i.e. normal to the plane of contact. So ignoring friction, the change in their velocities is normal to the plane of contact. There is no change to their velocities parallel to that plane.
This means that for what you are trying to do you can reduce the collision to one dimension.

 

[berkeman]

[Mentor Note: Because this question is more related to the OP's work, he has started a new thread in the technical Classic Physics forum about it: https://www.physicsforums.com/threa...re-masses-change-after-the-collision.1079687/ ]