In this paper, we present a novel convex optimization approach to address the minimum-time speed planning problem over a fixed path with dynamic obstacle constraints and point-wise speed and acceleration constraints. The contributions of this paper are three-fold. First, we formulate the speed planning as an iterative convex optimization problem based on space discretization. Our formulation allows imposing dynamic obstacle constraints and point-wise speed and acceleration constraints simultaneously. Second, we propose a modified vertical cell decomposition method to handle dynamic obstacles. It divides the freespace into channels, where each channel represents a homotopy of free paths and defines convex constraints for dynamic obstacles. Third, we demonstrate significant improvement over previous work on speed planning for typical driving scenarios such as following, merging, and crossing.