Terminal velocity is the constant speed reached when the net force on an object moving through a fluid becomes zero.
The object still has forces acting on it, but those forces balance. For a falling object, weight acts downward, while
buoyancy and drag act upward at terminal speed. For a rising object, buoyancy acts upward, while weight and drag act downward.
Forces in a fluid
The weight of the object is:
Weight.
\[
\begin{aligned}
W &= mg
\end{aligned}
\]
The buoyant force equals the weight of the displaced fluid:
Buoyancy.
\[
\begin{aligned}
B &= \rho V g
\end{aligned}
\]
The signed effective driving force is:
Effective driving force.
\[
\begin{aligned}
F_{\mathrm{drive}} &= W-B
\end{aligned}
\]
In this calculator, positive \(F_{\mathrm{drive}}\) means the object tends to fall downward. Negative
\(F_{\mathrm{drive}}\) means buoyancy is stronger than weight and the object tends to rise upward.
Quadratic drag model
Quadratic drag is commonly used for objects moving at moderate or high Reynolds number, where drag is proportional to
\(v^2\). The drag magnitude is:
Quadratic drag.
\[
\begin{aligned}
D &= \frac12 \rho C_d A v^2
\end{aligned}
\]
At terminal velocity, drag balances the magnitude of the effective driving force:
Quadratic terminal velocity.
\[
\begin{aligned}
\frac12 \rho C_d A v_t^2 &= |W-B| \\
|v_t| &= \sqrt{\frac{2|W-B|}{\rho C_d A}}
\end{aligned}
\]
The sign is added separately: positive means downward, negative means upward.
Stokes drag model
Stokes drag applies to small spheres in slow, laminar flow. The drag is linear in speed:
Stokes drag.
\[
\begin{aligned}
D &= 6\pi\eta r|v|
\end{aligned}
\]
For a sphere of radius \(r\), the displaced volume is:
Sphere volume.
\[
\begin{aligned}
V &= \frac43\pi r^3
\end{aligned}
\]
Substituting the sphere volume into the force balance gives the classic Stokes terminal velocity:
Stokes terminal velocity.
\[
\begin{aligned}
v_t &= \frac{2r^2g(\rho_s-\rho)}{9\eta}
\end{aligned}
\]
If \(\rho_s>\rho\), the value is positive and the sphere falls. If \(\rho_s<\rho\), the value is negative and the sphere rises.
Reynolds number check
The Stokes model is valid only when inertial effects are small. A common diagnostic is the Reynolds number:
Reynolds number estimate.
\[
\begin{aligned}
Re &= \frac{2\rho r|v_t|}{\eta}
\end{aligned}
\]
Stokes drag is most reliable for \(Re \ll 1\). If \(Re\) is not small, a quadratic-drag model or a more advanced drag-correlation
model is usually more appropriate.
| Model |
Main relation |
Best used for |
| Effective driving force |
\(F_{\mathrm{drive}}=W-B\) |
Determines downward or upward motion tendency |
| Quadratic drag |
\(|v_t|=\sqrt{2|W-B|/(\rho C_d A)}\) |
Moderate or high Reynolds number motion |
| Stokes drag |
\(v_t=2r^2g(\rho_s-\rho)/(9\eta)\) |
Small spheres in slow laminar flow |
| Neutral buoyancy |
\(W=B\) |
No terminal motion from rest in the ideal model |
Terminal velocity models assume a steady fluid, constant fluid properties, and a drag law appropriate for the chosen regime.
Real objects may tumble, change orientation, or experience drag coefficients that vary with Reynolds number.
