General Navigation Problems — Page 514, Lesson 519

I want to walk you through a classic timing-to-a-beacon problem, because it traps a lot of pilots. The trap is this: you're asked to adjust your arrival time, and your instinct is to just scale your Mach number by the ratio of the new ground speed to the old one. That gives the wrong answer, and I'll show you exactly why. First, let's set up the situation. You're flying at a True Mach Number of 0.77, in a Static Air Temperature of minus 55 degrees Celsius. At 1000 hours you are 150 nautical miles from a DME — that's your Distance Measuring Equipment, giving you slant range to the beacon. Your ETA, your Estimated Time of Arrival, is 1017. ATC asks you to slow down so you arrive at 1021 instead. The question is: what new True Mach Number do you need if you make the speed change when you still have 150 nautical miles to go? Let me define the terms before we touch numbers. True Mach Number, TMN, is the ratio of your True Airspeed to the local speed of sound. True Airspeed, TAS, is your actual speed through the air mass. Ground speed, G/S, is your speed over the ground — that's TAS combined with the wind component. If you have a headwind, ground speed is less than TAS; if a tailwind, ground speed is greater. And here's the key point I want you to hold onto: what you actually adjust in the cockpit is Mach Number or CAS, Calibrated Airspeed, and those are directly proportional to TAS, not to ground speed. The wind is invisible to your airspeed instruments. So let's work the numbers. First, your current ground speed. You have 150 nautical miles to go, and you're covering that in 17 minutes — from 1000 to 1017. 150 nautical miles in 17 minutes gives you 530 knots ground speed. Now, to arrive at 1021 instead, you need to cover that same 150 nautical miles in 21 minutes. That's a new required ground speed of 429 knots. Now here's the temptation. You might think, "I'll just scale the Mach number by the ratio of the new ground speed to the old one." That would be 0.77 times 429 over 530, which gives you Mach 0.62. That is wrong, and I want you to see why. That calculation ignores the wind entirely. It assumes ground speed and TAS change in the same proportion, which they don't when wind is present. So let's do it properly. We need to find the wind component first. Your old ground speed is 530 knots. Your TAS at Mach 0.77 and minus 55 degrees Celsius is 442 knots. So the wind component is the difference: 530 minus 442, which is plus 88 knots. A positive wind component means a tailwind — the wind is pushing you along, adding 88 knots to your TAS to give you that 530 ground speed. Now, your new required ground speed is 429 knots. To find the new required TAS, we reverse the sign of the wind component. Instead of plus 88, we use minus 88. So the new required TAS is 429 minus 88, which gives 341 knots. That's the TAS you actually need to fly to achieve that 429 ground speed into the same wind. Now we can scale the Mach number properly. The new Mach number is the old Mach number, 0.77, times the ratio of the new TAS to the old TAS — 341 over 442. That gives you Mach 0.59. And that's the correct answer — option d. Let me just recap the logic so it sticks. The wind component is constant — it doesn't change just because you slow down. So you have to strip the wind out of your ground speed to get the TAS you need, then convert that TAS back to a Mach number. The naive ratio of ground speeds gives you Mach 0.62, which is wrong because it treats the wind as if it scaled with your speed. The correct answer, Mach 0.59, accounts for that constant 88-knot tailwind. That's the whole lesson: timing problems are ground speed problems, but you fly by TAS, so the wind component is the bridge you must cross in both directions.