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Step 1: FBD
Draw a free-body diagram particles A and B, along with the bar, altogether. Is angular momentum for this system conserved? Why, or why not?
Step 2: Kinetics - Angular impulse/momentum
Write down the angular momentum for A, B and the bar individually, and then add together. For a particle A, for example, recall that angular momentum is given by HO = m rA/O x vA. For a rigid body, HO = IOω. You will also need the coefficient of restitution equation relating the radial (normal components here) components of velocity of A and B during impact.
Step 3: Kinematics
For writing down the velocity of points A and B, a polar description is recommended. For example, vA = R_dot eR + Rωeθ.
Step 4: Solve
Solve your equations from Steps 2 and 3 for the velocity of P. Do not forget that A will have both radial and transverse components of velocity.