General Navigation Problems — Page 514, Lesson 517

Let’s pick this up right where the wind-component question left off, because the first thing I want to show you is why a zero head/tail wind component doesn’t happen when the wind is at right angles to track. We had a TAS of 180 knots, and we found a 5-knot headwind whether we tracked 090°(T) or 270°(T). Draw the triangle. On a track of 090°(T), the wind is blowing from the north, so it’s pushing you southward, perpendicular to your track. Your ground speed ends up less than your TAS — 175 knots instead of 180. In trigonometrical terms, the hypotenuse is longer than the opposite or adjacent side. That’s the key: the TAS is the hypotenuse, and the ground speed is one of the shorter sides, so ground speed is always less than TAS when there’s any crosswind component. Now reverse the track to 270°(T). You get a mirror image of the same triangle. Still 180 knots TAS, still 175 knots ground speed — still a 5-knot headwind. So answer c was correct. Here’s the subtle point. Strictly speaking, a zero head/tail wind component does not occur when the wind is at right angles to track. And you can show, with the same drawing exercise, that it also doesn’t occur at right angles to heading. For practical purposes this is academic, but the precise truth is this: zero wind component is present when the wind is at right angles to the bisector between heading and track. That bisector is the line halfway between where you point the nose and where you actually go. So if the wind is perpendicular to that bisector, the along-track component is zero. Now let’s move to a new problem — calculating Rhumb Line track angles. You already know how to calculate distance in nautical miles for a change of latitude, and how to calculate distance for a change of longitude using the departure formula. Now we add the tangent relationship to find the track angle. Here’s a typical example. Question: what is the Rhumb Line track from A at 45°00'N 010°00'W to B at 48°30'N 015°00'W? The options are 315°(T), 330°(T), 215°(T), and 150°(T). First, draw these diagrams as though they were a Mercator chart — a rectangular graticule. On a Mercator chart, a straight line is a Rhumb Line. So we draw a rectangle, put A and B in their positions, and the straight line between them is our Rhumb Line track. The change of latitude from 45°00'N to 48°30'N is 3°30', which is 210 nautical miles. That’s our north-south distance. For the east-west distance, we use the departure. Strictly speaking, it should be the departure at mid-latitude — here that’s 46°45'N. But in these examples the changes of latitude are small, so it doesn’t make much difference. In fact, on this particular example, you get an answer slightly nearer the right one if you use the departure at 45°N. The departure formula is: departure equals change of longitude in minutes times cosine of latitude. Change of longitude from 010°00'W to 015°00'W is 5 degrees, which is 5 times 60, or 300 minutes. Cosine of 45° is 0.7071. So departure equals 300 times 0.7071, which is 212.13 nautical miles. Now we have a right-angled triangle. The opposite side is the change of latitude, 210 nautical miles. The adjacent side is the departure, 212 nautical miles. The tangent of angle x equals opposite over adjacent, which is 210 over 212, or 0.99. So angle x is 44.7°. Now, what is the track? From A, heading westwards along the parallel of latitude is 270°(T). We add x to that. 270 plus 44.7 gives 314.7°(T). That rounds to 315°(T), which is answer a. So the full method: find the change of latitude in nautical miles, find the departure using change of longitude in minutes times cosine of latitude, take the tangent as opposite over adjacent, get the angle, and then add or subtract it from the cardinal direction along the parallel. That gives you the Rhumb Line track angle.