I am working on finding the general solution to a disc colliding with a thin spinning rod in two dimensions floating in free space. The collision is perfectly elastic. The width of the rod is negligible.
Shown below is the situation and known variables. Omega is the angular velocity of the rod spinning about its center. Theta is the orientation of the rod with respect to horizontal. l is the length of the rod and d is the diameter of the disc. r is the coordinate for the center of mass of each object.
There are 5 variables that change after the instantaneous collision: Vx1, Vy1, Vx2, Vy2, and omega, so there must be 5 equations to completely determine the final state. I can only think of 4: 1) conservation of energy, 2) conservation of linear momentum x and 3) y, and 4) conservation of angular momentum. I think the 5th equation has to do with the orientation of the ball and rod like how far the ball strikes from the rod's center and what the rod's current angle is, but I cannot think of it. What is the 5th equation to make it generally solvable? Am I missing something somewhere else?
Context: I am working on a computer simulation and need to solve the equation generally to work with any conditions. I am fully aware that the equations will be disgusting.