= Groundspeed
\b[Groundspeed] is the magnitude of an <aircraft>'s horizontal <velocity> relative to the ground. It determines travel time along a ground track, whereas aerodynamic <lift> and <drag> depend on motion relative to the air. Let $\mathbf v_{g,h}$ and $\mathbf v_{a,h}$ be the horizontal ground-relative and air-relative <velocity vectors>, and let $\mathbf w_h$ be the horizontal wind <velocity>:
$$
\mathbf v_{g,h}=\mathbf v_{a,h}+\mathbf w_h,\qquad V_g=\lVert\mathbf v_{g,h}\rVert.
$$
For climbing or descending flight this horizontal convention distinguishes <groundspeed> from the magnitude of the full three-dimensional ground-relative <velocity>.<Relative Velocity by NASA Glenn Research Center>[Ground Reference, Wind Speed, Ground Speed and Airspeed; velocity-frame relation]{ref}
In an original horizontal-flight example, an <airspeed> of $50\,\mathrm{m/s}$ directed due north and a wind blowing east at $15\,\mathrm{m/s}$ give $\mathbf v_{g,h}=(15,50)\,\mathrm{m/s}$ in east/north coordinates. Thus $V_g=\sqrt{2725}\approx52.20\,\mathrm{m/s}$, with the track about $16.70\,{}^\circ$ east of north. Adding scalar speeds would wrongly give $65\,\mathrm{m/s}$.
To keep a due-north track in the same wind, the <aircraft> must instead supply an air-relative westward component of $15\,\mathrm{m/s}$. At the same <airspeed>, its northward component becomes $\sqrt{50^2-15^2}\approx47.70\,\mathrm{m/s}$, so \b[the on-track groundspeed is smaller than the airspeed]. The required <wind correction angle> is $\arcsin(15/50)\approx17.46\,{}^\circ$ west of north. Heading and track are distinct directions.
Timed positions from a ground-based tracker estimate <groundspeed>, not <airspeed>. Inferring the latter also needs the wind <velocity>. Conversely, a <Pitot-static tube> does not measure travel over the ground. The same distinction explains why a headwind changes the ground range of a steady <glider (aircraft)> without changing its air-relative <glide angle>.
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