Mental Math

Taking a look around at the world around us, one would would be hard pressed to find an area of our daily lives that has not been touched in some way by technology. We can be accessible to anyone at nearly any time, have access to a nearly unlimited supply of information from the comfort of our homes, and can be guided from point A to B without ever having looked at a city map. These advances have all been designed to make our modern lives easier and more convenient. Aviation has also seen huge advances in systems designed to allow pilots to operate the aircraft more safely, efficiently and economically. Systems such as GPS have taken much of the guesswork and mental math out of flying an aircraft, in much the same way they have allowed motorists to confidently navigate city streets with which they are not familiar, without having to look at a map. With the prevalence of technology it would be easy to be led to the point of being convinced that conducting these tasks manually are needless skills, antiquated relics of a bygone era.

For one summer while saving money for flight school I worked at a tree farm at which, among other tasks, I delivered trees. In a time prior to economical GPS systems, we navigated the city using our eyes, and maps. While initially I became quite adept at doing U-turns in our two ton delivery truck, with time I acquired the ability to get around efficiently in areas I did not know well. While it took some time to get to the point of being able to get around easily in unfamiliar areas, this skill has remained with me to this day, and has prevented me from getting lost more times than I can count.

Most of the aircraft in which I learned how to fly had yet to be equipped with reliable GPS systems, and those that did possess these systems were of little use beyond mild amusement from watching their blinking lights. As a result, those who learned how to fly and navigate at that time, did so by reference to maps, the ground below us, airspeed and heading, and by using our head. Inevitably there would be times where we would become unsure of our position, and have to draw upon our training, and the resources around us in order to safely get to our destination. Like learning how to navigate in the city, it took some time and was not always easy, although the benefits of such skills far outweighed the time and effort spent on learning and refining them.

When I started flying commercially, the goal became to navigate from A to B as efficiently as safely possible. This typically means using GPS extensively to allow the crew to proceed directly from point A to B, as opposed to using older NDB and VOR airways which do not always run directly between the point of departure and destination. As a result, VOR and NDBs are used mostly as a means to confirm position while enroute and to serve as a backup in the case of a GPS failure. Even while VOR and NDB navigation has taken a backseat to GPS which can tell you at a glance exactly where you were above the earth and what speed you were traveling over it, VOR and NDB remain the primary means of navigation on many non-precision approaches, although this too is changing. In addition to being able to tell a pilot their position and speed, many units also provide features such as turn anticipation, which tell a pilot when to turn in order to correctly intercept the next leg of their flight. GPS has made it easier to be constantly aware of ones position, has allowed for big steps forward in situational awareness, and has made precise navigation accessible, intuitive and reliable.One can not help but admire the cold precision that modern autopilots can maintain while tracking routes programmed into navigation systems capable of guiding aircraft from gear up right to touchdown. However, just as with other tools that have been designed to make our lives easier, the technology found in many of today's cockpits are not without a downside. The moment that these systems are used in place of, and not to complement the skills that we as pilots have worked hard to earn, we will find ourselves on a slippery slope where easier does not always mean better.

The ability to work out calculations of time, distance and speed has been one of the first to fall victim to these advanced systems in some cockpits.

While at first, the mental math required to come up with some of the figures that avionics can give us may seem daunting, in reality they are no more complex than any of the other mental math we use while flying. As an example, lets take a flight from Edmonton to Fort Mcmurray flying the Beech 1900. While the numbers will differ with aircraft, the principal upon which the calculations are based, remain the same.

Upon getting into the airplane, initially there will be little in the way of calculations that haven't already been done on the flight plan. For example, the rough track to be flown as well as the time enroute will have already been computed, and when briefing passengers, one needs only to quickly glance at the flight plan to give them an idea how long they will be aboard. After take-off, on the departure from the airport area, it is important to know not only the distance to the destination, but also how many miles are between you and the departure airport. This information becomes increasingly important when departing out of uncontrolled airports where it is the responsibility of the pilot for traffic separation, at least initially.

For example when departing an airport such as Rainbow Lake over which the airspace is uncontrolled to 18'000 feet, if an inbound flight calls that they are 8 miles southeast of the airport, heading northwest, you must be able to quickly determine your distance from the airport and the inbound aircraft to confirm that your paths will not cross at the same time and altitude.

On departure, we maintain a speed of 180 knots in the Beech 1900 as we climb to our cruising altitude. This is a convenient number as 180 knots works out to 3 miles traveled every minute. Glance at your watch when you turn enroute, and know that you are already about a half a mile to a mile from the airport when you do so, and you only need to glance at your watch again to get your distance from the airport. If it has been 2 minutes since turning enroute, you are somewhere around 6-7 miles from the airport. If you have to do a 180 degree turn to head to your destination after taking off, then you simply look at your watch as you are passing the airport after turning towards your destination and note the time.

On the climb-out out of Edmonton through 8000 feet, Edmonton Centre asks whether we can be at or above FL 190, (19000 Feet) in 10 minutes in order to accommodate arriving traffic from the north. We are currently climbing at about 1400 feet per minute, though this will slowly decrease as we climb into the thinner air above. With a nearly full load on a standard climb profile we can average about a 1000 foot per minute climb rate to our cruise altitude. Given that we have 11'000 feet to gain in order to comply with Edmonton Centres' request, with an average climb rate of 1000 feet per minute, it will take us 11 minutes to get above FL 190. With this information in hand we can now either inform Centre that we are unable to comply, or we could climb more aggressively in order to meet the climb requirement.

Enroute, once we have leveled off at our cruise altitude of FL 250, we are told, again by Edmonton Centre, to cross the LEXON intersection, one of the points in our flight plan, at a time of 1645. Like the climb requirement we have just dealt with, crossing time restrictions are a relatively rare occurrence, typically only found on the arrival into busier centres.

It is currently 1625, so now we have to figure out what speed we will have to fly in order to arrive over LEXON in 20 minutes. Looking at the distance readout to LEXON shows that we have 70 miles to go. First lets note that 20 minutes is one third of an hour. With that information in mind, we know that the distance we will travel in 20 minutes will be one third of our groundspeed. So if we are currently cruising at our flight planned speed of 270 knots, in 20 minutes we will have traveled one third of that figure, or 90 miles. We can also use this calculation in reverse for our particular situation; If we multiply the 70 miles we have to go to LEXON, by three, we get a required speed of 210 knots in order to meet our crossing time.

If we had to cross LEXON in 10 minutes, which is 1/6th of an hour, we would multiply our current distance to LEXON by 6 in order to determine the required groundspeed.

As we get closer to LEXON, we are told by Centre that our crossing restriction is canceled and we can now proceed direct to OTRIL, which is the final approach fix for the ILS onto Runway 25 at Fort McMurray. We are coming up on 100 miles from the field, which is the point at which we aim to have a plan established and the briefing on the descent and approach completed. First we must determine at what point we must begin the descent. Turboprop aircraft, like the Beech 1900, will typically travel 3 miles forward, for every thousand feet lost during the descent. This afternoon, we are cruising at 25'000 feet and Fort McMurray has an elevation of around 1200 feet, which for the sake of easy math, we will round down to 1000 feet. Thus we have 24'000 feet to lose, and will travel 3 miles for each 1000 feet, so by multiplying 24, which is the thousands of feet we must lose, by 3, we come up with a figure of 72 miles. Given that we will also have to slow the aircraft down as we approach the McMurray area, we will add somewhere between 3 to 5 miles to our descent figure in order to accomplish this. Now we will be starting our descent 77 miles back from the airport, in order to get down, and as we get closer, slow down. The only problem that we have left to solve is that we are not heading directly to the airport, but rather to the final approach fix, OTRIL. Looking at the approach plate for Runway 25, we determine OTRIL to be about 14 miles from the airport.

Subtracting 14 from our calculated distance to descend of 77, puts the top of descent at 63 miles from OTRIL.

Arriving into Fort McMurray from the south for Runway 25 can be challenging when the weather is low enough to require flying an instrument approach. The typical track flown inbound to OTRIL will have us approaching the runway at about a 110 degree angle. If a crew were to wait until the localizer, which gives the degrees of deviation off of runway center-line, started to move to begin the turn towards the runway, they would be well north of the approach path by the time they were pointed in the right direction. While being a sloppy way to start the approach, this also increases the possibility for a destabilized approach all the way down to landing. The alternative is to calculate how much distance the aircraft will require to turn 110 degrees, and start turning that distance back from OTRIL so that when we roll our wings level on the inbound track, we should be sitting very close to the extended runway center-line.

To arrive at this number we first must know what speed we will be intercepting the localizer at, which in the 1900 is 160 knots. We take 1% of that speed, or put another way, divide it by 100, which in our case works out to 1.6. We now take that 1.6 and divide it by 2 to give us a distance of .8 from OTRIL to start the turn. This would work if we were intercepting at an angle of 90 degrees, but since we have a larger turn to make, tacking on an additional .3 or .4 of a mile for that extra 20 degrees we need to turn should get us at least in the ballpark.

To make the above mental math come out right, the turn made must be rate one. While a few of the aircraft at that I fly still do have a turn and bank, or turn coordinator to indicate whether we are turning at rate one, (3 degrees per second) most do not. So along with the speed we will be intercepting the localizer at, we must also determine the angle of bank required. To do this we take 10 % of our airspeed and add 7. So for our 160 knot intercept speed, we divide by 10 to give us 16, to which we add 7, to come up with an angle of bank of 23 degrees. This method is not exact, but it will get you to within a degree or so of rate one at typical approach speeds up to about 200 knots.

Now as we are proceeding inbound on the localizer as we wait to intercept the glideslope from below, we can calculate roughly what rate of descent will keep us on the 3 degree slope down to the runway. To compute this number we only need to divide our groundspeed by 2, and add a 0 on to the answer. So for our 160 knot groundspeed, divided by 2 gives us 80, to which we will add a zero to the end to give us a figure of 800. At this speed, in order to follow a 3 degree slope downward, we will need a 800 feet per minute descent rate.

Finally we break out of the cloud bases to see the runway out the windshield, shortly afterward, touching down. As we exit the runway and finish the after landing checklist we prepare the airplane, and ourselves for the next departure. Now, aside from figuring out how long we were in the air and calculating our fuel load for the next leg, we can give our mind a rest.

"I believe any good pilot has a certain skepticism. If he or she isn't a skeptic, they are headed for trouble. This seems especially true with the computer -- and when I say computer I include FMS, autopilot and all. Being skeptical means a pilot refers to raw data to be certain the FMS etc., is doing its thing correctly. This is not always easy because as the computer develops it makes raw data more difficult to see, find and use." -Robert Buck