Richness of Arithmetical Structure to Deliver Alternative Equations of Distance in Uniform Acceleration Motion

The study introduces the arithmetical framework to derive classical kinematic equations without using calculus and provides comparable equations for displacement. The proposed method establishes a direct correspondence between kinematic quantities—velocity, acceleration, displacement, and time—and the elements of arithmetic progressions. Conventional calculus-based methods encounter constraints when applied to systems that suffer from accuracy; thus, it indicates that there are alternate mathematical tools and methodologies. A carefully chosen arithmetic-based model is developed, treating time as continuous when studied as a sequence and as discrete when regarded in a series. Further, acceleration is modeled as a deterministic continuous quantity, while velocity and displacement follow deterministic discrete patterns. Through transforming physical correlations directly to arithmetic patterns, this framework provides alternative distance equations. Based on standard summation, a comparative analysis of a constant-acceleration problem shows that the arithmetic model predicts displacement (47 m) with about 9% more accuracy than the classical calculus result (43 m). This confirms that the method is physically valid, and it not only makes calculations easier and reduces mistakes, but it also helps people understand motion under uniform acceleration better and gives more accurate results than the calculus-based model. The results imply that the arithmetic framework offers a unified way to connect discrete and continuous motion and gives classical mechanics a new mathematical base that makes calculations clearer. Moreover, it offers fresh methods for dealing with increasingly simplifying complex physical phenomena and advancing conceptual clarity.