The onset of Taylor-Gortler vortices in impulsively accelerating Couette flows was analyzed by using the energy method. This model considers the growth rate of the kinetic energy of the base state and also that of disturbances. In the present system the primary transient Couette flow is laminar, but for the Reynolds number Re>Rec secondary motion sets in at a certain time. For Re>Rec the dimensionless critical time to mark the onset of vortex instabilities, τc, is presented as a function of Re. It is found that the predicted τc-value is much smaller than experimental detectiontime of first observable secondary motion. Therefore, small disturbances initiated at τ c evidently require some growth period until they are detected experimentally. Since the present system is a rather simple one, the results will be helpful in comparing available stability models.