Mental arithmetic for pilots


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Commonly in flying a small plane (and in training to do so) it’s helpful to be able to to make approximate calculations of various factors in flight. These usually crop up during navigation exercises, particularly the ad-hoc diversion exercise which has to be planned and executed fairly rapidly from the pilot seat while handling the airplane. Other scenarios are during the approach and landing phases, when it’s helpful to be able to work out an approximate and appropriate rate of descent or determine when and where to begin that descent.

And, what can seem really really straightforward calculations to do in your head seated comfortably on the ground and indoors can turn into monstrous fear-inducing thermonuclear calculations. Or, even worse, appear to mock the pilot who knows the math is easy but because 90% of their attention is focused on flying a wayward airplane is unable to think of the answer.

So here are a few scenarios fleshed out, with “obvious” pointers to keep you on the right track, as it were.

Cross country navigation: time, speed and distance

Facts to remember: 60 knots ground speed is 1 nautical mile per minute flown. 120 knots ground speed is two nautical miles per minute flown. 90 knots groundspeed is one and a half nautical miles per minute flown.

Some worked examples:

20 miles in 10 minutes: 2 miles per minute, therefore 120 knots.

18 miles in 12 minutes: 1.5 miles per minute (18 is one and a half times as big as 12) therefore 90 knots.

If you know your ground speed is 120 knots, then four minutes is enough to go 8 miles. 6 miles takes three minutes. These numbers are always a factor of two apart. If you get mixed up, the bigger number is the distance and the smaller number is the time in minutes.

If you’re flying slowly with a moderate headwind you may find your ground speed is reduced to 60 knots – then the distance flown in nautical miles and the time taken in minutes, are the same.

Additionally, at 60 knots the airplane covers about 100 feet forward per second. We use this in checking the length of a proposed landing field in the precautionary landing exercise. Twenty seconds to cover a field means it’s about 2000 feet long.

Descents and climbs

100 feet per second is 60 knots, therefore 100 feet per minute is about one knot – sixty times slower. We can use this to think about angles of climb and descent. 100 feet per minute rate of climb at sixty knots airspeed is a climb gradient of 1 knot (upwards) in sixty (forwards) – therefore one in sixty.

If you recall the one-in-sixty-rule, which is that 1-in-60 is about one degree (1°) then you can see that climbing at 100 feet per minute at sixty knots airspeed is a climb angle of 1°. Now multiply by 5: five hundred feet per minute climb at sixty knots is about 5° climb angle.

We can use the same coming back down. Most light airplanes are comfortable approaching to land at 4 to 5° angle of descent. That works out at a rate of descent of 400 – 500 feet per minute at sixty knots ground speed. Or a little bit faster rate of descent if you’re flying a little bit faster.

ILS glideslopes are usually (but check the approach plate!) set at 3°. So you’d need a rate of descent of about 300 feet per minute if you’re grounding 60 knots, or 600 feet per minute if you’re grounding 120 knots. By simple comparison you’ll note that you can take half your groundspeed (from your GPS, or just estimate) and add a zero, to give you a rate of descent to stay on the glideslope.

What about visual descents, and when to start? 500 feet per minute descent is comfortable for passengers’ ears in an unpressurized airplane. If you’re cruising at 120 knots you need two minutes, or four miles distance, per thousand feet of descent to your nominal arrival altitude (for example, the circuit altitude of the destination airport.)

Example: if you’re cruising at 3500 feet and your circuit is at 1000 feet, then you have 2500 feet to descend, which is 2 and a half “units” of a thousand feet each – so you’d better start down two-and-a-half units of 4 miles – which is 10 miles – from your destination.

Some practice for you

Here are some simple questions for you to practice. If you want to add extra realism to the task of in-flight arithmetic, try solving them wearing headphones playing loud music while sitting in a rocking chair being violently rocked by a friend, and while stirring a bowl of water in your lap with one hand and peeling a carrot with the other.

  1. Your groundspeed is close to 90 knots. How long does it take to fly 21 nautical miles?
  2. What is your descent angle if your GPS groundspeed is 100 knots and your rate of descent is 750 feet per minute?
  3. It takes you 15 minutes to fly 22 nautical miles. What is your groundspeed?
  4. The airport to which you’re flying has a ground elevation of 750 feet. The circuit is 1000 feet above that. You’re 37 miles out, and cruising at 4500 feet ASL. When, and where, should you begin your descent?

Happy calculating!

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