Final answer:
The question involves applying physics principles such as energy and momentum conservation in scenarios including projectile motion and collisions, as well as the behavior of rolling objects.
Step-by-step explanation:
Projectile Motion and Momentum Conservation
The question pertains to the discipline of physics and specifically deals with concepts of projectile motion, momentum conservation, and energy conservation in the context of high school physics problems.
In the first scenario, given the heights 'a' and 'b', we would use energy conservation to find the velocity of the sphere as it leaves the curved surface. Then, by applying projectile motion equations, we can determine the speed at point C and the horizontal distance 'c'.
In the second scenario, the conservation of momentum is used to find the change in center-of-mass velocity when two masses collide and stick together.
In case of the billiard ball collision, we use conservation of momentum in two dimensions to find the velocity of the second billiard ball post-impact. Lastly, for rolling objects like the hoop and hollow sphere, we calculate the work required to stop them, and their final displacement on an incline, utilizing concepts of rotational kinematics and dynamics.