An important need for some dynamic systems is to design a periodic motion program that has a constant velocity segment for a specified time. A few examples of such systems are cam-follower systems used in continuous motion manufacturing, linear actuators, space-based scanners, and industrial robots. In this paper, the closed-form solution is given for a motion program that minimizes the cycle time subject to user-specified limits on positive and negative acceleration and jerk. The main benefit of minimizing cycle time is to maximize the throughput. Two motion programs that address the problem are presented and critically examined. For general applicability, the solution is presented in dimensionless form and an example is given to show its implementation to a typical problem. Conclusions regarding the profiles are drawn and given.