Terminal Velocity Calculator

Find the terminal velocity where drag force equals gravitational force: v_t = √(2mg / CdρA).

🪂 Drag 📐 v_t = √(2mg/CdρA) 🌬️ Air Resistance
Mass (m) kg
Drag coefficient (Cd)
Cross-section area (A) m²
Air density (ρ) kg/m³

Cd presets: skydiver spread (1.0), bullet (0.295), sphere (0.47), streamlined car (0.25)

⚠️ Enter valid positive numbers.

Understanding Terminal Velocity

Terminal velocity is the constant speed a falling object reaches when the drag force exactly equals the gravitational force. At this point, net force = 0, acceleration = 0, and the object falls at constant velocity. The formula is v_t = √(2mg / (Cd·ρ·A)), where m is mass, g = 9.8 m/s², Cd is the drag coefficient, ρ is air density, and A is the cross-sectional area.

The drag force is F_d = ½·Cd·ρ·A·v². As an object falls and speeds up, drag increases (proportional to v²) until it matches gravity (mg). At this balance point, speed stops increasing. A heavier object (higher mg) reaches a higher terminal velocity; a larger area or higher Cd (blunter shape) reaches a lower terminal velocity.

Familiar terminal velocities: a skydiver in spread-eagle position ≈ 53 m/s (190 km/h); in head-down position ≈ 90 m/s (320 km/h). A falling raindrop ≈ 9 m/s. A cat ≈ 12.5 m/s (they orient and spread out instinctively). A golf ball ≈ 30 m/s. A human body after atmospheric re-entry (Felix Baumgartner) reached 377 m/s at high altitude where air density is much lower.

Terminal velocity also applies to objects rising through fluid (bubbles in liquid), vehicles coasting on a flat road (engine thrust vs. drag), and particles settling in centrifuges. In all cases, the balance between driving force and drag sets the limiting speed.

Formula Reference Table

Solve ForFormulaNotes
Terminal velocityv_t = √(2mg / (Cd·ρ·A))Balance of gravity and drag
Drag forceF_d = ½·Cd·ρ·A·v²Increases as v increases
At terminal velocitymg = F_dNet force = 0
With altitudeρ decreases → v_t increasesHigher altitude = higher terminal velocity
Drag coeff (Cd)Sphere: 0.47, Skydiver: 1.0–1.3Shape-dependent constant
Air density (ρ)Sea level ≈ 1.225 kg/m³Decreases with altitude

3 Worked Examples

Example 1
Skydiver — Find Terminal Velocity

80 kg skydiver, Cd = 1.0 (spread-eagle), A = 0.7 m², ρ = 1.225 kg/m³.

  • v_t = √(2×80×9.8 / (1.0×1.225×0.7))
  • v_t = √(1568 / 0.8575)
  • v_t = √1828.5 = 42.8 m/s (154 km/h) — standard value is ~53 m/s with arms and legs extended more
✓ Terminal velocity ≈ 43–53 m/s (spread eagle)
Example 2
Raindrop — Tiny Drop

A 2 mm diameter raindrop: m ≈ 4.19 mg, A = π(0.001)² ≈ 3.14×10⁻⁶ m², Cd ≈ 0.47.

  • v_t = √(2×4.19e-6×9.8 / (0.47×1.225×3.14e-6))
  • v_t = √(0.0000821/1.81e-6) = √45.4 ≈ 6.7 m/s
  • Larger raindrops (3–5 mm) reach 8–9 m/s
✓ Raindrop terminal velocity ≈ 7 m/s (25 km/h)
Example 3
High Altitude — Felix Baumgartner

At 39 km altitude, ρ ≈ 0.004 kg/m³. 80 kg jumper, A = 0.7 m², Cd = 1.0.

  • v_t = √(2×80×9.8 / (1.0×0.004×0.7)) = √(1568/0.0028)
  • v_t = √560,000 = 748 m/s (but limited by sound barrier effects at ~340 m/s)
  • Actual max was 377 m/s = Mach 1.25 — thin air allows supersonic freefall
✓ v_t ≈ 748 m/s (air too thin for normal drag limits)

Real-World Applications

🪂
Skydiving
Skydivers control terminal velocity by body position. Spread eagle ≈ 53 m/s; head-down ≈ 90 m/s. Wingsuits dramatically increase A, reducing terminal velocity to 40 m/s or less.
☔
Meteorology
Raindrop size determines fall speed (2–9 m/s), affecting how far drops drift horizontally in wind. This is critical for precipitation measurement and weather modeling.
⚕️
Drug Delivery
Inhaled particles < 1 μm float indefinitely; 5–10 μm particles deposit in bronchi. Pharmaceutical aerosols are engineered to specific sizes for targeted lung deposition.
🚗
Automotive Design
Car top speed is set by terminal velocity when engine force = aerodynamic drag. Cd × A is the key metric; reducing it allows higher top speed for the same engine power.
🌊
Particle Sedimentation
Stokes' law (low Reynolds number version) governs settling velocity in centrifuges and wastewater treatment. Larger/denser particles settle faster — enabling size-based separation.

Common Mistakes to Avoid

⚠️
Using wrong cross-section area

A is the frontal area (cross-section perpendicular to motion), not surface area. For a sphere of radius r: A = πr². For a skydiver: roughly shoulder width × hip width.

⚠️
Using wrong Cd

Drag coefficient is highly shape-dependent and must be found for the specific geometry. A sphere (Cd = 0.47) is very different from a streamlined body (Cd = 0.04). When in doubt, use experimental values.

⚠️
Ignoring density variation with altitude

Air density decreases exponentially with altitude: ρ(h) ≈ 1.225 × exp(−h/8,500) kg/m³. At 10 km (cruising altitude), ρ ≈ 0.41 kg/m³ — terminal velocity is ~1.73× higher than at sea level.

⚠️
Expecting terminal velocity to be reached instantly

Objects reach terminal velocity asymptotically — 63% of v_t after one time constant, 95% after three. The time constant τ = m/(Cd·ρ·A·v_t/2). A skydiver takes ~10–15 seconds to approach terminal velocity.

⚠️
Confusing terminal velocity with escape velocity

Terminal velocity is the maximum fall speed in a medium. Escape velocity is the minimum speed to leave a planet's gravity. They are completely different concepts.

Frequently Asked Questions

What is the formula for terminal velocity? ▾
v_t = √(2mg/(Cd·ρ·A)). This comes from setting gravitational force (mg) equal to drag force (½·Cd·ρ·A·v²) and solving for v. The formula shows: more mass → higher v_t; larger area → lower v_t; denser air → lower v_t.
Why does a heavier object fall faster in air but not in vacuum? ▾
In vacuum, all objects fall at the same rate (g). In air, terminal velocity v_t ∝ √(m/A). A denser (higher m/A ratio) object reaches higher terminal velocity. A cannonball (high m, small A) falls much faster than a feather (low m, large A) for this reason.
What is Cd (drag coefficient)? ▾
Cd is a dimensionless number representing aerodynamic bluntness. Sphere: 0.47. Bullet: 0.295. Flat plate: 1.17. Streamlined teardrop: 0.04. Human (standing): 1.0–1.3. Cd is determined experimentally in wind tunnels or CFD simulations.
How do skydivers control their terminal velocity? ▾
By adjusting body position: more surface area exposed to airflow = more drag = lower terminal velocity. Spread eagle maximizes A; tracking (arrowhead) or head-down minimizes it. Wingsuits add artificial wing area, dramatically reducing v_t to wingsuit glide speed (~40–45 m/s).
Does terminal velocity exist in water? ▾
Yes — terminal velocity in water uses the same formula but with water density (ρ_water ≈ 1000 kg/m³ vs. air ρ ≈ 1.225). Since water is 816× denser, terminal velocity in water is √816 ≈ 28.6× lower than in air for the same object. A skydiver's equivalent 'terminal velocity' in water is ≈ 1.8 m/s.
Why did Felix Baumgartner break the sound barrier? ▾
At 39 km altitude, air density ρ ≈ 0.004 kg/m³ — about 300× less than sea level. With v_t ∝ 1/√ρ, terminal velocity is √300 ≈ 17× higher than at sea level. For an 80 kg jumper at sea level v_t ≈ 53 m/s; at 39 km: 53 × 17 ≈ 900 m/s — well above Mach 1 (340 m/s at sea level, ~300 m/s at altitude).
What is Stokes' Law? ▾
For very small, slow-moving particles (low Reynolds number < 1), drag is linear in velocity: F_d = 6πηrv (Stokes' Law). Terminal velocity = 2r²(ρ_particle−ρ_fluid)g / (9η). Used for micron-sized particles in centrifuges, blood cells settling, and pharmaceutical aerosols.
How does terminal velocity affect impact injuries? ▾
At terminal velocity, kinetic energy KE = ½mv_t². A 70 kg person at 53 m/s has KE = 98,105 J = 23.4 kcal. This energy must be absorbed on impact — which is why landing on hard surfaces from terminal velocity is invariably fatal. Water, with surface tension, provides little energy absorption at these speeds.

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