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Understanding the fundamental interactions in the complex behavior of one car moving in a sequence of traffic lights necessarily implies the inclusion of finite braking and accelerating capabilities. This characteristic is usually not considered in the standard cellular automaton models, where car interactions are the main concern. Therefore, here we develop a model which includes interactions and finite braking and accelerating capabilities, filling the gap between a standard cellular automaton model that considers car interactions but infinite braking and accelerating capabilities and the continuous one car model that includes finite braking and accelerating capabilities but does not consider, as the name indicates, car interactions. The proposed new model bridge these two seemingly different approaches in an effort to investigate how the traffic jams are produced. We found that, in the appropriate limits, we can reproduce the complex behavior of the one car continuous model and the dynamics close to the resonance induced by the interacting cars, forced by the traffic lights. In the processes of introducing car interactions, we observe how the average velocity decreases to finally obtain traffic jams, which are an emergent state in which the traffic lights control the generation of pulses of cars but do not control its average speed. This model is expected to improve our understanding of the complexity that appears in city traffic situations, as the finite braking and accelerating capabilities are necessary to describe the vehicle dynamics, the control strategy of traffic light synchronization, the motion of buses in segregated lights, and the whole urban design. [ABSTRACT FROM AUTHOR]