Motion planning in dynamically changing environments is one of the most complex challenges in autonomous driving. Safety is a crucial requirement, along with driving comfort and speed limits. While classical sampling-based, lattice-based, and optimization-based planning methods can generate smooth and short paths, they often do not consider the dynamics of the environment. Some techniques do consider it, but they rely on updating the environment on-the-go rather than explicitly accounting for the dynamics, which is not suitable for self-driving. To address this, we propose a novel method based on the Neural Field Optimal Motion Planner (NFOMP), which outperforms state-of-the-art approaches in terms of normalized curvature and the number of cusps. Our approach embeds previously known moving obstacles into the neural field collision model to account for the dynamics of the environment. We also introduce time profiling of the trajectory and non-linear velocity constraints by adding Lagrange multipliers to the trajectory loss function. We applied our method to solve the optimal motion planning problem in an urban environment using the BeamNG.tech driving simulator. An autonomous car drove the generated trajectories in three city scenarios while sharing the road with the obstacle vehicle. Our evaluation shows that the maximum acceleration the passenger can experience instantly is -7.5 m/s^2 and that 89.6% of the driving time is devoted to normal driving with accelerations below 3.5 m/s^2. The driving style is characterized by 46.0% and 31.4% of the driving time being devoted to the light rail transit style and the moderate driving style, respectively.