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Free Fall with Quadratic Air Resistance — result sheet
Estimate fall time, final speed, and terminal speed for an object dropped from rest under a constant quadratic-drag model.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: vₜ = √(2mg/(ρCᴅA)); y(t)=(vₜ²/g)ln cosh(gt/vₜ); t=(vₜ/g)acosh(exp(gy/vₜ²)); vfinal=vₜtanh(gt/vₜ).
With quadratic drag, downward acceleration decreases as speed increases and approaches a terminal speed. For an object released from rest, the analytic hyperbolic-tangent solution gives the time needed to cover a specified height and the speed at that time. This is more informative than a vacuum free-fall result when the drag coefficient, area, and air density are known approximately.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Drop height — m; minimum 0; maximum 1000000000
- Object mass — kg; minimum 1.0E-6; maximum 1000000000
- Frontal area — m²; minimum 1.0E-6; maximum 1000000
- Drag coefficient — minimum 1.0E-6; maximum 10
- Air density — kg/m³; minimum 1.0E-6; maximum 100
- Gravitational acceleration — m/s²; minimum 1.0E-6; maximum 1000
Worked example
| Input | Value |
|---|---|
| Drop height | 100 |
| Object mass | 75 |
| Frontal area | 0.7 |
| Drag coefficient | 1 |
| Air density | 1.225 |
| Gravitational acceleration | 9.80665 |
About 4.956 s fall time, 34.19 m/s final speed, and 41.42 m/s terminal speed.
Assumptions and limits
- The object starts from rest and falls vertically through still air with constant frontal area and drag coefficient.
- Drag follows Fᴅ=½ρCᴅAv², gravity is constant, and the object does not change orientation or break apart.
- The result is an idealized model; wind, turbulence, lift, ground effect, parachute deployment, and human or structural safety are not modeled.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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