A lower bound is derived for the length of an autoresonant accelerator designed to produce relativistic positive ions of a given energy. The ion kinetic energy is found to scale as the 2/3 power of the distance along the accelerator in the nonrelativistic limit and as the 1/3 power in the ultrarelativistic limit. For protons trapped in a negative‐energy Doppler‐shifted cyclotron wave that enters the accelerator with a phase velocity of 0.05c, we estimate that GeV energies can be attained only if the accelerator length exceeds 75 m, which is roughly 10 times greater than previous estimates made by the authors of this collective acceleration scheme.