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This paper presents an analysis of the acceleration of a fast boat using a simulation model. Mathematical equations of ship motion dynamics with two types of propeller capabilities are developed using MATLAB and Simulink as simulation tools. The equations are extended to include the acting thrust, resistance, propeller’s performance curves, and the PID governor curve for the acceleration manoeuvre. The application models the dynamic differential equations representing the vessel dynamics in one degree of freedom. MATLAB code was used to develop the ship acceleration as a multibody system. Modules of hydrodynamic forces, resistance, moments, and propeller performances were implemented to simulate the ship manoeuvring process. A comparison of the results for the boat’s propulsion performance with two different propellers and the characteristics of the PID governor, which controls the fuel dose in the gas turbines, was carried out. We present a summary including a comparative analysis of the results for the boat dynamics with and without the PID governor. The results obtained here confirm significant discrepancies between the results of numerical simulations with and without the PID governor.