Final answer:
In an inelastic collision, total momentum is conserved, but individual velocities are not. For two billiard balls colliding inelastically, the final velocity is the vector sum of the initial momenta divided by the total mass.
Step-by-step explanation:
The question involves a physics concept dealing with inelastic collisions. In a two-dimensional inelastic collision, the individual velocities of the colliding objects (in this case, billiard balls a and b) are not conserved separately, but the total momentum of the system is conserved. Given that ball a travels at 6 m/s at 65 degrees and ball b at 2 m/s at 120 degrees with both having a mass of 5 kg, we can calculate their velocities after collision based on the conservation of momentum. However, since they collide inelastically, they stick together, and the final velocity would be a single vector resulting from the vector sum of their initial momenta divided by the total mass. Specifically, the problem requires solving for the combined velocity vector after the collision, using the principles of conservation of momentum in the x and y directions. Without actual computation, we cannot give a specific numerical answer; the question asks us to apply these physical principles to find the resulting velocity.