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It is commonly accepted that a rod after having been accelerated and then left free contracts by the Lorentz factor , and also that a clock after having been accelerated and then returned to rest exhibits time dilation in accordance with the proper time formula . Although both issues have been debated numerous times, no models have been proposed to trace the dynamical nature of these phenomena. The purpose of the present paper is to propose a simple relativistic model of an elastic rod, which allows exact analytical solutions for a number of problems. The solutions of two problems are presented: (i) acceleration of the rod by a constant force applied at one of its ends during a finite time interval, resulting in the Lorentz contraction, (ii) acceleration and deceleration of a light clock, resulting in time dilation. [ABSTRACT FROM AUTHOR]