Nonholonomic wheeled mobile robots are often required to implement algorithms designed for holonomic kinematic systems. This creates a velocity tracking problem for the actual wheeled mobile robot. In this paper, we investigate the issue of tracking the desired velocity in the least amount of time, for a differential drive nonholonomic wheeled mobile robot. If the desired velocity is a constant, the Pontryagin Maximum Principle can be used to design a control. A control is designed for the cases when the wheel can be commanded speeds and torques. When the desired velocity is smoothly time-varying, we propose a hybrid structure and study its properties.