Free-Fall with Air Resistance Calculator
Terminal velocity, speed & distance as a function of time with air drag — complete with comparison vs vacuum free fall. 100% runs in the browser, no server.
Scope of this tool
- This tool = fall kinematics WITH air drag (quadratic drag ∝ v²): terminal velocity, v(t), y(t), and comparison vs vacuum.
- Not vacuum free fall (no air): use Free-Fall Distance (#422) and Free-Fall Velocity (#423).
- Not the drag force itself (Fd = ½·ρ·v²·Cd·A as a force/newtons): use Drag Equation Calculator (#1592).
- This model assumes quadratic drag (high Reynolds number, M ≪ 1) and constant Cd·A during the fall; it does not model changes in body orientation or Stokes drag (∝ v) for very small/slow objects.
General parameters
1 · Terminal velocity
2 · v(t) & y(t) profile vs vacuum
| t (s) | drag v (m/s) | drag y (m) | vacuum v (m/s) | vacuum y (m) | v reduction (%) |
|---|
3 · Time to reach x% of terminal velocity
4 · Fall time from height (with vs without air)
Verified examples (80 kg skydiver, Cd·A = 0.42 m², ρ = 1.225 kg/m³)
- Terminal velocity vt = √(2·80·9.80665/(1.225·0.42)) = 55.22 m/s (198.8 km/h ≈ 123.5 mph).
- At t = 5 s: v = 39.23 m/s, fall distance 109.22 m.
- Reaching 90% of vt (49.70 m/s) takes 8.29 s; 99% takes 14.90 s.
- Falling 100 m: 4.76 s with drag vs 4.52 s in vacuum; impact speed 38.03 m/s vs 46.69 m/s in vacuum — an 18.5% reduction.
All numbers were recomputed with 60-digit Decimal and frozen in vectors.json before the engine was written.
Frequently asked questions
What is the terminal velocity of a skydiver?
For an 80 kg skydiver in the spread eagle position (C_d·A = 0.42 m², sea-level air ρ = 1.225 kg/m³), terminal velocity is 55.22 m/s, equivalent to 198.8 km/h or 123.5 mph, computed from v_t = √(2·m·g/(ρ·C_d·A)). The head-down position (C_d·A ≈ 0.16 m²) raises v_t to about 90 m/s.
What is the formula for falling speed with air resistance?
With quadratic drag, v(t) = v_t·tanh(g·t/v_t), where v_t = √(2·m·g/(ρ·C_d·A)). Fall distance y(t) = (v_t²/g)·ln(cosh(g·t/v_t)). For small t (g·t ≪ v_t), tanh(x) ≈ x so v(t) ≈ g·t — the same as vacuum free fall.
How long does it take to reach 90% of terminal velocity?
The time is t = (v_t/g)·artanh(0.9). For the skydiver above (v_t = 55.22 m/s): t = 8.29 s, with a speed of 49.70 m/s. Reaching 99% takes 14.90 s — the approach to terminal is asymptotic.
How large is the effect of air resistance when falling 100 meters?
The skydiver (v_t = 55.22 m/s) falls 100 m in 4.76 s vs 4.52 s in vacuum (5.4% longer). Impact speed 38.03 m/s vs 46.69 m/s in vacuum — an 18.5% reduction.
What is the terminal velocity of a raindrop?
A 1 mm diameter raindrop (mass ≈ 0.524 mg, C_d·A ≈ 3.69e-7 m²) has a terminal velocity of 4.77 m/s (≈ 17.2 km/h). This is why rain is harmless even though it falls from clouds.
Does air density affect terminal velocity?
Yes, v_t is inversely proportional to √ρ. At 2,000 m altitude (ρ = 1.006 kg/m³), the skydiver’s v_t rises from 55.22 to 60.94 m/s — a 10.3% increase.
How does this tool differ from a regular free-fall calculator?
This tool models air drag (quadratic): speed is capped by terminal velocity and distance uses ln(cosh). The free-fall-distance (#422) and free-fall-velocity (#423) calculators use the vacuum model (y = ½g·t², v = g·t) with no air at all.
How does this tool differ from the drag equation calculator?
The drag equation calculator (#1592) computes the drag force F_d = ½·ρ·v²·C_d·A (in newtons) at a single speed. This tool computes the kinematics (v(t), y(t), v_t, fall time) from the equation of motion with drag — different physics.
When is the quadratic drag model valid?
This model is valid for high Reynolds numbers (Re ≳ 1000) and speeds far below the speed of sound — suitable for humans, balls, and raindrops. For very small, slow dust/particles, Stokes drag (∝ v, not v²) applies instead.
Is this calculator accurate for planning skydives?
This is an educational tool, not for planning real jumps. A human body’s C_d·A varies with posture and clothing, air density changes throughout the fall, and skydiving safety involves far more factors. Always follow a certified instructor.
See also
- Free-Fall Distance — vacuum free-fall distance (no air).
- Free-Fall Velocity — vacuum free-fall speed (no air).
- Drag Equation Calculator — drag force Fd = ½ρv²CdA + power.
Free tool by NRTechStudio