We performed numerical experiments on a freely rising and falling sphere in stagnant water for moderate particle Reynolds numbers (130Re(p)1091) using spheres of various diameters and various sphere-water density ratios (0.0016mD(p)0.004m and 0.08p/f1.92). The methodology to carry out the arbitrary Lagrangian-Eulerian (ALE) moving mesh simulation technique was developed using ANSYS CFX (R) CFD software and the results obtained were validated with the experimental results published in the literature. The sphere trajectories, dynamics of sphere movement, and angular velocities play a significant role in transient drag coefficient. We observed that after maintaining a sphere diameter and dimensionless density difference (/f) between water and a sphere at the same level as in the rising and falling sphere, the rising sphere attained terminal velocity faster than the falling sphere. The time required to keep the sphere inside the domain at a fixed distance from the rising and falling sphere decreased with increasing sphere diameter at a given /f.