# Wind Power Calculator (P = ½ρAv³)

Calculate wind turbine power output with P = 1/2 rho A v cubed. Enter wind speed, blade length, and efficiency. Shows theoretical, Betz limit, and practical power.

## What this calculates

Wind power depends heavily on wind speed because the energy scales with the cube of velocity. Doubling wind speed means 8 times more power. This calculator takes wind speed, blade length, air density, and turbine efficiency to estimate real-world power output, from a backyard turbine to a utility-scale wind farm.

## Inputs

- **Wind Speed** (m/s) — min 0
- **Blade Length (Rotor Radius)** (m) — min 0 — Small turbine: 1-3m, Utility: 50-80m
- **Air Density** (kg/m³) — min 0 — Decreases with altitude and higher temperatures
- **Turbine Efficiency** (%) — min 0, max 59.3 — Betz limit maximum is 59.3%

## Outputs

- **Swept Area** (m²) — Circular area swept by rotor blades
- **Total Wind Power (in swept area)** (W) — Total kinetic energy available in the wind
- **Betz Limit (59.3%)** (W) — Maximum extractable power (Betz limit)
- **Practical Power Output** (W) — Realistic output at the given efficiency
- **Practical Power Output** (kW) — Practical output in kilowatts
- **Daily Energy (24h at this speed)** (kWh) — Energy produced if wind stayed constant all day

## Details

## The Wind Power Equation

**P = 1/2 x rho x A x v cubed**

Where rho is air density (1.225 kg/m cubed at sea level), A is the swept area of the rotor (pi x r squared), and v is wind speed in m/s.

### The Cubic Relationship

This is the single most important thing about wind power: energy scales with the **cube** of wind speed.

| Wind Speed | Relative Power |
|-----------|---------------|
| 5 m/s | 1x (baseline) |
| 10 m/s | 8x |
| 15 m/s | 27x |
| 20 m/s | 64x |

A site with average 8 m/s winds produces more than twice the energy of a site with 6 m/s winds.

### The Betz Limit

German physicist Albert Betz proved in 1919 that no wind turbine can extract more than 59.3% of the wind's kinetic energy. If it captured 100%, the air would stop and pile up behind the rotor. Modern utility turbines achieve 35-45% overall efficiency (including generator and gearbox losses), which is impressively close to the theoretical maximum.

### Turbine Size Comparison

| Type | Blade Length | Rated Power |
|------|-------------|-------------|
| Small residential | 1-3 m | 0.4-10 kW |
| Community | 10-25 m | 50-500 kW |
| Utility onshore | 50-70 m | 2-5 MW |
| Offshore | 80-115 m | 8-15 MW |

### Air Density Effects

Air density decreases with altitude and temperature. At 1,500 m elevation on a hot day, density might be 1.05 kg/m cubed instead of 1.225, reducing output by about 14%. Wind farms in highlands account for this in their projections.

## Frequently Asked Questions

**Q: What is the Betz limit?**

A: The Betz limit states that no wind turbine can convert more than 59.3% of the kinetic energy in wind into mechanical energy. Albert Betz derived this in 1919 from conservation of mass and energy. If a turbine extracted all the energy, the air would stop moving and block incoming wind. The 59.3% figure is the sweet spot where air slows to one-third its original speed.

**Q: Why does doubling wind speed give 8 times the power?**

A: Power depends on velocity cubed (v x v x v). Doubling v means 2 x 2 x 2 = 8. This happens because faster wind delivers both more mass per second (proportional to v) and each unit of mass carries more kinetic energy (proportional to v squared). The combined effect is v cubed.

**Q: What wind speed do turbines need to start generating?**

A: Most turbines have a cut-in speed around 3-4 m/s (7-9 mph). Below that, there is not enough energy to overcome friction. They reach rated power at about 12-15 m/s and shut down at 25 m/s (56 mph) to prevent damage. The ideal average wind speed for a wind farm site is 6-9 m/s.

**Q: Can I use this for small home turbines?**

A: Yes. A small home turbine with 1.5 m blades in 6 m/s average wind with 30% efficiency produces about 23W of average power, or roughly 0.55 kWh per day. That is enough for a few LED lights or charging devices, but not for powering a whole household. Small turbines work best as supplements to solar panels.

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Source: https://vastcalc.com/calculators/physics/wind-power
Category: Physics
Last updated: 2026-04-08