Terminal Velocity Paradoxes in Viscous Media: A Theoretical Perspective

This study examines the dynamics of two spheres falling independently in a viscous fluid, highlighting conditions under which a lighter sphere can achieve a higher velocity than a heavier one. Through theoretical modeling and simulations, the motion of spheres with varying densities and radii, released simultaneously in a uniform viscous medium, was analyzed. The investigation considers gravitational, buoyant, and drag forces, with the spheres moving under identical initial conditions and without mutual interaction. The results confirm the well-established case where the heavier sphere exhibits a greater terminal velocity. However, an intriguing phenomenon is identified: under specific conditions, a lighter sphere can surpass its counterpart in terminal velocity. Additionally, when spheres of equal weight are compared, the denser sphere consistently attains a higher terminal velocity. The study reveals non-trivial time-dependent acceleration patterns, with alternating dominance between heavier and lighter spheres before terminal velocities are reached. Furthermore, the order of impact with the ground is shown to depend on the release height, illustrating a complex interplay of forces. These findings offer novel insights into fluid dynamics, with implications for education and engineering applications.