Terminal Velocity Calculator – Drag, Settling and Fall Speed
The Terminal Velocity Calculator works out the fastest speed a falling object can reach in a fluid, the point where the upward drag force exactly cancels its weight and the acceleration drops to zero. Enter a mass, frontal area, drag coefficient and fluid density — or pick from the shape and fluid presets — and the tool returns the terminal velocity, the Reynolds number that tells you which drag law applies, and a full step-by-step derivation.
What Terminal Velocity Actually Means
A falling object starts out accelerating at g, but drag grows with the square of speed. Once ½ρv²C_dA = mg the net force is zero and the object keeps falling at a constant speed. Rearranging that balance gives the central formula:
v_t = √(2mg / (ρ · A · C_d))
m object mass (kg)
g gravitational acceleration (m/s²)
ρ fluid density (kg/m³)
A cross-sectional area facing the flow (m²)
C_d drag coefficient (dimensionless)Two consequences follow immediately. Terminal velocity does not depend on the drop height — the height only decides whether the object has had time to get there. And heavier objects do fall faster in air, because drag depends on size and shape rather than on mass, so a denser object needs more speed before drag can match its weight.
Quadratic Drag Versus Stokes Settling
The square law only holds when inertia dominates the flow. For dust, pollen, fine sand or emulsion droplets, viscosity dominates instead and drag becomes linear in speed. That regime uses Stokes' law:
v_t = (ρ_p − ρ_f) · g · d² / (18µ)The Reynolds number Re = ρvd/µ decides which model to trust. Below Re ≈ 1 Stokes settling applies; above Re ≈ 1000 the quadratic law is right; in between the drag coefficient itself changes with speed and both models are only approximate. The calculator computes Re for you and flags the awkward transitional band rather than quietly returning a number.
Shape Presets and Typical Drag Coefficients
Drag coefficients are experimental values, not derived constants. Common reference figures are a sphere at 0.47, a cube at 1.05, a flat plate at 1.28, a streamlined teardrop at 0.04, a skydiver belly-to-earth around 1.0, a head-down dive near 0.7 and a round parachute canopy about 1.5. A typical 80 kg skydiver with 0.7 m² of frontal area therefore levels off near 43 m/s (about 154 km/h), while the same jumper head-down with a quarter of the area exceeds 90 m/s.
How Long It Takes to Get There
With constant C_d, the approach to terminal velocity has a closed-form solution, v(t) = v_t · tanh(g·t / v_t). The characteristic time τ = v_t / g sets the pace: an object reaches half of its terminal velocity in about 0.55τ, 95% in roughly 1.83τ, and never quite reaches 100%. For our skydiver that means about 8 seconds and 217 metres of fall before drag has almost fully taken over.
Buoyancy, Altitude and Other Corrections
In dense fluids the displaced fluid is not negligible, so the weight term becomes the effective weight (m − ρ_f·V)g. A steel ball in water settles noticeably slower than the uncorrected formula suggests, and an object less dense than the fluid rises instead of falling. Air density matters just as much: because v_t scales with 1/√ρ, thin air at 8 km altitude raises terminal velocity by more than 50% compared with sea level, which is exactly why high-altitude jumps are so much faster.
Practical Uses
Beyond homework, the same balance sizes a parachute canopy for a target landing speed, predicts raindrop and hailstone fall speeds in meteorology, estimates sedimentation rates in chemical engineering, and tells model-rocket and drone builders how big a recovery device needs to be. It also marks the boundary where an ideal free-fall calculation stops being believable — if the vacuum answer exceeds the terminal velocity computed here, drag is already the dominant effect.