The Triangle of Velocities and Dead Reckoning - ATPL 061 — General Navigation

The wind components of the last lesson told you what the wind does; this lesson is about what the pilot does back. Its centrepiece is the oldest diagram in air navigation, the triangle of velocities, and around it sit the working angles of every leg: drift, the wind correction angle, the track error, and the one-in-sixty rule that turns miles off track into degrees of correction. It closes with dead reckoning, the art of knowing where you are when nothing outside confirms it.

Drift and the wind correction angle

Point the nose one way and the wind carries you another. Drift is the angle between the heading flown and the track actually made over the ground, named left or right as seen from the aircraft: a wind from the left drifts the aircraft to the right. The actual track follows directly: track = heading plus right drift, or minus left drift. A heading of 262 with 8° of right drift makes a track of 270. To stop the wind winning, the pilot aims off: the WCA (wind correction angle) is the angle between the desired track and the heading, held into the wind, sized so the drift it invites exactly cancels. Wanting a track of 180 against a wind expected to drift the aircraft 7° left, the pilot steers 187, and the drift carries the aircraft back onto the line.

When the guess and the air disagree, the difference gets its own name. The TKE (track error) is the angle between the required track and the track actually flown, and it is the actual drift measured against the drift the WCA allowed for. Steering 082 for a required track of 090 (a WCA of 8° for an expected 8° of right drift), a crew that actually experiences 12° of right drift is making a track of 082 plus 12, which is 094: a TKE of 4° right. The crosswind component behind all of this is the one the last lesson's recipes produce: by trigonometry, wind speed × sin(angle off track), or by MDR (mental dead reckoning), the angle in sixtieths of the wind. One instance carried through this lesson: a 24 kt wind 45° off the track is 24 × sin 45°, about 17 kt of crosswind, or by the sixtieths, 45/60 of 24, 18 kt.

The triangle of velocities

Every wind problem in this subject is one picture. The TOV (triangle of velocities) adds three vectors nose to tail: the air vector (heading and TAS), the wind vector (the W/V, the wind velocity), and their sum, the ground vector (track and GS). Drawing it correctly includes labelling it correctly, and the convention is counted in arrowheads:

one arrowhead marks the air vector, heading and TAS;
two arrowheads mark the ground vector, track and GS;
three arrowheads mark the wind vector, the W/V.

The triangle is solved, not admired. On the mechanical computer's wind face the standard problem, heading and GS from track, TAS and W/V, runs: set the wind direction under the index and mark the wind dot up from the centre; then set the required track under the index and slide the TAS arc under the dot. The wind dot then sits on the drift lines at the WCA, and the GS reads under the centre. The same face solves the reverse problems with their own settings. To find the W/V from heading and TAS plus track and GS: set the heading under the index and the TAS under the grommet, mark the dot where the observed drift line crosses the GS arc, then rotate the dot to the vertical below the grommet: the wind direction sits under the index and the dot's distance from the grommet is the wind speed. To find track and GS from heading, TAS and W/V: this time mark the wind dot down from the grommet, on the side the wind blows toward, then set the heading under the index and the TAS under the grommet: the dot then shows the drift on its line and the GS on its arc, and track = heading ± drift. The dot changes sides with the problem because the grommet changes meaning: it carries the ground vector's tip in the first problem and the air vector's tip in this one. The mental solution uses the last lesson's components: the crosswind sets the WCA, the along-track component sets the GS. For track 090, TAS 120 kt, wind 360/20: the wind is all crosswind from the left, the WCA works out at 10° left, heading 080, and with nothing on the nose but the cost of the correction itself, the GS settles just under the TAS at about 118 kt, exactly the triangle in the figure.

Speed factor and the one-in-sixty rule

The mental arithmetic of this subject runs on one conversion: the SF (speed factor) is the speed divided by 60, the aircraft's miles per minute, 2 for 120 kt, 2.5 for 150 kt. Its first job is the wind correction angle without a computer:

An 18 kt crosswind at a TAS of 90 kt (SF 1.5) demands a WCA of 18 over 1.5, which is 12°. The formula is the 1:60 rule in disguise, and the rule itself is the navigator's protractor: at 60 NM, 1 NM sideways subtends about 1°, so miles off track and degrees of correction convert through the distance. An off-course fix is corrected in two parts:

the TE (track error) is 60 × distance off / distance flown, the angle the flown track has diverged from the required one, the same angle the drift section named the TKE, now measured from distances;
the CA (closing angle) is 60 × distance off / distance to go, the extra angle needed to converge on the next waypoint;
the heading change to fly direct to the waypoint is TE + CA.

The same fix also measures the wind for you: flown with no correction, the divergence is pure drift, so the average drift angle since the last waypoint is the track error itself: 60 × distance off / distance flown. Six NM left of track after 90 NM is an average drift of 4° left, and that number, fed back as a WCA, is what keeps the next leg straight.

Dead reckoning

Between fixes, position is an act of arithmetic. A DR (dead reckoning) position is deduced from the last known position by applying the track and ground speed (or heading, TAS and wind) over the time elapsed: from a fix at 1010, on track 090 at GS 120 kt, the 1025 DR position sits 30 NM further down the 090 line. The DR position is a prediction, and its value is measured against reality: the difference between a DR position and an actual fix is the flight's error budget made visible: along-track difference means the GS was wrong, cross-track difference means the drift was wrong, and both feed the revised wind, WCA and log entries that keep the rest of the flight honest. Refining performance from exactly this comparison is what the VFR navigation module turns into technique.

What the exam asks

The one-in-sixty family and the WCA arithmetic are the bankers here; the triangle's labelling convention is a gift question for anyone who has drawn it once. The overview, following the learning objectives:

Calculate XWC by trigonometry and MDR; explain and apply drift and WCA; calculate track from heading and drift, and TKE from WCA and drift

Draw and correctly label the TOV; resolve it for heading and GS and for track and GS by computer and by MDR, and for W/V by computer

Define the speed factor; calculate WCA = XWC/SF; calculate the heading change at an off-course fix by the 1:60 rule, and the average drift from the fix

Determine a DR position and evaluate the difference between a DR position and a fix

Drift and the wind correction angle

The triangle of velocities

Check you understood

Speed factor and the one-in-sixty rule

Dead reckoning

What the exam asks