How to Improve the Kalman Filter for Tracking the Periodic Motion of a Car?
You basically have 4 models here:

Accelerating to constant speed.
Moving at constant speed.
Decelerating to zero speed.
Standing.

So the basic solution is building the 4 models and switching using Hard Switch between them.
Yet there is a smoother framework to handle smooth transition between them called Interacting Multiple Model (IMM) Kalman Filter.
Using the IMM Framework (Which is basically adding weighing step for the models) with the knowledge of the Input as described in other answers will give you a much better results.