This work describes the design of an adaptive cruise control (ACC) applied to a realistic automotive simulation model, considering both upper- and lower-level controllers. The upper-level controller provides a desired acceleration/deceleration to a host vehicle to maintain a safe distance related to a leader vehicle or to track a desired cruise speed otherwise. On the other hand, the lower-level controller provides control signals to the throttle and brake pedals of the host vehicle aiming to track the desired acceleration/deceleration setpoint from the upper-level controller. For the upper-level controller, we consider a control framework that unifies the stability/tracking objective, expressed as a control Lyapunov function (CLF), the safety constraint, expressed as a control barrier function (CBF), and the comfort constraint, by means of a quadratic programming (QP), where safety must be prioritized. The lower-level controller is implemented considering a model-free control. The results, obtained by numerical simulations, demonstrate that the safe distance between the vehicles is ensured and the desired cruise speed is tracked adequately. Therefore, the stability/tracking objective and the safety and comfort constraints, related to the upper-level controller, are properly satisfied. The lower-level controller tracks the desired acceleration/deceleration demanded by the upper-level controller with good performance.