You hop on the carousel horse. the attendant flips the switch and you move up and down, around and around. when the ride ends, the attendant again flips the switch. the up and down motion stops, as you continue to move in a circular motion, gradually slowing to a stop. why?

The reason why we gradualy slow untill we stop is acceleration. In this example we have two types of movement: vertical and circular.

The vertical movement does not have great speed. So when  we stop moving we do not experience great acceleration. Newton's second law states: F=m*a. This means that we do not experience great force.

The circular movement is a bit different. No matter where we stand on the carousel we have same angular speed. This means that we pass same angle in the same amount of time. But we also have linear speed. This speed depends on the distance from the center of carousel. If we are close to the center we have smaller speed than when we stand at the far end of carousel. If we stopped without slowing gradually we would experience great acceleration thus meaning we would experience great force. Sometimes this force could cause injuries.