Distance and Speed - ATPL 061 — General Navigation

Navigation runs on two currencies, miles and knots, and this lesson mints both. The minutes of latitude the last lesson counted become distances you can fly; the airspeed on the instruments becomes, step by step, a speed over the ground; and at the end the two meet in the small, constant arithmetic of every leg: speed, distance, time.

The nautical mile and its relatives

The unit of this whole subject is built from the grid itself: 1 NM (nautical mile) is equal to 1.852 km, which is the average distance of 1 minute of latitude change on the WGS-84 ellipsoid. Because meridians are all the same size, that holds anywhere on Earth: a minute of latitude is a nautical mile, whether counted at the equator or at 80°N. Longitude offers the same convenience exactly once: 1 minute of longitude change at the equator is approximately equal to 1 NM, because the equator is the one parallel that is a full-size great circle. Away from it the parallels shrink, and a minute of longitude shrinks with them, which is the whole story of the next section.

The exam also expects fluent conversion between the units the aviation world actually mixes:

1 NM = 1.852 km = about 6,076 ft;
1 SM (statute mile) = 1.609 km = 5,280 ft, the road mile, used for visibility in some countries;
1 m = about 3.28 ft, and 1 ft = 12 in (inches).

One worked line as a template: a distance of 216 km is 216 divided by 1.852, about 117 NM; the same 216 km in statute miles is 216 divided by 1.609, about 134 SM. The nautical mile is the largest of the three mile-like units, so converting into NM always makes the number smaller.

Distances on the grid

Along a meridian the grid does the work: the distance between two positions on the same meridian is the change of latitude converted to minutes, one NM per minute: from 45°30'N to 52°00'N is 6°30', which is 390 minutes, so 390 NM. Two positions on opposite (antipodal) meridians lie on one great circle over the pole, and the same counting runs over the top: add each position's distance from the pole: from 70°N on one meridian to 75°N on the anti-meridian is 20° up to the pole and 15° down the far side, 35° in all, so 2,100 NM.

Along a parallel the mile-per-minute rule fails, because every parallel except the equator is smaller than a great circle. The east-west distance corresponding to a change of longitude is named the departure, and it carries the shrinkage as a cosine:

At 60°N, where the cosine is exactly one half, 10° of longitude is 600 minutes times 0.5, which is 300 NM, half its equatorial size. The same formula runs backwards to find a new position from a distance flown: 240 NM due north is 240 minutes, exactly 4° of latitude; 240 NM due east at 60°N is 240 divided by 0.5, which is 480 minutes, 8° of longitude. North-south distances convert at face value; east-west distances are divided by the cosine before they become degrees.

Air distance

A leg has two lengths: the ground distance the chart shows, and the distance the aircraft travels through the moving air to cover it. The second is the air distance, measured in NAM (nautical air miles), and the two are connected through the two speeds:

Into a headwind the ground speed drops below the TAS, the ratio exceeds one, and the air distance is greater than the ground distance: a headwind stretches the leg through the air, a tailwind shortens it. A 350 NM leg flown at TAS 460 kt against a GS of 400 kt costs 350 times 460 over 400, about 403 NAM. Altitude enters through the same ratio: climbing changes the TAS and usually the wind, so the same ground leg can cost a different number of air miles at a different level, and air distance, not ground distance, is what fuel burn follows.

From CAS to TAS to Mach

The airspeed indicator reads CAS (calibrated airspeed), and the air thins with altitude, so the true speed through the air runs ahead of it. The precise conversion is the mechanical computer's job: set the outside air temperature against the pressure altitude in the airspeed window, and the TAS then reads on the outer scale opposite the CAS on the inner. With −5°C set against 10,000 ft, opposite CAS 160 kt the outer scale shows a TAS in the high 180s. The mental version is the rule of thumb: TAS is about 2 per cent greater than CAS for every 1,000 ft of altitude. At 10,000 ft that is 20 per cent: CAS 160 kt is roughly TAS 192 kt, the thumb rounding a few knots high of the computer's figure. High-speed flight brings the third scale: the Mach number is the TAS divided by the local speed of sound, so

At the standard −56.5°C of the high cruise the speed of sound is about 573 kt, so Mach 0.80 is a TAS of about 458 kt; the same sum reversed turns a TAS into a Mach number. What the exam most likes to probe is how the three speeds move against each other, and the key is that each conversion has altitude inside it. Climbing at constant CAS, the thinning air pushes TAS up, and the falling temperature drags the speed of sound down, so the Mach number rises even faster: constant CAS in the climb means TAS and M both increase. Climbing at constant Mach the logic reverses: the falling speed of sound pulls TAS down with it, and CAS falls faster still: constant M in the climb means TAS and CAS both decrease (above the tropopause, where the ISA temperature stops falling, constant M becomes constant TAS). Descents mirror the climbs, and in level cruise the three simply stay locked together by the local temperature.

Ground speed

The wind stands between TAS and the speed that actually covers the chart. Its first-order effect splits along the track:

the HWC (headwind component) or TWC (tailwind component) is the part of the wind blowing along the track: by trigonometry, wind speed × cos(angle off the track), and by MDR (mental dead reckoning), the sixtieths rule applied to the complement, (90° − angle)/60 of the wind speed;
the XWC (crosswind component) is the part blowing across it: by trigonometry, wind speed × sin(angle off the track), and by MDR, roughly the angle itself in sixtieths of the wind speed.

A 30 kt wind 40° off the nose carries a headwind component of 30 times cos 40°, about 23 kt, and a crosswind of 30 times sin 40°, about 19 kt. The MDR versions land close enough for mental work: the crosswind is 40 sixtieths of 30, 20 kt, and the headwind is the complement's share, 50 sixtieths of 30, 25 kt. Both shortcuts cap at the full wind, since no component can exceed it: within about 30° of the nose call the wind all headwind, beyond about 60° all crosswind. Then GS = TAS − HWC, or GS = TAS + TWC: at TAS 120 kt against that 23 kt component the ground speed is about 97 kt. The crosswind's cost returns in the next lesson as the wind correction angle; what matters here is its side effect: the larger the WCA (wind correction angle) the aircraft must hold, the smaller the share of TAS left pointing along the track, so GS falls as WCA grows even before any headwind is counted. The full triangle, solved on the mechanical computer, is the next lesson's subject.

With GS in hand, every timing question is one proportion: time in minutes = 60 × distance / GS. At GS 210 kt, 70 NM takes 60 times 70 over 210, which is 20 minutes. The same proportion runs in every direction the exam can turn it:

average GS between two fixes is 60 × the distance between them / the minutes between them: 34 NM in 12 minutes is a GS of 170 kt;
revised GS to make a time is 60 × the distance to go / the minutes remaining: 96 NM to cross in 32 minutes demands 180 kt.

The paperwork closes the loop: when the wind or the ground speed changes, the flight log's remaining legs are re-entered with the revised GS, and the times and fuel figures move with them: a lower GS means later ETAs (estimated times of arrival) and more fuel burned per leg, a higher GS the reverse. A log that still shows the planned figures after the wind has changed is a record of a flight that no longer exists.

What the exam asks

This is a calculating lesson: the unit conversions, the departure cosine, the NAM ratio and the 60-times proportion carry nearly all the marks. The overview, following the learning objectives:

State the NM/km relationship and the minute-of-latitude and minute-of-longitude facts; convert between NM, km, SM, ft and in

Calculate distances along meridians, over the pole between antipodal meridians, and along parallels; find a new position from north-south and east-west distances

Convert ground distance to air distance with NAM = NM × TAS/GS and evaluate the effect of wind and altitude

Calculate TAS from CAS by the 2%-per-1,000-ft rule, TAS from Mach and back, and deduce how CAS, TAS and M move in climb, cruise and descent

Calculate head/tail and crosswind components by trigonometry and MDR; apply them to GS; perform GS/distance/time, revised-GS and average-GS calculations, and revise the flight log