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In the article the mathematical model of the vehicle motion with elastic tires, which describes the phases of the initial rectilinear motion on the longitudinal slope of the road is shown. The mathematical model is valid for vehicle with two axles, one of which is driving axel and provided that the stopped vehicle on a slope is kept from rolling only by braking the non-driving axle. The aim of the study is to determine the equations to describe the process of vehicle initial rectilinear motion during moving uphill, which take into account the tires deformation energy on the stationary vehicle braking wheels. The presented equations described in the article are reflect the dynamic state of the stationary vehicle on a slope and the phase of its initial moving off uphill after brining the engine torque through the transmission to the wheels of the drive axle. The system of equations describing the vehicle dynamic behavior after brining torque to the drive wheels takes into account their false slip. The system of equations is obtained from the general dynamic equation in the form of balance of the elementary work of external forces and moments with the absence of external slipping of the driving wheels. The mathematical model of the vehicle initial rectilinear motion on a longitudinal slope of a highway described in the article can be used in theoretical basis for analyzing and synthesizing control algorithms for automatic driver assistance systems during vehicle initial motion.