Examining the motion of the roller coaster qualitatively, we see that there is a period of deceleration that starts when the leading car enters the loop. The deceleration continues until the entire loop is traversed. The coaster then continues at constant speed until the tail of the train enters the loop. There is a period of acceleration that ends when the tail of the train leaves the loop. The coaster then continues at constant speed, v, equal to the speed before the loop was encountered.
To simplify the analysis, I will ignore the fact that the roller coaster is neither a point mass nor is any car a point mass. That is to say, I will ignore the energy of rotation of the individual cars which make a rotation as they traverse the loop. I will treat the roller coaster as a length of rope. So, sometimes I will...