Abstract: This article uses a second-order matrix nth power to calculate the general expression of the velocity after the nth collision in a one-dimensional complete elastic collision. From this, the velocity after any collision is obtained. The collision velocity is divided into two cases. Based on the analysis, the corresponding relationship between the actual total number of collisions N and the pi is obtained. On this basis, the expression of the approximate calculation of the pi with the total number of collisions N and the mass ratio k of the two elastic sliders is derived.

Key words: one-dimensional elastic collision, pi, second-order matrix