This work presents a distributed velocity planning strategy for multi-vehicle cooperation along pre-defined paths. Specifically, we consider a class of tasks where multiple vehicles must navigate given paths with conflict zones (e.g., merging and crossing) as fast as possible without any inner collisions. Given the paths, the biggest challenge for velocity planning is to create collision avoidance constraints without complete spatio-temporal information. To overcome the challenge, a scheme is proposed to project collision-related information from coordinate space to 1-D space with geometry-based safety guarantees. To enhance the ability to deal with medium-and large-scale problems, the alternating direction method of multipliers (ADMM) is introduced. Unlike classic ideas of applying distributed optimization, we formulate ADMM in a semi-centralized, semi-parallel fashion instead of a fully distributed fashion. In this way, a trade-off between overall performance and computational efficiency can be achieved. We evaluate our distributed planning strategy through simulations in multiple cases¹¹Video of the results is available at https://youtu.be/GR6BwFTLErw. The results demonstrate that our method not only plans collision-free paths but also balances between overall performance and computational load.