A small spherical ball of radius r and density rho falls freely under gravity through a height h before entering a tank of water (density sigma, visc…

Mass of the ball $m = \frac{4}{3}\pi r^3 \rho$. Initial kinetic energy just as it enters water is $K = mgh = \frac{4}{3}\pi r^3 \rho g h$. The net upward force on the ball submerged at depth $H$ is $(Buoyant - Weight) = \frac{4}{3}\pi r^3 \sigma g - \frac{4}{3}\pi r^3 \rho g = \frac{4}{3}\pi r^3 g (\sigma - \rho)$. Work done against net upward force brings the ball to rest: $W = \left(\frac{4}{3}\pi r^3 g (\rho - \sigma)\right) H = K = \frac{4}{3}\pi r^3 \rho g h$. Cancelling terms gives $H(\rho - \sigma) = \rho h \implies H = \frac{h\rho}{\rho - \sigma}$.