Terminal Velocity Calculator

Compute the maximum falling speed (terminal velocity) when drag force equals weight. Visualize the velocity‑time curve, determine vt = √(2mg / ρACd), explore real‑world objects from skydivers to raindrops.

Object mass in kilograms
Earth standard: 9.81 m/s²
Air at sea level ≈ 1.225 kg/m³, water ≈ 1000 kg/m³
Cross‑sectional area facing the flow
Sphere ≈ 0.47, Skydiver spread ≈ 1.0, streamlined ≈ 0.04
? Skydiver (spread: 80kg, 0.8m², Cd=1.0)
⚾ Baseball (0.145kg, area 0.0043m², Cd=0.3)
? Raindrop (2mm radius, ~3.3e-5kg, Cd=0.5)
? Steel sphere (5cm radius, 4.1kg, Cd=0.47)
? Feather (0.005kg, 0.01m², Cd=1.2)
Client‑side physics: All calculations and graphs are performed locally. No data is transmitted to any server.

Physics of Terminal Velocity: When Drag Balances Weight

Terminal velocity is the constant maximum speed a falling object reaches when the resistive force (drag) equals the gravitational force. At this equilibrium, net acceleration becomes zero. For most macroscopic objects moving through a fluid, drag follows quadratic law: Fdrag = ½·ρ·v²·Cd·A. Setting Fdrag = mg yields the famous formula:

vt = √( 2·m·g / (ρ·Cd·A) )

Valid for turbulent flow (Re >> 1), typical for most human‑scale objects in air.

The derivation originates from Newton's second law: m dv/dt = mg - ½ ρ Cd A v². The analytical solution yields v(t) = vt·tanh(gt / vt). This hyperbolic tangent function beautifully describes approach to terminal speed.

Scientific & Practical Importance

  • Skydiving & parachute design: Terminal velocity determines fall speed before deployment (≈ 55 m/s belly‑to‑earth). Parachutes increase area, reducing vt to ~5 m/s for safe landing.
  • Meteorology: Raindrop terminal velocity (8–10 m/s) governs precipitation intensity and kinetic energy. Hailstones can achieve much higher speeds causing damage.
  • Engineering: Sedimentation tanks, particle separators, and free‑fall systems rely on terminal velocity estimates for design.
  • Spacecraft re‑entry: Although high‑speed regime differs, terminal velocity concepts assist drogue chute calculations.

Step‑by‑Step Calculation Logic

Our calculator follows these validated steps:

  1. Collect mass, gravity, fluid density, projected area, and drag coefficient.
  2. Validate all inputs > 0 (gravity > 0). If any non‑positive, display warning.
  3. Compute terminal velocity: vt = sqrt(2 * m * g / (ρ * Cd * A)).
  4. Determine characteristic time τ = vt/g.
  5. Calculate time to 95% terminal speed: t95 = τ * atanh(0.95).
  6. Approximate Reynolds number for qualitative flow regime: Re = ρ vt L / μ, using characteristic length L ≈ √A (simplified, ambient air viscosity μ ≈ 1.81e-5 Pa·s).
  7. Generate velocity‑time graph from t=0 to t = max( 5 s, 1.3 * t95 ) with adaptive sampling.

All results are displayed in SI units (m/s, N, s). The interactive graph dynamically updates when parameters change.

Influence Factors & Misconceptions

Parameter Effect on Terminal Velocity Typical value variation
Mass (m) Increasing mass → higher vt (square root dependence) Skydiver 70–100 kg → vt ~55–65 m/s
Projected area (A) Larger area → lower vt (parachute effect) Parachute A=50 m² → vt ≈ 5 m/s
Drag coefficient Cd Streamlined shape → lower Cd → higher vt Sphere Cd=0.47, teardrop ~0.04
Fluid density ρ Denser fluid (water vs air) → lower vt Water ρ=1000 → vt 10× smaller
Case Study: Skydiver Belly‑Fly vs Head‑Down

A typical skydiver with mass 80 kg, projected area 0.8 m² and Cd=1.0 has terminal velocity ≈ 44 m/s. By changing to a head‑down position, area reduces to ~0.2 m² and Cd ≈ 0.7, resulting in vt ≈ 95 m/s — nearly double. This extreme speed requires specialized suits. Our calculator demonstrates such sensitivity, useful for sport skydiving and wingsuit design.

Limitations & Advanced Considerations

The quadratic drag model is accurate for Reynolds numbers > 1000 (turbulent regime). For very slow, small objects (dust, pollen), Stokes' law (linear drag) dominates. The calculator includes an approximate Re number to help users identify regime. For precise engineering, also note buoyancy force (Archimedes), which becomes significant when fluid density is comparable to object density. In air, buoyancy is negligible for solids but matters for balloons. Additionally, compressibility and Mach number effects are omitted.

Trusted Physics Reference – This tool implements Newtonian mechanics and fluid drag principles as documented in Fundamentals of Fluid Mechanics (Munson) and University Physics (Young & Freedman). Reviewed by the GetZenQuery tech team (June 2026). All equations align with public domain resources and NIST standards.

Frequently Asked Questions

Yes, directly: vt ∝ √m. A heavier object experiences larger gravitational force and requires higher speed for drag to match weight, increasing terminal velocity.

For a belly‑to‑earth position: ≈ 55 m/s (200 km/h). In a head‑down freefly position: can exceed 90 m/s (320 km/h).

Small mass combined with relatively large area and high drag coefficient yields very low terminal velocity (~0.5–1 m/s). Air resistance dominates early.

Absolutely. Adjust fluid density (ρ) to water ~1000 kg/m³, and set appropriate drag coefficient. The formula works for any Newtonian fluid.

The velocity curve uses the exact analytical solution v(t)=vt·tanh(gt/vt). It is mathematically precise for quadratic drag. Numerical precision is double‑float.
Authoritative references: NASA Terminal Velocity; Encyclopædia Britannica; Drag Coefficient of Common Shapes, ESDU 81043.