A potential acceleration of a quantum open system is of fundamental interest in quantum computation,quantum communication,and quantum metrology.In this paper,we investigate the "quantum speed-up capacity" which reveals the potential ability of a quantum system to be accelerated.We explore the evolutions of the speed-up capacity in different quantum channels for two-qubit states.We find that although the dynamics of the capacity is varying in different kinds of channels,it is positive in most situations which are considered in the context except one case in the amplitude-damping channel.We give the reasons for the different features of the dynamics.Anyway,the speed-up capacity can be improved by the memory effect.We find two ways which may be used to control the capacity in an experiment:selecting an appropriate coefficient of an initial state or changing the memory degree of environments.