A Direct Relation for Identifying Equal-DistanceSecondsin Vertical Motion

In vertical motion under uniform gravity, the distances covered during correspondingone-second intervals on the way up and on the way down are equal in magnitude, providedair resistance is neglected and the body returns along the same vertical path. This paperpresents a concise index relation, referred to here as the Kumar Relation, for identifyingthe descending second that corresponds to a selected ascending second. If the time fromlaunch to the highest point is an integer T seconds and the selected ascending interval isthe t-th second, the matching descending interval, counted from launch, isN = T + (T − t) + 1 = 2T − t + 1.The relation follows from reflecting the time interval about the instant at which thevertical velocity is zero. It is checked against the standard nth-second displacementequation and illustrated with a numerical example. The method does not replace thegeneral equations of motion; rather, it provides a quick way to map interval indices whenthe ascent time is known. The paper also states the assumptions and limits of the relationand compares its use with the conventional calculation.Keywords:vertical motion; uniformly accelerated motion; nth-second displacement; time symmetry;equal-distance intervals; Kumar Relation.