Changing pseudonyms are commonly accepted as one approach to protect location privacy in VANETs, but the effectiveness of most changing pseudonyms schemes have never been investigated rigorously. In this paper, we present a detailed analytical model based on a kind of simplest scheme, called Random Changing Pseudonyms (RCP), to quantify the level of location privacy provided by this scheme. Furthermore, according to the proposed model, we analyze and compare two special distributions, i.e., uniform discrete and age-based distributions. The results show that RCP scheme with uniform discrete distribution is better than with age-based distribution on protecting location privacy in VANETs. The accuracy of this model is verified by a set of simulations.