We restrict purely kinetic k-essence.Assuming that the equation of state is a power law of the kinetic energy,i.e.ω =ω0Xα,we find that α must be positive to obtain accelerated phases,constrained from the conditions for stability and causality.In this case the k-essence behaves like a phantom.We also study the evolutions of the equation of state and the speed of sound with numerical simulations.In the last decade a convergence of independent cosmological observations have suggested that the Universe is experiencing accelerated expansion.An unknown energy component,dubbed as dark energy,has been proposed to explain this acceleration.