What is the velocity v of a metallic ball of radius r falling in a tank of liquid at the instant when its acceleration is one-half that of a freely falling body? (The densities of metal and of liquid are ρ ρ and σ σ respectively, and the viscosity of the liquid is η η ).
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$Net force on the ball = downward force - upward force = \frac{mg}{2} \frac{4}{3} \pi r^{3} (\rho - \sigma) - 6 \pi \eta r v = \frac{mg}{2} \frac{4}{3} \pi r^{3} (\rho - \sigma) g - 6 \pi \eta r v = \frac{1}{2} \left( \frac{4}{3} \pi r^{3} \rho \right) g r^{2} (\rho - \sigma) g - \frac{9}{2} \eta v = \frac{1}{2} r^{2} \rho g \frac{9}{2} v \eta = r^{2} (\rho - \sigma) g - \frac{1}{2} r^{2} \rho g = \frac{1}{2} r^{2} \rho g - r^{2} \sigma g \frac{9}{2} v \eta = \frac{1}{2} r^{2} g (\rho - 2 \sigma) v = \frac{r^{2} g}{9 \eta} (\rho - 2 \sigma)$
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