Abstract: An algorithm for time integration of motion equations using linear finite elements with adaptive time-step size had been proposed by the authors, in which a formula for the calculation of adaptive time-step size h based on maximum norm, characterized by containing a term of 5/2 order of h, was also presented. This paper gives a mathematical proof for the proposed formula. In addition, the numerical examples of both single and multiple degreed systems are obtained to verify that 5/2 order is optimal.