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Navigation
Ground speed and course from NED velocity
Speed over the ground and direction of travel from the north and east velocity components.
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- v1
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- Result
- Vg [m/s]
vN and vE are the north and east components of the velocity over the ground, as a satellite receiver or inertial unit reports them. The ground speed is the length of the horizontal velocity vector, and the course χ = atan2(vE, vN) is its direction measured clockwise from north. The course is the direction of travel, which in a crosswind differs from the heading of the aircraft. The course is shown among the derived results.
Speed over the ground and direction of travel from the north and east velocity components. The course is measured clockwise from north.
Calculator
- Course, atan2(vE, vN), clockwise from north18.43 °
- Ground speed in km/h113.8 km/h
- Ground speed in knots61.47 kt
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.
- With 30 m/s to the north and 10 m/s to the east the ground speed is 31.62 m/s on a course of 18.43°, which is east of north. Reversing the east component to −10 m/s gives 341.6°, west of north.
- Course and heading are different things: in a crosswind an aircraft pointing along its heading moves over the ground along its course. The difference is the drift angle (see the wind calculation).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| vN North velocity | −60 to 60 m/s | 10 | 60.83 | 50.83 | +2.4 m/s | +7.23 % |
| vE East velocity | −60 to 60 m/s | 30 | 67.08 | 37.08 | +2.4 m/s | +2.65 % |
Sweep of East velocity
| East velocity (m/s) | Ground speed (m/s) |
|---|---|
| −60 | 67.08 |
| −45 | 54.08 |
| −30 | 42.43 |
| −15 | 33.54 |
| 0 | 30 |
| 15 | 33.54 |
| 30 | 42.43 |
| 45 | 54.08 |
| 60 | 67.08 |
Sweep one variable
Variables and how each one contributes
Ground speed
What the equation gives you. Everything else feeds it.
North velocity
The velocity component towards north in m/s. Negative values are towards south. Raises the ground speed, by less than the same percentage because the east component still contributes. Raising it by 10 % (to 33) moves the result by +9.04 %, from 31.62 to 34.48 m/s.
East velocity
The velocity component towards east in m/s. Negative values are towards west. Raises the ground speed, and turns the course towards east. Raising it by 10 % (to 11) moves the result by +1.04 %, from 31.62 to 31.95 m/s.
Worked example with the values above
- Squares: vN² + vE² = 900 + 100 = 1,000.
- Ground speed: Vg = √1,000 = 31.62 m/s.
- Course: atan2(10, 30) = 18.43°, clockwise from north.
Assumptions and limits
- Local north–east axes, with the velocity measured over the ground (not through the air).
- A horizontal velocity only. A vertical component, when present, is not included in the ground speed.
Common mistakes
- Using atan2(vN, vE), which gives an angle from east, counter-clockwise, rather than a compass course.
- Reporting a negative course. Add 360° so that the course lies between 0° and 360°.
Statistics and uncertainty
If each velocity component has independent noise of σ, the ground speed has noise of about σ and the course has noise of about σ/Vg in radians. With σ = 0.1 m/s that is 0.18° at 31.6 m/s but 5.7° at 1 m/s, so the course is poorly defined at low speed.
Where it is used
- Turning GNSS velocity output into speed and track, and comparing it with heading and airspeed.
- Checking the direction and speed of a moving target given its velocity components.
Maths you need
- Pythagoras' theorem: length of a vector from its components
- The atan2 function: angle of a point, with the quadrant resolved