Car-following models describe how one vehicle follows the preceding vehicles. In order to better model and explain car-following dynamics, this paper categorizes the state of a traveling vehicle into three sub-processes: the starting (acceleration) process, the car-following process, and the stopping (deceleration) process. The starting process primarily involves vehicle acceleration behavior. The stopping process in volves not only car-following behavior but also deceleration behavior. This paper regards both the stopping process and the starting process as spring systems. The car-following dynamics during the starting process and the stopping process is modeled in this paper. The parameters of the proposed models, which are represented in the form of trigonometric functions, possess explicit physical meaning and definitive ranges. We have calibrated the model of the starting process using data from the Traffic Engineering Handbook, and obtained reasonable results. Compared with traditional stimulus-response car-following models, this model can better explain traffic flow phenomena and driver behavior theory.