Under the pseudopotential approximation, for a single Paul trapped ion which interacts with a flashing ratchet potential formed by two laser standing waves, we apply the method of integral-equation to construct the exact solution of the classical equation, and use the numerical technique to plot the orbits in phase space and calculate the current defined by the average velocity. By combining the analytical results with numerical ones,we investigate the regular and chaotic motions of the system. Compared to the case of the flashing lattice potential, two important ratchet effects are found here: First, the flashing ratchet potential changes the chaotic region in the parameter space such that we can make the regular motion of ion chaotic or vice versa by adjusting the parameters of the second laser. Secondly, by analyzing the evolutions of the average current versus the laser parameters, we find that the presence of the flashing ratchet potential can lead the trapped ion to be transported in averagy along a single direction. When the potential strength is increased to the chaotic region, the current decreases evidently and its direction varies, indicating the onset of chaotic motion.