A general collision where both objects move initially and both objects move at the end has two unknown to solve for (two different final velocities).
To be able to find those final velocities, we would need to know something about the energy of the system. The general case is beyond the level of this course to solve in full but the case where kinetic energy is exactly conserved can be solved.
This is the completely elastic collision . Even in this special scenario, we will further simplify by considering the case where object 2 is initially at rest. Then the final velocities of object 1 and 2 are
and
As described in your book there are three very important special cases.
Here is a special sequence of two perfectly elastic collisions both of type "Case C" that lead to dramatic effects at the end.
The second collision in the demo involves the tennis ball going down with speed v colliding with the basketball going up with speed v. This is not of the type where object 2 is at rest. Here is the trick though. Just go to the reference frame where object 2 does not move. To do that with must subtract a velocity v from both both objects. In that reference frame, the basketball is not moving and the tennis ball is moving down with speed 2v. Then do the collision (which is Case C, the tennis ball bounce back with speed approximately 2v upward). Then we add back the velocity we subtracted at the beginning for a final velocity of 3v. This is illustrated in this image.