Terminal Velocity
Estimate terminal velocity v = √(2mg/(ρ·A·Cd)).
v ≈ — m/s
Terminal velocity: v = √((2·m·g)/(ρ·C_d·A))
Terminal velocity is reached when aerodynamic drag equals weight, so net force (and acceleration) becomes zero. In the laminar regime (low Reynolds), Stokes drag applies: F_d = 6π·μ·R·v. In the turbulent regime, drag is quadratic: F_d = (1/2)·ρ·C_d·A·v², yielding v_term = √((2·m·g)/(ρ·C_d·A)). A 1 cm steel ball in water settles at ~2 m/s. A skydiver in belly-to-earth position reaches roughly 55 m/s (200 km/h); in a head-down dive, ~85 m/s (310 km/h). A raindrop tops out near 9 m/s — without air, even a pin would fall faster than any raindrop. Heavier or smaller-area objects have higher v_term; lighter or larger-area objects (parachutes, leaves) have lower v_term.
Applications
Skydiving and parachute design, ballistics of unguided bombs (gravity bombs), industrial spray atomization, sedimentation analysis (centrifugation, decantation in water treatment), meteorology of precipitation, and dust dispersion in occupational hygiene.
FAQ
Why does a parachute slow a fall so dramatically? It increases area A by ~50× and drag coefficient C_d, which reduces v_term by roughly √50 ≈ 7×, from ~55 m/s to ~5–7 m/s — safe for landing.
Does mass affect terminal velocity? Yes — v_term scales with √m for fixed area and shape. That's why a heavier skydiver in the same posture falls faster than a lighter one.
How long until terminal velocity is reached? Typically 5–10 seconds and 300–500 m for a human skydiver in belly position — depends on initial conditions and posture changes.
Related Tools
Settling Velocity (Stokes)
Calculate the settling (terminal) velocity of a particle in laminar regime by Stokes' Law, v_s = g·d²·(ρ_s − ρ_w) ÷ (18·μ), from the particle diameter d (m), the solids ρ_s and water ρ_w densities (kg/m³) and the fluid dynamic viscosity μ (Pa·s). The settling velocity is the speed at which an isolated particle SINKS in a still fluid, when weight (minus buoyancy) balances drag. Stokes' Law (1851) holds for the LAMINAR regime (small particles, particle Reynolds < ~1) — fine sand, silt, clay — and has the remarkable property that velocity grows with the SQUARE of diameter: particles twice as large sink four times faster. This calculation is fundamental in many fields: particle settling and separation (settling tanks, thickeners, water and effluent clarifiers), grain-size classification by sedimentation (pipette or hydrometer test), sediment transport in rivers and reservoir deposition, and hydraulic transport (the particle settling velocity sets the critical deposition velocity in the pipe). For large particles (higher Reynolds), Stokes' Law fails and Newton's terminal velocity (turbulent regime) is used. Enter the particle diameter, the densities and the viscosity.
Settling Velocity (Newton)
Calculate the settling (terminal) velocity of a particle in turbulent regime by Newton's law, v_t = √(4·g·d·(s − 1) ÷ (3·C_d)), from the particle diameter d (m), the solids relative density s = ρ_s/ρ_w and the drag coefficient C_d (dimensionless, ≈ 0.44 for spheres in turbulent regime). While Stokes' Law holds for SMALL particles (laminar regime, particle Reynolds < 1), Newton's law holds for LARGE, dense particles — gravel, crushed stone, coarse sand — that sink fast, generating TURBULENT flow around them (particle Reynolds > ~1000). In this regime, drag is no longer proportional to velocity (Stokes) but to its SQUARE, and the terminal velocity grows with the SQUARE ROOT of diameter (not the square, as in Stokes) — large particles sink fast, but the size dependence is milder. The drag coefficient C_d ≈ 0.44 is roughly constant in this range (the 'Newton region' of the sphere drag curve). This calculation is fundamental in designing coarse-particle classifiers and separators, sizing settling basins for coarse solids, coarse-sediment transport and hydraulic transport of gravel and granular ore. For the intermediate range between Stokes and Newton, transition correlations are used. Enter the diameter, the relative density and the drag coefficient.
Parachute Terminal Velocity
Compute terminal velocity Vt = √(2·m·g/(ρ·Cd·A)).
Hull Speed
Compute the hull speed of a displacement vessel, V ≈ 2.43·√(LWL), in knots, from the waterline length (LWL, in meters). It is the theoretical limit of a hull's economical speed: as the boat approaches it, it gets trapped in its own bow wave and the required power soars. That is why sailboats and displacement craft rarely exceed it. Enter the waterline length.
Belt Centrifugal Tension
Calculate the centrifugal tension in a belt, T_c = m·v², from the mass per unit length m (kg/m) and the belt velocity v (m/s). When the belt wraps a pulley at high speed, its own mass, making the turn, generates a CENTRIFUGAL force tending to 'throw' the belt outward, LIFTING it off the pulley. This creates an additional tension throughout the belt (the centrifugal tension), the same at all points and not contributing to power transmission — it only 'steals' part of the belt's gripping capacity against the pulley. Centrifugal tension grows with the SQUARE of velocity, so it is negligible at low speeds but becomes important in fast belts. The effect is harmful: by lifting the belt off the pulley, centrifugal tension REDUCES the normal contact force and thus the friction available to transmit power — there is an OPTIMAL velocity above which increasing speed reduces transmissible power (the belt starts to 'float'). So belt speed has a practical limit (typically 25-30 m/s for conventional V-belts, more for special belts). Centrifugal tension must be added to the tensions to get the total tight- and slack-side tensions. Enter the mass per unit length and the velocity.
Greenshields Speed
Calculate the speed of a traffic stream by the linear Greenshields model, v = v_f·(1 − k ÷ k_j), from the free-flow speed v_f, the current density k and the jam density k_j (vehicles/km). The result, in the unit of v_f, shows speed falls linearly with density: on an empty road (k = 0), vehicles travel at free-flow speed; as density rises, speed decreases, reaching zero at total jam (k = k_j). It is the most classic macroscopic traffic flow model, the basis of the parabolic flow-density relationship. Enter the free-flow speed, the current density and the jam density.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.