A nonlinear differential equation of the force of direct central quasistatic impact of elastic bodies bounded in the area of their contact by rotation surfaces is compiled. To determine the coefficients of the equation and the order of its degree nonlinear force, we used the well-known solution of the axisymmetric contact problem of elasticity theory, constructed in due time by I. Ya. Shtaerman, for the case of dense static contact of bodies, when the order of their boundary surfaces is not lower than the second. In the case of the second order, it goes into the well-known static solution of G. Hertz, whose assumption in the theory of shock is also taken here in the formulation of the dynamics problem. A closed analytic solution of the composite differential equation with respect to the force of impact as a function of time is constructed. It is expressed through Ateb-sine. This function also describes the process of motion of the centers of mass of bodies in the stages of their compression and expansion. Compact formulas are derived for calculating the maximum values of the impact force, the approach of the centers of mass of the bodies and the duration of the impact. Thanks to the use of the Ateb-sine and its approximation by elementary functions, it was possible to obtain a fairly simple scan of the fleeting process of mechanical shock in time. It is shown that well-known dependencies that describe the impact of elastic balls follow from the derived formulas. Examples of calculations are given in which the influence of various factors on the main characteristics of the impact is investigated. It is noted that the theory set forth concerns only the impact of bodies with low velocities, when plastic deformation does not occur during their dynamic compression. To extend the theory beyond the limits of elasticity, it is necessary to determine a constant for the stage of compression in the impact force equation not by calculation, but by experimental method. Then, during compression and decompression of bodies, the impact force will be described by different analytical expressions, and the speed recovery coefficient will become less than one, which is consistent with practice.

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