General Navigation Problems — Page 514, Lesson 517

Let’s pick up with the idea that a wind blowing exactly at right angles to your track does not give you a zero headwind or tailwind component. That’s the surprising bit we just proved with the 5-knot headwind on both 090°(T) and 270°(T). Now I want to show you the general rule, then move into a new calculation: Rhumb Line track angles. First, the general rule. If the wind is at right angles to your track, you still get a small headwind or tailwind component. The same is true if the wind is at right angles to your heading. Strictly speaking, the zero wind component occurs only when the wind is at right angles to the bisector between heading and track. That’s the line exactly halfway between the two. For practical flying this is an academic point, but it’s the precise definition, and you should know it. Now let’s switch to Rhumb Line track angles. You already know how to calculate distance in nautical miles from a change of latitude, and how to calculate distance from a change of longitude using the departure formula. Now we add the tangent relationship to find the track angle itself. Here’s a typical example. 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), or 150°(T). The key is to draw the diagram as though it were a Mercator chart — a rectangular graticule, where meridians and parallels are straight lines crossing at right angles. On such a chart, a straight line is a Rhumb Line, which crosses every meridian at the same angle. First, the change of latitude from 45°00'N to 48°30'N is 3°30', which is 210 nautical miles. That’s your north-south distance. For the east-west distance, we use the departure. Strictly speaking, we should use the departure at mid-latitude, which here is 46°45'N. But because the change of latitude is small, it doesn’t make much difference. In fact, on this example, using the departure at 45°N gives an answer slightly closer to the correct one. The departure formula is: departure equals change of longitude in minutes times cosine of latitude. The change of longitude from 010°00'W to 015°00'W is 5 degrees, which is 5 times 60, or 300 minutes. Multiply that by the cosine of 45°, which is 0.7071. That gives 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 the angle x equals opposite over adjacent, which is 210 over 212, or 0.99. Therefore angle x is 44.7°. To get the track, we start from the track westwards along the parallel of latitude, which is 270°(T). We add angle x to that, giving 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, then add or subtract it from the cardinal track along the parallel. That gives you the Rhumb Line track angle.