Terminal Velocity Calculator
Estimate terminal velocity in air or water using either quadratic drag (general objects) or Stokes drag (small spheres at low Reynolds number).
Calculation Steps
Mechanical • Fluid Mechanics
Terminal Velocity Calculator
Terminal velocity is the steady fall speed reached when drag balances the object’s effective weight (weight minus buoyancy). This guide shows how to use the calculator, which drag model applies, and how to sanity-check results for real engineering scenarios.
Quick Start
Use the Terminal Velocity Calculator to estimate the steady falling speed of an object through a fluid. The calculator supports two drag regimes: quadratic drag (most real objects in air/water) and Stokes drag (tiny spheres in laminar flow). Follow these steps to get a correct, defensible result.
- 1 Select “Mode”. Choose Quadratic drag for general objects (skydivers, tools, debris, raindrops at moderate size). Choose Stokes drag only for very small spheres at low Reynolds number (fine particles, micro-bubbles).
- 2 Pick “Solve For”. Most users solve for terminal velocity \(v_t\). If you have a measured \(v_t\) and want to back-calculate drag coefficient \(C_d\) (quadratic mode) or viscosity \(\mu\) (Stokes mode), select those instead.
- 3 Enter object properties. For quadratic mode: enter mass \(m\), displaced volume \(V\), projected area \(A\), and drag coefficient \(C_d\). For Stokes mode: enter radius \(r\) and particle density \(\rho_{obj}\). Use realistic geometry—projected area is the “frontal” area the fluid sees.
- 4 Enter fluid properties. Set the fluid density \(\rho_f\) for air/water/oil. In Stokes mode, also input viscosity \(\mu\). If you’re unsure, use typical values (air at sea level \(\rho_f\!\approx\!1.2\,\text{kg/m}^3\), water \(\rho_f\!\approx\!1000\,\text{kg/m}^3\)).
- 5 Confirm gravity. Leave \(g\) at Earth standard unless modeling another planet or high-precision lab work.
- 6 Read the main result and quick stats. The calculator reports the solved variable and shows quick stats like weight, buoyancy, and net driving force. These help you verify the physics: at terminal speed, drag should equal \(W-B\).
- 7 Sanity-check the regime. If you used Stokes mode, check the Reynolds number in quick stats. If \(Re \gtrsim 1\), Stokes drag is not valid— switch to quadratic drag.
Choosing Your Method
Terminal velocity depends on the drag model. In practice you’ll choose among three approaches: a quadratic, high-Reynolds “engineering” model; a low-Reynolds Stokes model; or a numerical/empirical approach when the flow is complicated.
Quadratic Drag (Most Cases)
\[ (mg-\rho_f V g)=\tfrac{1}{2}\rho_f C_d A v_t^2 \]
- Valid for moderate to large Reynolds numbers.
- Works for almost all objects in air/water at real speeds.
- Simple closed-form solution for \(v_t\).
- Requires a reasonable \(C_d\) estimate.
- Assumes steady, uniform flow and constant \(C_d\).
- Accuracy drops for tumbling or changing posture.
Stokes Drag (Tiny Spheres)
\[ v_t=\frac{2}{9}\frac{r^2 g(\rho_{obj}-\rho_f)}{\mu} \]
- Very accurate for laminar flow around small spheres.
- No drag coefficient needed.
- Great for settling velocity of fine particles.
- Only valid when \(Re \lesssim 1\).
- Assumes a rigid sphere and Newtonian fluid.
- Fails for fluffy particles or non-spherical shapes.
Empirical / Numerical
\[ C_d=f(Re,\text{shape})\quad \text{or}\quad \text{CFD / drop tests} \]
- Best for complex shapes, porous objects, or transitional flow.
- Can handle changing orientation or compressibility.
- Lets you refine \(C_d\) from experiments.
- More effort: requires data or simulation.
- Not necessary for early-stage sizing.
- May still carry uncertainty in real conditions.
A simple decision rule:
- If the object is bigger than a grain of sand and falling faster than a few cm/s, use quadratic drag.
- If it’s a tiny sphere settling slowly in a viscous fluid and \(Re\) stays under ~1, Stokes drag is safe.
- If the geometry/flow is complicated or critical, validate \(C_d\) empirically.
What Moves the Number
For terminal velocity, a handful of variables dominate. These are the “levers” you should focus on when interpreting outputs or doing sensitivity checks.
The driving force is \(W-B=(m-\rho_f V)g\). Heavier objects fall faster; larger displaced volume increases buoyancy, especially in water.
In quadratic drag, \(v_t \propto 1/\sqrt{A}\). Doubling frontal area drops terminal velocity by about 29%. Posture changes for bodies can swing \(A\) dramatically.
\(v_t \propto 1/\sqrt{C_d}\). Streamlined shapes (low \(C_d\)) fall faster. Bluff bodies (high \(C_d\)) slow down quickly. Typical ranges are 0.1–2.0.
Higher \(\rho_f\) increases drag and buoyancy. The same object can have terminal velocity 20–30× lower in water than in air.
In Stokes drag, \(v_t \propto r^2\). A particle 2× larger settles 4× faster, all else equal.
\(v_t \propto 1/\mu\). Warm fluids (lower viscosity) allow faster settling. Oils vs. water can change \(v_t\) by orders of magnitude.
Worked Examples
These examples mirror how engineers typically use terminal velocity. Each one shows the equation path used in the calculator.
Example 1: Human falling in air (Quadratic drag)
Given:
- Mass \(m = 80\,\text{kg}\)
- Displaced volume \(V = 0.07\,\text{m}^3\) (about 70 L)
- Projected area \(A = 0.70\,\text{m}^2\) (belly-to-earth)
- Drag coefficient \(C_d = 1.0\)
- Air density \(\rho_f = 1.225\,\text{kg/m}^3\)
- Gravity \(g = 9.81\,\text{m/s}^2\)
Compute net driving force: \[ W-B=(m-\rho_f V)g \] \[ W-B=(80-1.225\cdot 0.07)\cdot 9.81 \approx 784.9\,\text{N} \]
Solve for terminal velocity using quadratic drag: \[ v_t=\sqrt{\frac{2(W-B)}{\rho_f C_d A}} \] \[ v_t=\sqrt{\frac{2(784.9)}{1.225\cdot 1.0\cdot 0.70}} \approx 42.7\,\text{m/s} \]
Interpretation: \(42.7\,\text{m/s}\) is about \(96\,\text{mph}\), consistent with a stable belly-to-earth skydive. If posture changes to “head-down” (smaller \(A\) and \(C_d\)), terminal velocity can exceed \(70\,\text{m/s}\).
Example 2: Fine sand particle settling in water (Stokes drag)
Given:
- Particle radius \(r = 0.10\,\text{mm} = 1.0\times10^{-4}\,\text{m}\)
- Particle density \(\rho_{obj}=2650\,\text{kg/m}^3\)
- Water density \(\rho_f=1000\,\text{kg/m}^3\)
- Water viscosity \(\mu=1.0\times10^{-3}\,\text{Pa·s}\)
- Gravity \(g=9.81\,\text{m/s}^2\)
Apply Stokes terminal velocity: \[ v_t=\frac{2}{9}\frac{r^2 g(\rho_{obj}-\rho_f)}{\mu} \]
Substitute: \[ v_t=\frac{2}{9}\frac{(1.0\times10^{-4})^2 (9.81)(2650-1000)}{1.0\times10^{-3}} \] \[ v_t\approx 0.036\,\text{m/s} \] That’s about \(3.6\,\text{cm/s}\).
Check Reynolds number: \[ Re=\frac{2r\rho_f v_t}{\mu} \] \[ Re=\frac{2(1.0\times10^{-4})(1000)(0.036)}{1.0\times10^{-3}}\approx 7.2 \]
Interpretation: \(Re \approx 7\) is above the Stokes limit. The Stokes estimate is still a useful first cut, but for accuracy you should switch to quadratic drag (or a transitional \(C_d(Re)\) model). This is exactly why the calculator surfaces Reynolds number as a sanity check.
Common Layouts & Variations
Terminal velocity shows up across aerospace, civil, environmental, and mechanical engineering. The table below lists common configurations, typical modeling choices, and what to watch for.
| Use case / object | Typical fluid | Best model | Notes | Pros | Cons |
|---|---|---|---|---|---|
| Skydiver / human body | Air | Quadratic drag | \(C_d\) 0.7–1.3; posture changes \(A\) | Matches field data well | High orientation uncertainty |
| Raindrops (1–5 mm) | Air | Quadratic drag | Drop deformation affects \(C_d\) | Simple & accurate enough | Need size-dependent \(C_d\) |
| Steel ball in oil | Viscous oil | Stokes drag (if \(Re<1\)) | Great for viscosity measurement | High precision in laminar range | Breaks at higher speeds |
| Sand/gravel settling | Water | Quadratic or transitional | Often \(Re\) 1–1000 | Captures real settling | Needs shape factor / \(C_d\) |
| Light debris in flood flow | Water | Quadratic drag | Buoyancy is significant | Easy risk estimate | Debris tumbling changes \(A\) |
| Micro-particles in air filtration | Air | Stokes drag | Usually \(r\) in microns, low \(Re\) | Direct settling velocity | Sphericity assumption |
Specs, Logistics & Sanity Checks
Terminal velocity estimates are often used in safety checks, device sizing, and field assessments. Here’s what to verify before you rely on the number.
1) Check units and geometry
- Projected area \(A\) must be perpendicular to the fall direction.
- Volume \(V\) should reflect displaced fluid, not “solid only.”
- Use consistent units; the calculator converts internally to SI.
2) Validate your drag inputs
- Typical \(C_d\): sphere 0.47, cylinder cross-flow ~1.1, flat plate normal ~1.9.
- Roughness, porosity, and tumbling can raise \(C_d\).
- For unknown shapes, bracket with \(C_d\) 0.5–1.5.
- Don’t apply Stokes unless \(Re\) is small.
3) Watch buoyancy in liquids
- In water and oils, buoyancy can remove a big chunk of weight.
- If \(\rho_{obj}\approx \rho_f\), terminal velocity can be very small.
- Negative driving force means the object floats—no terminal fall speed exists.
4) Regime and environment effects
- Temperature changes both \(\rho_f\) and \(\mu\) (especially gases).
- Altitude lowers air density → higher \(v_t\).
- Compressibility matters above ~Mach 0.3; use a refined \(C_d\) model there.
- In gusty winds, “terminal velocity” is a mean value, not an instant speed.
Frequently Asked Questions
What exactly is terminal velocity?
Terminal velocity \(v_t\) is the steady falling speed where the drag force equals the net driving force. In quadratic drag this balance is: \[ (mg-\rho_f V g)=\tfrac{1}{2}\rho_f C_d A v_t^2 \] Once this condition is met, acceleration becomes zero and speed stops increasing.
When should I use quadratic drag vs. Stokes drag?
Use quadratic drag for nearly all macroscopic objects in air or water. Use Stokes drag only for tiny spheres in laminar flow where Reynolds number \(Re \lesssim 1\). If quick stats show \(Re\) above ~1, switch to quadratic drag.
How do I estimate drag coefficient \(C_d\)?
\(C_d\) depends on shape and Reynolds number. For a first estimate, use published values: sphere \(\approx 0.47\), cube \(\approx 1.05\), flat plate normal \(\approx 1.8\)–1.9, human belly-to-earth \(\approx 1.0\). If uncertain, bracket with a range and check sensitivity.
Why does buoyancy matter, and what volume should I use?
Buoyancy reduces the driving force by \(\rho_f V g\). Use the object’s displaced volume—essentially the outer volume that pushes fluid aside. In water, buoyancy can greatly lower \(v_t\).
What if the calculator gives no result or a negative driving force?
If \(m-\rho_f V \le 0\) (quadratic) or \(\rho_{obj}-\rho_f \le 0\) (Stokes), the object is neutrally buoyant or floats. There is no terminal fall velocity in that case.
Can terminal velocity change during a fall?
The theoretical \(v_t\) for a given shape is constant, but real objects can change orientation or deform, altering \(A\) and \(C_d\). That effectively changes terminal velocity during the fall.
Is this calculator accurate for very fast or compressible flows?
At very high speeds (roughly above Mach 0.3 in air), compressibility affects drag and \(C_d\) becomes Mach-dependent. The quadratic model is still a useful estimate, but for precision you should use a compressible \(C_d(Re,Ma)\) correlation or CFD.