Real falling objects reach a terminal velocity where drag balances weight: about 55 m/s (200 km/h) for a skydiver belly-down, but only a few m/s for a raindrop. This calculator solves the exact equations of motion for drag proportional to speed squared.
How to use the Free Fall with Air Resistance Calculator
- Mass
- Altitude (fall height)
- Gravitational acceleration g
- Air resistance from…
- Air resistance coefficient k (kg/m)
- Cross-section area
- Drag coefficient C_d
- Medium density ρ
- Speed for drag force
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose whether to enter the air resistance coefficient k directly or to build it from area, drag coefficient and air density. The calculator also gives the drag force at any speed you choose.
Inputs explained
| Field | What to enter |
|---|---|
| Mass | Units: kg, g, mg, t, lb, oz |
| Altitude (fall height) | Units: m, cm, mm, km, in, ft … |
| Gravitational acceleration g | Units: m/s², ft/s², g, cm/s² |
| Air resistance from… | Choose one: Coefficient k, Area, drag coefficient and density |
| Air resistance coefficient k (kg/m) | A number |
| Cross-section area | Units: m², cm², mm², km², in², ft² |
| Drag coefficient C_d | A number |
| Medium density ρ | Units: kg/m³, g/cm³, g/mL, lb/ft³, lb/in³ |
| Speed for drag force | Units: m/s, km/h, mph, ft/s, knots, km/s |
What each result means
- Terminal velocity = sqrt([Mass]×[Gravitational acceleration g]÷[k])
- Time of fall = [Terminal velocity]÷[Gravitational acceleration g]×ln(exp([Altitude (fall height)]×[k]÷[Mass]) + sqrt(exp(2×[Altitude (fall height)]×[k]÷[Mass]) – 1))
- Speed at the ground = [Terminal velocity]×(exp(2×[Gravitational acceleration g]×[Time of fall]÷[Terminal velocity]) – 1)÷(exp(2×[Gravitational acceleration g]×[Time of fall]÷[Terminal velocity]) + 1)
- Drag force at the chosen speed = [k]×[Speed for drag force]^2
- Air resistance coefficient k (kg/m) = [k]
Example values
How the main result changes when Mass changes and everything else stays at the example values above:
| Mass (kg) | Time of fall (s) |
|---|---|
| 37.5 | 31.042 |
| 56.25 | 26.356 |
| 75 | 23.699 |
| 93.75 | 21.977 |
| 112.5 | 20.771 |
| 150 | 19.199 |
Formula
k = ½ρC_dA, v_t = √(mg/k)
t = (v_t/g)·arcosh(e^{hk/m}), v = v_t·tanh(g t / v_t)
Worked example
Inputs — Mass: 75 kg; Altitude (fall height): 1,000 m; Gravitational acceleration g: 9.80665 m/s²; Air resistance from…: Area, drag coefficient and density; Cross-section area: 0.7 m²; Drag coefficient C_d: 0.7; Medium density ρ: 1.225 kg/m³; Speed for drag force: 30 m/s.
Results — Terminal velocity = 178.214 km/h; Time of fall = 23.699 s; Speed at the ground = 178.184 km/h; Drag force at the chosen speed = 270.112 N; Air resistance coefficient k (kg/m) = 0.30013.
Key terms
- Drag coefficient
- A shape factor for air resistance.
Frequently asked questions
What is terminal velocity?
The speed at which drag equals weight, so the object stops accelerating.
How fast does a skydiver fall?
About 200 km/h belly-down, faster head-down.
Why use the speed squared?
At everyday speeds and sizes, air drag grows roughly with the square of speed.
Does a heavier object fall faster with air resistance?
Yes, for the same shape – it needs more drag to balance its weight.