We demonstrate that the synchronization effect observed [Pedersen et al., Phys. Rev. Lett. 87, 055001 (2001)], when a bunch of ions oscillates between two mirrors in an electrostatic ion beam trap, can be explained as a negative mass instability. We derive simple necessary conditions for the existence of a regime in which this dispersionless behavior occurs and demonstrate that in this regime, the ion trap can be used as a high resolution mass spectrometer.