In 《 SECTION 6.5, where we looked at the drag force on objects moving through fluids, we calculated the terminal speed of an object falling in air but not that of an object sinking in a liquid. The reason is that the buoyant force of air can usually be neglected, but the buoyant force of a liquid cannot.

FIGURE 14.22 shows an object sinking in a liquid. There are two upward forces: the drag force $\vec{F}{}_{\text{drag}}$ and a buoyant force $\vec{F}{}_{\text{B}}$. When the object has reached terminal speed and is no longer accelerating, these forces must balance the gravitational force. Thus

where, in the last step, we used $V_{\text{f}}=V_{\text{o}}$ for an object that is submerged. There’s often little difference between ${\rho }_{\text{o}}$ and ${\rho }_{\text{f}}$ for objects falling in liquids, so the terminal speed is often much lower than in air.

FIGURE 14.22 The forces on an object sinking in a liquid.

Recall that the drag force depends on the Reynolds number Re. At high Reynolds number, $Re>1000$, the drag is quadratic in $v\text{:}$ $F_{\text{drag}}=\frac{1}{2}{C}_{\text{d}}{\rho }_{\text{f}}A{v}^2$. At low Reynolds number, $Re<1$, the drag on a sphere of radius $r$ is linear in $v\text{:}$ $F_{\text{drag}}=6\pi {\eta }_{\text{f}}rv$, where ${\eta }_{\text{f}}$ is the viscosity of the fluid. Exact values depend on density and viscosity, but as a rough rule objects smaller than $100\mu \text{m}$ fall in water with linear drag while objects larger than a few mm experience quadratic drag. There’s no simple expression for drag for in-between cases. Equation 14.15 can be solved for $v_{\text{term}}$ after we have identified the correct model for $F_{\text{drag}}$.