Free Fall with Air Resistance Calculator
This calculator solves the physics of free fall through a medium (air, water, etc.) where drag force is proportional to velocity: Fdrag = k · v. It computes terminal velocity, the time to reach the ground, and the velocity at impact.
Physics Behind the Calculator
When an object falls through air, two forces act on it: gravity (mg downward) and aerodynamic drag (kv upward). The resulting motion satisfies the differential equation:
m · (dv/dt) = mg − k·v
This equation has an exact analytical solution, giving the following key results:
Key Formulas
| Quantity | Formula | Description |
|---|---|---|
| Terminal velocity | vt = √(mg / k) | The maximum speed reached when drag equals gravity |
| Fall time | t = (vt/g) · arccosh(ehg/vt²) | Exact time from height h to ground |
| Impact velocity | vmax = vt · tanh(g·t / vt) | Velocity when the object hits the ground |
| Resistance coefficient | k = ρ · A · Cd / 2 | Can be entered directly or computed from components |
Parameters Explained
- Height (h) — The initial drop height above the ground. Enter in metres (metric) or feet (imperial).
- Mass (m) — The mass of the falling object. Enter in kilograms or pounds.
-
Air resistance coefficient (k) — A combined drag parameter in kg/m.
You can enter it directly, or tick "Calculate k from components" to derive it from:
- ρ (rho) — fluid/medium density, e.g. 1.225 kg/m³ for air at sea level
- A — cross-sectional area of the object in m² (or ft²)
- Cd — dimensionless drag coefficient (e.g. 0.47 for a sphere, 0.8 for a skydiver)
Typical Values
| Object | Mass | k (kg/m) | Terminal velocity |
|---|---|---|---|
| Skydiver (spread-eagle) | 80 kg | 0.24 | ~57 m/s (205 km/h) |
| Skydiver (head-down) | 80 kg | 0.10 | ~88 m/s (317 km/h) |
| Baseball | 0.145 kg | 0.0013 | ~33 m/s (118 km/h) |
| Raindrop (large) | 0.00034 kg | 0.000025 | ~9 m/s (32 km/h) |
| Feather | 0.003 kg | 0.05 | ~0.77 m/s (2.8 km/h) |
Units
The calculator supports two unit systems. Switch between them with the "Unit System" selector at the top of the form. All internal physics calculations are performed in SI units (metres, kilograms, m/s), and results are converted for display.
| Quantity | Metric | Imperial |
|---|---|---|
| Height | m | ft |
| Mass | kg | lb |
| Velocity | m/s, km/h | ft/s, mph |
| Area | m² | ft² |
| k coefficient | always kg/m | |
Notes and Limitations
- The model assumes drag force is linear in velocity (F = kv). At high speeds the quadratic drag (F = ½ρACdv²) is more realistic; use the component input to approximate it by entering the correct k at typical speeds.
- g = 9.81 m/s² (standard gravity) is used throughout.
- The formula uses the exact closed-form solution via the inverse hyperbolic cosine (arccosh), with an asymptotic approximation applied automatically when the fall height is very large to avoid numerical overflow.
- The result shows what percentage of terminal velocity the object has reached upon impact. When it is ≥ 99%, the object has effectively reached terminal velocity before hitting the ground.