A gymnastic swing back and forth on the end of a 4 m long roap that is attached to the ceiling of a gymnasium. If, at the highest point of each swing, she is 1.5 m from the celing, what is her maximum speed?

Use conservation of energy principles \Delta KE + \ Delta PE = 0. At the top of the are she starts with PE= mgh and velocity = 0, hence her initial KE = 0. The potential energy gets turned into kinetic energy as she swings downward and reaches the lowest point of the arc. So the change in PE as she moves from top to botom is -mgh, and the change in KE is (1/2) mv^2.