The rigid-body motion in a minimum time, between two specified positions in a vertical plane, is considered for the specified value of the initial mechanical energy. The problem is formulated and solved in a closed form, which is a contribution of this paper, considering non-linear differential equations of the two-point boundary value problem of Pontryagin’s maximum principle. It is shown that the solution thus obtained also represents global minimum time for motion. In the spirit of the classical brachistochrone problem of the particle the realization of this motion is also achieved exclusively by ideal mechanical constraints but without the action of active forces. Parametric equations of a moving and fixed centroid, as well as the laws of change of the tangential and normal component of the constraint reaction, which occurs in rolling without slip, are obtained in the analytical form. Based on these laws, the dependence of the coefficient of sliding friction on time is obtained. Maximum value of this coefficient must be smaller than the Coulomb coefficient of friction for the case of real rough surface. If this is not satisfied, an appropriate optimal control problem is formulated, whose solution yields constant value of the friction coefficient equal to the Coulomb coefficient, over some of the time intervals of motion.