Considering the multibody method for the dynamic analysis of mechanical systems, this paper aims at constructing a simple computer model describing the dynamics of a fixed-wing aircraft performing a longitudinal motion. To this end, a simplified model of an aerial vehicle was analyzed without control surfaces, featuring an axial thrust, and having limited aerodynamic actions. By using the Digital DATCOM software, the modeling of the aerodynamic coefficients was then taken into account considering also the elevator as a control surface. First, aircraft dynamics was studied in general in the context of multibody dynamics. Then, the case study considered as the illustrative example of the paper was analyzed, namely the Cessna 172 Skyhawk aircraft. By modeling the externally applied actions and the aerodynamic coefficients, the fundamental mechanics behind the take-off phase of the flight was subsequently analyzed. In the paper, the equations of motion describing the dynamic behavior of the illustrative example were driven using a Lagrangian formulation approach. The dynamic model of the demonstrative example was then implemented in a computer code constructed in the MATLAB environment. By doing so, the goal of this process was to develop as accurately as possible a virtual model of the Cessna 172 Skyhawk aircraft. As shown in the paper using numerical simulations, the computer model of the case study analyzed in this work is able of simulating the vertical motion of the aircraft. In the conclusive part of the manuscript, a discussion about the dynamic behavior of the aircraft found in this investigation and some comments on several topics for future research are reported as well.