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Terminal Velocity Calculator

Terminal velocity is the fastest speed a falling object can reach when air resistance balances gravitational force. At this equilibrium, acceleration stops and the object descends at constant speed. Engineers, skydivers, and physicists use terminal velocity calculations to predict how fast anything from raindrops to meteorites fall through different media.

Last updated:

Creators Davide Borchia, PhD
2,460 people find this calculator helpful

Understanding Terminal Velocity

When an object falls, two competing forces act on it: gravity pulling downward and drag resistance pushing upward. Early in the fall, gravity dominates and the object accelerates. As speed increases, drag grows stronger. Eventually, drag equals the gravitational force exactly. At that moment, net force becomes zero and acceleration stops—the object enters terminal velocity, a steady state of constant descent speed.

This phenomenon applies everywhere: a raindrop reaches terminal velocity within seconds of falling from a cloud, a skydiver stabilises at terminal velocity within about 12 seconds, and even dust particles settle through air at terminal velocity. The speed achieved depends on four things: how heavy the object is, its shape, its cross-sectional area facing the wind, and the density of the medium it falls through.

The Terminal Velocity Equation

Terminal velocity occurs when drag force and weight balance perfectly. The equilibrium condition leads to this formula:

Vₜ = √(2mg / ρACd)

  • Vₜ — Terminal velocity (m/s)
  • m — Mass of the object (kg)
  • g — Gravitational acceleration (9.81 m/s² on Earth)
  • ρ — Density of the fluid medium (kg/m³; ~1.2 for air at sea level)
  • A — Cross-sectional area perpendicular to motion (m²)
  • Cd — Drag coefficient (dimensionless, depends on shape)

Key Factors Controlling Terminal Velocity

Object properties: Heavier objects reach higher terminal velocities because gravity must overcome more inertia. A doubled mass roughly increases terminal velocity by 41% (√2). Similarly, streamlined shapes have lower drag coefficients. A sphere has Cd ≈ 0.47, while a teardrop shape might be 0.04—producing vastly different fall speeds.

Cross-sectional area: Larger projected areas catch more air and increase drag significantly. This is why a skydiver falling feet-first (small area) reaches about 60 m/s, but spreads wide in a belly-to-earth position and slows to 40 m/s. A parachute's enormous area reduces terminal velocity to a safe 5–7 m/s for landing.

Medium density: Terminal velocity is inversely proportional to fluid density. Objects fall much faster through air than water. A human reaches 60 m/s in air but only 3 m/s in water, both calculated with the same formula but different ρ values.

Worked Example: Skydiver Terminal Velocity

A skydiver with mass 75 kg falls in a belly-to-earth position with cross-sectional area 0.18 m² and drag coefficient 0.7. Air density is 1.2 kg/m³.

Vₜ = √(2 × 75 × 9.81 / (1.2 × 0.18 × 0.7))
Vₜ = √(1471.5 / 0.1512)
Vₜ = √9737 ≈ 98.7 m/s ≈ 355 km/h

This matches real skydiving data. If the diver assumes a head-down position (area 0.08 m²), the denominator shrinks and terminal velocity climbs to ~140 m/s, explaining why headfirst falls are much faster.

Common Pitfalls and Practical Notes

Avoid these mistakes when calculating or interpreting terminal velocity.

  1. Confusing terminal velocity with average velocity — Terminal velocity is the final, steady-state speed—not the average speed during the fall. If a skydiver jumps from 4000 m and reaches terminal velocity after just 10 seconds, they spend most of their descent at constant speed, so average velocity is much higher than initial velocity but lower than terminal velocity.
  2. Neglecting shape and orientation changes — Drag coefficient is not fixed for a given object—it changes dramatically with orientation and surface texture. A baseball's Cd varies between 0.2 and 0.5 depending on spin rate and seam orientation. Always verify your Cd value matches the exact configuration you're modelling.
  3. Ignoring air density variations — Air density drops with altitude: at sea level it's ~1.2 kg/m³, but at 10,000 m it's ~0.4 kg/m³. A skydiver's terminal velocity increases as they descend because density rises, reducing drag and allowing faster speeds. Use local conditions, not global averages.
  4. Forgetting that real objects don't instantly reach terminal velocity — Terminal velocity is a limiting speed, reached asymptotically. Most objects get within 5% of it after 12–30 seconds of falling. Very light objects (feathers, parachutes) take much longer. The calculator gives the final equilibrium speed, but actual motion during the first few seconds involves acceleration.

Frequently Asked Questions

At what point does a falling object reach terminal velocity?

An object approaches terminal velocity asymptotically as drag force builds up. Heavier, denser objects like steel balls reach equilibrium quickly—within 5–10 seconds of falling. Lighter objects like feathers or parachutes take much longer because their lower mass-to-drag ratio means acceleration is gentler. In practice, most everyday objects are within 95% of terminal velocity after 12–30 seconds. Once achieved, the object descends at constant speed until environmental conditions change.

Why does a skydiver fall faster head-down than belly-to-earth?

Head-down orientation dramatically reduces cross-sectional area facing the air. Belly-to-earth might present 0.18 m²; head-down shrinks this to 0.05–0.08 m². Since terminal velocity is inversely proportional to area, this reduction increases speed by 50–80%. A belly-to-earth skydiver reaches ~55 m/s, while head-down exceeds 90 m/s. Drag coefficient also changes slightly with body position, but area is the dominant factor. This is why accuracy matters when inputting dimensions.

What is terminal velocity in water versus air?

Objects fall roughly 7–8 times slower in water than air because water density is ~800 times higher. A steel ball might fall at 40 m/s in air but only 5–6 m/s in water. The formula includes fluid density (ρ) in the denominator, so higher density directly reduces terminal velocity. This is why humans can jump safely into water from heights that would be fatal on land, and why parachutes work: even in air, their enormous area increases the effective ρ term.

How do you measure the drag coefficient for a custom object?

Drag coefficient is determined experimentally through wind tunnel testing, computational fluid dynamics (CFD) simulations, or dropping tests with high-speed video. Standard values exist for common shapes: spheres (0.47), cylinders (0.5), and streamlined bodies (0.04–0.1). If your object doesn't match these, engineers use CFD to calculate Cd. For rough estimates, you can time a drop from a known height and back-calculate Cd if you measure the actual fall speed, though this requires precise measurements and corrections for non-terminal-velocity effects.

Why does a baseball have a lower drag coefficient than a smooth sphere?

A baseball's drag coefficient (~0.33) is lower than a smooth sphere's (0.47) due to its seams. Paradoxically, the rough surface reduces drag by promoting a thin turbulent boundary layer that separates later than on a smooth ball. This is called the Magnus effect and drag crisis. Spin also modifies Cd: a spinning baseball experiences different aerodynamics than a stationary one. Golf balls exploit this with dimples, lowering Cd to ~0.26. Smooth spheres are actually aerodynamically worse than textured ones.

Can terminal velocity be exceeded?

In principle, no—terminal velocity is defined as the speed where forces balance. However, objects can exceed it transiently if pushed by a sudden force or if the medium suddenly becomes less dense (like a parachutist entering thinner air at high altitude). Once conditions stabilise, drag will increase and speed drops back to terminal velocity for the new conditions. Meteorites and space capsules enter the atmosphere at hypersonic speeds well above their equilibrium terminal velocity, but decelerate as drag builds until reaching a lower, stable speed corresponding to their terminal velocity at that altitude.

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