To increase the efficiency of road transport and reduce exhaust emissions, research has been intensified in assisted control systems, where adaptive cruise control (ACC) and cooperative adaptive cruise control (CACC) stand out. While ACC uses data from onboard sensors and radars, CACC also uses data from inter-vehicle wireless communication. In this context, this paper proposes a new control law for ACC systems that provides a competitive performance compared to CACC, despite not requiring wireless communication between vehicles. The new control law formulation employs both the vehicle acceleration and the relative speed, which generalizes similar ones. The design methodology formulated as a convex optimization problem ensures string and internal stability as well closed-loop pole placement if a set of linear matrix inequalities (LMIs) is satisfied. At the end of the paper, simulations demonstrate the effectiveness of the proposed methodology.