I am a little confused about how mechanical energy conservation operates when it comes to things like predicting velocity. I know that if conservative forces are the only forces acting on a body, then we can say that mechanical energy is conserved. This is simple to see when we have lateral up and down motion, but when it comes to predicting the velocity of a pendulum or a roller coaster (neglecting all friction) I'm not sure how the law operates. For example, given the initial peak height of the roller coaster, I can predict the velocity at any point, despite the fact that there are various loops and curves. And for a pendulum, the motion is in an arc. Despite these complexities, the same equations used for these situations are used for simple free-falling situations. Could someone give me a deeper understanding of how these equations are able to make predictions about velocity and such in complex situations like riding a roller coaster?