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Ground speed from airspeed and wind
Ground speed when the aircraft flies at a given airspeed and heading in a wind of known speed and direction.
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- Vg [m/s]
Velocity over the ground is the velocity through the air plus the velocity of the air. With the aircraft heading ψ (clockwise from north) at true airspeed Va, and a wind of speed W blowing from direction δw, the north and east ground-speed components are Va cos ψ − W cos δw and Va sin ψ − W sin δw. The ground track and the drift angle are shown among the derived results.
Ground speed when the aircraft flies at a given airspeed and heading in a wind of known speed and direction. Ground velocity is air velocity plus wind velocity, so in a crosswind the track differs from the heading. The wind direction δw is the direction it blows from.
Calculator
- Ground track, clockwise from north99.35 °
- Drift angle, track − heading9.354 °
- Headwind component, W cos(δw − ψ) (negative is a tailwind)7.071 m/s
- Crosswind component from the right, W sin(δw − ψ)−7.071 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Flying heading 90° at 50 m/s in a 10 m/s wind from 45° gives a ground speed of 43.51 m/s, 6.49 m/s slower than the airspeed, and a track of 99.35°. That is a drift of 9.35° from the heading.
- The wind splits into 7.07 m/s headwind and 7.07 m/s of crosswind from the left. To make good the heading as a track, the aircraft would have to point about 8.13° into the wind (the wind correction angle).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Va True airspeed | 20 to 250 m/s | 14.74 | 243 | 228.3 | +10% | +11.4 % |
| ψ Heading (clockwise from north) | 0 to 360 ° | 40.02 | 59.99 | 19.97 | +10% | +3.1 % |
| δw Direction the wind blows from | 0 to 360 ° | 40.07 | 59.95 | 19.89 | +10% | −1.42 % |
| W Wind speed | 0 to 40 m/s | 35.36 | 50 | 14.64 | +10% | −1.32 % |
Sweep of Wind speed
| Wind speed (m/s) | Ground speed (m/s) |
|---|---|
| 0 | 50 |
| 5 | 46.6 |
| 10 | 43.51 |
| 15 | 40.8 |
| 20 | 38.55 |
| 25 | 36.84 |
| 30 | 35.76 |
| 35 | 35.36 |
| 40 | 35.66 |
Sweep one variable
Variables and how each one contributes
Ground speed
What the equation gives you. Everything else feeds it.
True airspeed
The true airspeed in m/s, the speed relative to the air. Raises the ground speed by almost the same amount, since the wind is small next to it. Raising it by 10 % (to 55) moves the result by +11.4 %, from 43.51 to 48.45 m/s.
Heading (clockwise from north)
The heading in degrees, clockwise from north: the direction the aircraft points. Turning the heading changes how much of the wind is a headwind or a tailwind. Raising it by 20° (to 110°) moves the result by +7.25 %, from 43.51 to 46.66 m/s.
Wind speed
The wind speed in m/s. Changes the ground speed by the component of the wind along the heading, and turns the track by the crosswind part. Raising it by 10 % (to 11) moves the result by −1.32 %, from 43.51 to 42.93 m/s.
Direction the wind blows from
The direction the wind blows from, in degrees clockwise from north (a westerly wind is from 270°). Rotating the wind direction changes how much of the wind opposes or aids the aircraft. Raising it by 20° (to 65°) moves the result by −5.41 %, from 43.51 to 41.15 m/s.
Worked example with the values above
- North component: Va cos ψ − W cos δw = 50 × 6.123×10−17 − 10 × 0.7071 = −7.071 m/s.
- East component: Va sin ψ − W sin δw = 50 × 1 − 10 × 0.7071 = 42.93 m/s.
- Vg = √(50 + 1,843) = 43.51 m/s on a track of 99.35°.
Assumptions and limits
- Steady, uniform wind and a constant true airspeed and heading, with a level flight path (no climb).
- Heading and track measured in the same reference (true north, not magnetic).
Common mistakes
- Using the direction the wind blows towards instead of where it comes from. A wind 'from 270°' blows towards the east.
- Using indicated airspeed. The wind triangle needs true airspeed.
Statistics and uncertainty
An error in the assumed wind translates almost one for one into an error in ground speed along the track: a 2 m/s error in the wind component along the heading is 4.6 % of the ground speed here, which is also the error in time and fuel for a leg flown on that estimate. Winds vary with altitude and time, so a forecast value is a mean with considerable spread.
Where it is used
- Planning legs, time and fuel in a wind.
- Separating the effects of airspeed, heading and wind in recorded navigation data.
Maths you need
- Vector addition: ground velocity = air velocity + wind velocity
- Compass directions: angles clockwise from north