Figure 20.
(a) The task is to move the mass from x = 0 to xmax in time tmax, starting and ending at rest. (b) The optimal strategy is OABC, OA is instantaneous, AB is constant speed and BC is instantaneous. The series of horizontal lines are the energy contours. A non-optimal path will necessarily have to intersect a higher energy contour than AB, thus requiring greater positive work.