Nugatory said:
The orientation of the axes is a choice that you make, so there’s nothing to calculate. Just set ##\kappa## to zero and proceed.
Be aware that if your objective is to calculate the angle at which the two objects rebound after the collision, then the problem is underconstrained - energy and momentum are conserved no matter what the rebound angle is.
The original diagram before changing to the zero momentum frame (ZMF) is attached as the first figure to this reply. In order to understand how to switch to the zero momentum frame (ZMF) in two dimensions, I viewed a video by Dr. Ben Yelverton (see reference video below.). The second attachment is a snippet of how he setup the problem. In the first attachment it seems to me that if the momentum of stream #1 is greater than that of stream #2, the center of mass of the incoming streams would move along a line as shown in my original post. I know I didn't state that the drawing assumed that the momentum of stream #1 is greater than stream #2 in the original post.
However, it does seem that I could rotate the resulting transformation to be on the x-axis. But that should then rotate the outgoing streams as well (which outgoing streams I am not considering at the moment because I am trying to understand how this ZMF transformation works).
The third diagram is from a solution for a two dimensional problem [
https://phys.libretexts.org/Bookshe..._Collisions_in_Center-of-Mass_Reference_Frame].
However, here, the incoming spheres are initially along the x-axis (one at rest). The solution continues on as follows:
Note: I know this is solution is for the angles at which the objects rebound, but I thought I could use this technique for the incoming stream angle as shown in my original post.
$$v_{1,f}^{'} = v_{1,f} - v_{cm}$$
[note: I don't know how to add the vector symbols above variables in latex].
Then, the next equations are,
$$v_{1,f}cos\theta_{1,i} = v_{1,f}^{'}cos\phi_{cm} - v_{cm}$$
$$v_{1,f}sin\theta_{1,i} = v_{1,f}^{'}sin\phi_{cm}$$
NOTE: I don't know what the latex representation for the center of mass angle shown on the drawing is, so I substituted in ##\phi##.
First off, I thought to get the final velocity, that ##v_{cm}## is added in the equation (but perhaps this is more a question of convention). An even bigger question is: shouldn't component velocities of ##v_{cm}## be used from my original post drawing for both equations? And that's where I got confused. Also, I don't know why the two equations above use ##\theta_{1,i}## when the drawing shows ##\theta_{1,f}##.
The momentum equation here is of no help because it equals zero which means that the angle will cancel out on both sides.