Desired Track (DTK) #3:
Headwork, Part One

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Among other things, aviation can be referred to as a numbers game: heading, altitude, airspeed, distance, time. Fortunately, today's cockpits have a lot of equipment available to compute and display the important ones. But what's behind some of these numbers? How can we compute some of them on our own, in our head, when we want to or need to? In this first of a two-part series, AVweb Executive Editor Joseph E. (Jeb) Burnside explores how to use

Numbers are very important in aviation. They help pilots answer constant questions like, "How much longer till we get home?" and "Whaddya mean your plane can't carry this steamer trunk?" They help remind us of the differences between, for example, a C205A and a C-5A, or between a PA-28-235 and a PA-28-236. They also tell us how fast (or slow) we are going, whether we have enough fuel to reach our destination and, for some, how much longer until forced retirement.

Some numbers are easy to obtain: Indicated airspeed, for example, or altitude are available after only a glance at the instrument panel. Other numbers, like true airspeed or the altitude above ground at which you can expect clouds to form, require some help, like a TAS ring on the airspeed indicator, an E6-B "whiz wheel" or a complete weather briefing. Of course, a lot of information is only a knob twist or button push away, given the sophistication of today's avionics. And that stuff never fails, right?

Right. So, what do you do if you left your whiz wheel in your car's trunk, you've drained the last bit of juice from your calculator's batteries and the circuit breaker on your brand-new panel-mounted GPS navigator won't reset? That's when you need to do some headwork.

Of course, you remember how to perform all the little calculations you need to answer all the myriad questions that can arise and successfully complete your flight, don't you? No? Maybe what you need are a few "gouges" — rules of thumb which, while perhaps not always spot-on in accuracy, are at least "good enough for government work" (and haven't we all seen how good that can be lately?)

How Fast?

True Airspeed ...

I like to fly high. The air is smoother, there's less traffic, ATC radar coverage is better and I have more gliding options if that fan up front that keeps me cool stops turning. I also like to monitor my plane's performance and the effects of wind, turbulence, altitude and weight at various stages of a flight. Yet, my plane doesn't have one of those nifty doodads on the airspeed indicator that I can use to directly read true airspeed. Instead, I have to drag out the whiz wheel and spin a few dials to come up with the answer. There's an easier way.

The rule of thumb, or gouge, to determine true airspeed is relatively simple: To determine TAS, add 2 percent for each 1,000 feet of altitude to the indicated airspeed. In other words, if you're indicating 100 knots at 10,000 feet MSL, your true airspeed is 120 knots. Put another way:

TAS = IAS * [1 + (0.02 * altitude in feet / 1,000)]

Yes, I know that temperature enters into this calculation, but only to determine true altitude. As I noted above, these are simply gouges, designed for "quick and dirty" calculations in your head, not for more important uses like certification or winning bar bets.

Oh, by the way: This works for either mph or for knots.

... Dynamic Hydroplaning ...

Ever wonder how fast you need to be going on a wet runway to encounter dynamic hydroplaning? This is the kind of hydroplaning in which the tire is lifted up off the runway by the water on its surface, not the more exotic kinds, like reverted rubber or viscous hydroplaning, which can occur at much lower speeds.

The two key variables in hydroplaning are tire pressure and speed. In essence, a tire will lose contact with the runway when its speed is equal to nine times the square root of its pressure. If you're lucky enough to be flying some heavy iron with a tire pressure of, say 100 psi, that tire will hydroplane at around 90 knots: The square root of 100 is 10; times 9 equals 90.

For lighter aircraft with lower tire pressures, that speed if much lower, of course. For example, the main-gear tires on my Debonair carry a maximum inflation pressure of 40 psi. The square root of 40 is ... what? Well, again, using the rule that we're only interested in rules of thumb here, it's around 6.3; 6 * 6 is 36, and 7 * 7 is 49, so it's got to be between 6 and 6.5 (it's actually 6.32455 and a bunch of trailing digits, but we're in a cockpit, remember?) In this case, 9 * 6.3 = 56.7, which — conveniently — is about the speed at which I want to be touching down. 

If we assign to "Vhp" the speed at which dynamic hydroplaning will occur and agree that "SQRT" is the mathematical function that derives a square root, we get:

Vhp = 9 * [SQRT(100)]

Again, this is a rule of thumb. Don't have your attorney or insurance company send me a nasty email after you lose control at 85 knots.

... And Angle of Bank

One aspect of instrument flying that requires some understanding can be the standard-rate turn and its relationship to airspeed. While it's easy to use the turn coordinator (TC) or turn-and-bank indicator (T&B) to establish and maintain a standard-rate, three-degrees-per-second turn, some students have trouble grasping the fact that the angle of bank required to establish a standard-rate turn increases with airspeed. In other words, the bank angle required to fly a standard-rate turn is shallower when slowed for a holding pattern or maneuvering on an approach than when in cruise when, presumably, the airspeed is higher.

Of course, the rate of heading change is inversely proportional to the airspeed: Put another way, the faster you go, the less rapid will be any attempt to change heading. Put still another way, the radius of the turn is greater when the airspeed is greater; that radius is smaller when the airspeed is slower.

But you knew all that. How do we compute, in our head, the bank angle required to establish a standard-rate turn? It's relatively easy...

There are two formulae that can be used. Presuming "BA" is our shorthand for "bank angle," they are:

BA = TAS * 0.15

   or

BA = [(TAS * 0.10) + (TAS * 0.10) / 2]

Putting this into use, let's presume we're flying along at 140 KTAS. (Note that in the case of a verified vacuum or DG failure and at low altitude, I'd probably say "screw it" and use indicated airspeed which, as discussed above, won't be far from TAS at all.) We want to establish a three-degree-per-second, standard-rate turn to change our heading. Here's how it's done with the first formula:

21 = 140 * 0.15

Okay, I used my calculator for that. Let's do it again, using the second formula, and do it in our heads:

14 = (TAS * 0.10)

7 = 14 / 2

21 = 7 + 14

In other words, take the TAS, move the decimal point one place to the left, then add half of the TAS. Piece o' cake! So, at 140 KTAS, the bank angle required to establish a standard-rate turn is 21 degrees. At 90 KTAS, it's 13.5 degrees.

I can hear you now: "Why is this of any interest at all to me; I can establish the correct bank angle on the TC or T&B and turn to the appropriate heading, then roll out, right?" Yes, that's true — when everything is working. Presuming you have no backup systems and no moderately useful HSI mode on your portable, battery-powered GPS, how will you even begin to know when to roll out of a turn to a new heading that dark and stormy night when the only thing you have left is your back-up attitude indicator and your whiskey compass? You'll establish a standard-rate turn and hold it for the amount of time required to change to the new heading, that's how.

Once we know the bank angle at 140 KTAS, and once we determine the number of degrees we have to turn to settle on our new heading, it's a simple thing to turn to a desired heading without a DG: Just divide the total heading change by three, roll into the standard-rate turn, hold it for a couple of seconds shy of the number of seconds required for the heading change, and roll out. The controller will be amazed. So will your flight instructor.

How Far? How Long?

Diameter ...

When was the last time you shot a night, circling approach in IMC? Yeah, me either, but wouldn't it be nice to know, quickly, the diameter of the standard-rate turns you'll be making as you maneuver to land and be able to compare that value to the prevailing visibility? It'd be nice to know if you'll be able to keep the runway in sight when circling, wouldn't it? It's actually a lot easier than finding the airport.

The diameter of a standard-rate turn (one featuring three degrees per second of heading change), expressed in nautical miles, is equal to your true airspeed divided by 100. As a formula, it looks like this, when we assign "diameter" to D:

D = TAS / 100

Of course, when using this to calculate whether we'll be able to keep the runway in sight at night when the reported visibility is 1-1/2 statute miles, we probably need to find the radius of the turn, since we hope the runway will actually be in the center of the turn.

Oh, before we go any further, you do know the difference between a statute mile and a nautical mile, right? A statute mile is 5,280 feet, while a nautical mile is 6,076 feet.

Presume you're in an aircraft conducting a circling approach at 100 KIAS at an airport with a field elevation of 2,500 feet MSL and maintaining a constant standard-rate turn, you'll be flying a circle which is 1.05 nm in diameter. Here's the way the numbers work:

KTAS = IAS * [1 + (0.02 * altitude in feet / 1,000)]

KTAS = 105

D = KTAS / 100

   [or 105 / 100]

D = 1.05 nm.

If the local airport is reporting one statute mile of visibility and you're circling at 105 KTAS in a standard-rate turn with the runway in the middle of the turn, you probably will be able to keep the runway in sight, since the radius of the turn is half of its diameter or, in this case, .525 nm.

You've got a lot of recent practice flying patterns that close to a runway and at the MDA, right?

... And Distance/Time to Station

This one's in the FAA's Instrument Flying Handbook, FAA-H-8083-15, which was recently revised. Let's say you're droning along on your trip to somewhere as you fiddle with a CDI, which is tuned to a distant VOR. As a means to double-check your position (like me, you never get lost, of course...), you wonder how far away that VOR is. Well, believe it or not (and if you've read this far, you should believe it) there is a relatively quick way to determine your distance to the station.

Essentially, what you need to do is fly more or less perpendicular to the VOR and note two values: The bearing change, or the radial on which you started and the radial on which you ended measuring, and the time it took for the change to occur.

Here's the formula:

          DTS = TAS * Minutes / Bearing Change

where DTS = "Distance To Station."  Here's the way it works:

As you're flying along, center the VOR needle and note the time and the radial. Then, after a few minutes, center the needle again and note the radial. The difference is, of course, the value we need for the variable Bearing Change. Presuming we're doing 140 TAS, that it's been five minutes and the difference between the first bearing and the second is ten degrees, here's the math:

DTS = (140 * 5) / 10

DTS = 700 / 10

DTS = 70

Thus, we're 70 miles (nautical or statute, depending on which airspeed measurement we're using) from the VOR.

Interestingly enough, we can use the same basic information to determine how long it would take us to get to the station. Here's that formula, where TTS is "Time To Station."

TTS = (60 * Minutes) / (Bearing Change)

We've already done the hard part, when we determined the time and the bearing change. Of course, if you've been following along, you know that, at 140 TAS and 70 miles to the station, it's going to be ... well, you'll have to wait for it.

TTS = (60 * 5) / 10

TTS = 300 / 10

TTS = 30

So, we're 70 miles from the station, doing 140 TAS and we're 30 minutes from the station, presuming there's no wind, of course. Of course, 70 is exactly half of 140, so it's a fairly easy mental calculation to come up with half of the distance traveled in an hour at 140 — 70 miles — as the distance to the station.

One final note: This works with VOR, NDB, Localizer or any other electronic navaid you can use to determine a bearing change.

Until next time ...

So ends Part One of this two-part series on numbers, gouges, shortcuts, whatever you want to call them. If you have some favorite "gouges," feel free to share them with me, via e-mail.

See you next month!