In this paper, we employ the ideas of obfuscation, i.e., concealment, and power variability to propose a new location privacy scheme, the fake point, which can be used to ensure users' privacy in NEMO-based VANETs. The proposed scheme thwarts location privacy attacker who exploits positioning systems to localize users. To the best of our knowledge, our scheme is the first to apply obfuscation techniques for a user's exact location, rather than a wide area applied with current location privacy schemes. Our analysis shows that the fake point scheme ensures higher MNN's location privacy. In addition, the extensive simulations demonstrate that our proposed scheme can practically be implemented due to the high possibility of having many MNNs select the same fake point, and hence increasing the attacker's confusion.