This paper investigates the accuracy and robustness of car-following (CF) and adaptive cruise control (ACC) models used to simulate measured driving behaviour of commercial ACCs. To this aim, a general modelling framework is proposed, in which ACC and CF models have been incrementally augmented with physics extensions; namely, perception delay, linear or nonlinear vehicle dynamics, and acceleration constraints. The framework has been applied to the Intelligent Driver Model (IDM), the Gipps model, and to three basic ACCs. These are a linear controller coupled with a constant time-headway spacing policy and with two other policies derived from the traffic flow theory, which are the IDM desired-distance function and the Gipps equilibrium distance-speed function. The ninety models resulting from the combination of the five base models and the aforementioned physics extensions, have been assessed and compared through a vast calibration and validation experiment against measured trajectory data of low-level automated vehicles. When a single extension has been applied, perception delay and linear dynamics have been the extensions to mostly increase modelling accuracy, whatsoever the base model considered. Concerning models, Gipps-based ones have outperformed all other CF and ACC models in calibration. Even among ACCs, the linear controllers coupled with a Gipps spacing policy have been the best performing. On the other hand, IDM-based models have been by far the most robust in validation, showing almost no crash when calibrated parameters have been used to simulate different trajectories. Overall, the paper shows the importance of cross-fertilization between traffic flow and vehicle studies.