Pilot's Nightmare Copyright ©2004 by Paul Niquette.  All rights reserved. very pilot, it seems, will eagerly express his or her own opinion about how to deal with this puzzle.  Aviation books and websites provide a wide selection of empirical solutions.  A focus group with aviators might go something like this...

 When that noisy contraption up front goes quiet, you immediately want to minimize your rate of descent and prolong your time aloft, so I think 15o is the right choice. Maybe so, but the more time you take in the turn back toward the airport the more altitude you will give up in the glide.  I would go for 30o. Your best bet in such an emergency is to balance rate of turn with rate of descent.  That calls for a bank angle of 45o. A 60o angle of bank will let you reverse course a whole lot quicker without increasing your rate of descent by all that much. Never mind about descent rate.  The idea is to get that turn over with as fast as possible, which means rolling your plane into a 75o bank.

They cannot all be right.  Let's get to work (if you don't dig trig, you can skip on down to here).

Figure 1 depicts a horizontal view of an airplane in straight and level flight. It is being acted upon by four forces (shown as vectors, each having a direction as indicated by the arrows and a magnitude expressed here in pounds). By inspection of Figure 1, we see a balance of forces such that for straight and level flight, L = W, which keeps the plane aloft, and T = D, which keeps the plane moving through the sky at a steady airspeed v.  That's airspeed, not groundspeed.

The aerodynamic forces are proportional to the square of airspeed  The most fundamental attribute of an airfoil is its "lift-to-drag ratio" ('L/D', pronounced rather hurriedly 'elloverdee') and for a typical single-engine plane, 6 < L/D < 11.

Figure 2 provides a horizontal view of an airplane in a left turn, still in level flight. To remain aloft, W must now be balanced by the vertical component of L, which calls for a value of L itself larger than W.  The expression "Requisite Lift" gives emphasis to this reality.  When b = 0o, cos b = 1 and L = W, as we saw in Figure 1.  At any other angle of bank, cos b < 1, which makes L > W.

For example, when b = 60o, cos b = 1/2, and L = 2 W.  The airplane, along with the passengers inside, "feel" twice as heavy as when they were taxiing for take-off. Vectors can be simplified by strategic selection of vantage point.  Thus, in Figure 2, neither D nor T can be "seen" (T points straight out of your screen, and D points straight in).  That allows us to exclude D and T in the analysis below.

By inspection of Figure 2, we see that for a turn in level flight, L = W / cos b.  Meanwhile, the Lift vector L is manifesting a horizontal component, L sin b, which produces a turning maneuver -- a circular flight path, creating a balancing Centrifugal Force F = L sin b.

Sophisticated solvers don't bother to distinguish "centrifugal force" (fugitive from the center of a circular path) and "centripetal acceleration" (that which is capable of drawing a mass toward the center), the equivalence having been settled three centuries ago by Newton's Second Law of Motion. o solve the Pilot's Nightmare puzzle, we must first figure out how long it will take to turn back to the airport at a given angle of bank b.  For that we need to ascertain the Rate-of-Turn w (a typographic approximation of omega, the traditional symbol for "angular velocity").  Elementary physics teaches variations of the expression F = W/g w v.  The denominator g is the acceleration of gravity (32.2 feet per second per second), such that W/g represents the mass of the airplane flying a circular course at tangential velocity v (in feet per second) and at rate-of-turn w (in radians per second, where radians = 180o).

Since L = W / cos b and F = L sin b, then F = W sin b / cos b = W tan b.  Since W/g w v = W tan b, the value of W gets canceled out. Solving for rate of turn, we have w = g tan b / v radians per second.

To turn back toward the airport and line up with the runway, the gliding flight path will require more than 180o of turning.  As suggested in the sketch, your strategy calls for 270 in one direction (left, say) plus 90o in the opposite direction (right) for a total of 360o of turning, or 2 radians.  That maneuver will require 2 / w seconds, and during all that time the plane will be losing altitude. n the expression w = g tan b / v, we find out merely what every pilot knows -- that for a given angle of bank b, there is reciprocity between airspeed v and rate of turn w:  The faster the plane flies the slower its rate of turn.  "High speed maneuverablity" is an oxymoron.  To make a quick turn, you must slow the airplane as well as bank steeply.  There are limits, however...
Fly too slow and the plane stops flying.  Oh right, and as b approaches 90o, cos b approaches zero.  Look what that does to the requisite Lift.  Since L = W / cos b, the force exerted by the wings on the airplane -- and therefore the apparent weight of passengers inside -- will become astronomical.
The Pilot's Nightmare puzzle postulates an engine failure during flight.  Without propulsion, of course, a plane will be gliding -- but not necessarily diving (movie-makers take note).

Figure 3 depicts a horizontal view of an unpowered airplane gliding in a straight line at airspeed v with its Direction of Flight sloping downward at Angle of Descent a.  As with v in Figure 1, a is relative to the local parcel of air in which the plane is flying, not necessarily with respect to the ground (with a head-wind equal to v, for example, the plane would be descending vertically).  The Pilot's Nightmare puzzle, by the way, assumes landing in a tail-wind, but as shown in Figure 3, the plane has Vertical Descent Rate u = v sin a (in feet per second), where u does not depend on relative wind.  Whatever the windspeed, at a given airspeed, the larger the value of a the faster will be the rate of descent u and the sooner the plane will run out of altitude. Elementary trigonomery reveals that with the appropriate value of a, there can be a component of Weight aligned with the Direction of Flight, W sin a, which will exactly balance aerodynamic Drag D to keep the plane in motion.  That component of weight acts as a replacement for engine Thrust T seen in Figure 1.  In a gliding maneuver, there will still be a component of Weight that acts perpendicular to the wings W cos a, and that must be balanced by L.  "Requisite Lift" appears again in Figure 3 to give emphasis to this reality.

As you might expect, there is an optimum airspeed for gliding:

• not too fast or precious altitude will be thrown away in -- well, in a dive;
• not too slow or the flight will be quickly terminated in a stall -- or worse, in a spin. ven a student pilot knows that his or her best "Angle of Descent" will be achieved at "minimum drag speed," which is typically about 30% above stall speed.  Accordingly, in an emergency, v will almost certainly be established by the pilot at about 1.3 times stall speed.  The aerodynamic Drag D will thus be minimized.  For a straight-ahead glide, with b = 0, let's call the minimum-drag airspeed v0.

By inspection of Figure 3, then, we see that L = W cos a, and since cos a ? 1, L is less than the aircraft weight W.  The previous sentence does not have an exclamation point.  It appears instead in the answer to the following question: How can the plane stay aloft if Lift is less than Weight?  Aha, there is a vertical component of Drag D that actually helps to hold the airplane up!

The casual solver will note in passing that when the Angle of Descent a approaches 90o, cos a approaches zero making Lift totally useless for keeping the plane aloft.  Meanwhile, since sin a approaches one, D approaches W, and the plane would be executing a rather steep dive at an airspeed called, somewhat ominously, "terminal velocity."
It may not be obvious but if you look closely at Figure 3, you will see that in a power-off glide at v0, the lift-to-drag ratio L/D = cos a / sin a or L/D = 1 / tan a.
A gliding plane with an L/D = 10 will therefore have an Angle of Descent a of about 6o.  If the plane has a stall speed of, say, 50 nautical miles per hour (kts), minimum drag speed would be about 65 kts (v0 = 110 feet per second). The Vertical Descent Rate calculated as u = v0 sin a will be about 12 feet per second.  From 1,000 feet AGL, the pilot would have not quite a minute and a half (83 seconds) before the flight is concluded.  Fine, but that assumes no turning.
Figure 4 depicts the situation postulated in the Pilot's Nightmare puzzle -- a gliding turn.  As in Figure 2, we have made a strategic choice for our vantage point.  Our head-on view this time, though, is looking "uphill" along the descending direction of flight.  Thus, neither D nor W sin a (the dead-stick replacement for T) can be "seen" in Figure 4. In place of W, which always acts downward vertically, Figure 4 shows W cos a, which acts downward perpendicular to the wings as derived in Figure 3.  The Lift vector L, which acts perpendicular to the wings, is shown to have a requisite value L = W cos a / cos b.  That's what it takes to keep the plane aloft.  You will notice that whenever b > a, cos b ? cos a, and L > W much as we saw in Figure 2.

Gliding, say, at Angle of Descent a = 6o and at an Angle of Bank b = 50o, the requisite Lift L = 1.5 W.   The wings are carrying about half again the gross weight of the airplane, which raises the question, how does the pilot increase the value of L?  Answer: increase airspeed, of course.  But how much?
As described above, L increases according as the square of v.  To compensate for banking, we observe that the requisite Lift must be increased from its straight-flight value, L = W cos a, by a ratio that looks like this: (W cos a) / (W cos a cos b) or more simply 1 / cos b.  Airspeed, then must be increased from its straight-flight value v0 by a ratio that looks like this: v2 / v02.  Thus, for a gliding turn, the requisite airspeed will be given by v = v0 (1 / cos b)1/2  (feet per second).  Hmm, this determination turns out to be independent of the Angle of Descent a as well as W (a counter-intuitive reality, wouldn't you say?). ithout an engine, the only way to increase v might seem to require steepening the angle of descent a.  Doing so will increase the rate of descent u all right, but that would happen anyway just as a result of increasing v.  In fact, since D varies with v2, the force of Drag will increase by the same ratio as Lift, 1 / cos b.  Pilots are cautioned in early flying lessons that the speed at which a plane will stall increases with angle-of-bank by that same ratio; so too will the minimum drag speed increase.  That means, the pilot can maintain the same angle of descent a in the gliding turn as for straight-flight, while the rate of descent u increases with v.  How's that for a counter-intuitive reality?

Applying simple algebra, we are now able to derive the following two expressions:

w = (g / v0 ) (cos a sin b) / (cos b)1/2 (radians per second)
u = (v0 sin a) / (cos b)1/2 (feet of altitude per second)
Thus, the time to complete the turning maneuver (270o + 90o = 360o = 2 radians) will be given by  2 / w = 2 v0 (cos b)1/2 / g cos a sin b  (seconds).  Multiplying that value of maneuvering time by the rate of descent u gives us the altitude lost in executing the gliding turn.  Let's call that h.
h = 2 v02 tan a / g sin b (feet of altitude)
Here are specifications for my favorite airplane as exemplars for our calculations:

 Stall Speed 50 kts 85 ft / sec Minimum Drag Speed v0 65 kts 110 ft / sec Lift-to-Drag Ratio L / D 10:1 10 lb / lb Angle of Descent a 0.1 radian 5.7 degrees

So, then, using that formula for h, what angle of bank would you choose?   The following table gives h for the values of b offered in the Pilot's Nightmare puzzle:

 Angle of Bank b (degrees) Airspeed v (ft / sec) Rate of Turn w (radians / sec) Rate of Descent u (ft / sec) 2 -Turning  Time (sec) Altitude Loss  h (ft) 15 111.5 0.077 11.1 81.6 909 30 117.8 0.157 11.8 40.0 471 45 130.4 0.246 13.0 25.6 333 60 155.0 0.358 15.5 17.6 272 75 215.5 0.555 21.5 11.3 244

At b = 15o, you will take over 80 seconds to complete your gliding turn and will lose over 900 feet.  You can cut that altitude loss almost in half by choosing 30o -- and half again at 60o.  Obviously, the greater the value of b, the lesser the value of h.  You might as well, therefore, just go ahead and roll your plane on over to a 75o angle of bank.  You will complete your maneuver in a little over 11 seconds, with an altitude loss of only 244 feet.

Watch out, though...

Any maneuver using a bank angle greater than 60o is officially classified as "aerobatic" and can do structural damage to most models of light aircraft.  A puzzle-solver might not be expected to know that, perhaps, but it is required knowledge for all pilots -- and for good reason: At a 75o angle of bank, your plane and passengers would be pulling upwards of 4 gs.  Your banking stall speed will be 100 kts -- twice that of level flight.  The minimum drag speed you must carry will be 130 kts (215.5 feet per second).  So maybe you are not about to stall; however, truth be known: The structure of the airframe would actually be less challenged if your plane did stall, thus releasing the "Grasp of Bernouli."  A stalled wingfoil therefore makes sure that your airplane will arrive at the crash-site...in one piece.
Best check your aircraft owner's manual before horsing around in the sky at a 75o angle of bank.  Besides, choosing 60o only costs you 272 feet in altitude loss.

Oh, but there's more to think about.  Much more... ne of aviation's many counter-intuitive aspects is -- tah-dum -- "adverse yaw."  For surviving the Pilot's Nightmare you must execute a gliding maneuver comprising two turns in order to line up with the runway of departure.  Now, an airplane cannot change its bank angle instantaneously.  The pilot applies aileron forces to institute a rolling motion, but -- hey, the next sentence deserves its exclamation point:  While an airplane is actually rolling, it is not turning!  That's right, it is flying straight, as a consequence of adverse yaw, which, for a coordinated turn, must be balanced by applying opposite rudder.  Obviously, the greater the value of b, the longer the rolling takes and the longer the turning is postponed.

For our purposes here, let us assume a rather generous roll-rate of 15o per second (for a slowly gliding plane, remember).  Rolling the plane from wings level into a bank angle b of, say, 60o for the course reversal will take you about four seconds.  The opposite turn requires first rolling the wings level and then rolling into the opposite bank angle to get lined up with the runway -- another eight seconds.  Finally, there's rolling the wings level before touch-down, which takes yet another four seconds. Figure 5 on the right depicts the flight path of your airplane.  The engine failure is shown occuring after climbing for some distance x beyond the departure end of the runway.  Bank angle b produces turns with a radius of curvature r.  The requirement to balance adverse yaw is revealed by the straight flight segments with lengths y1 , y2 , y3 estimated as follows (assuming a roll-rate of 15o per second):

y1 = (b/15)(v + v0 ) / 2, which assumes climbing at minimum drag speed v0 ;
y2 = (b/15) v, which assumes no change in airspeed during roll-in and roll-out;
y3 = (b/15)(v + v0/ 1.3) / 2, which assumes a full-stall landing at v0/ 1.3
The total altitude lost while balancing adverse yaw can be estimated as (y1 + 2y2 + y3) tan a.

Figure 5 also shows cumulative horizontal heading changes, which turn out (no pun intended) to be less than the total of 360o (2 radians) you must use if there were no such thing as adverse yaw (as shown in the puzzle sketch).   What you want to do is take the shortest path back to the airport on a course shaped like a tear-drop.  By simple trigonometry, we find that there is an angle given by... = arc tan (r / y2 ).
By inspection of Figure 5, we learn that the first turn requires a total of  radians and the second turn requires radians in the opposite direction for a total of + 2 radians to bring the plane back to the airport lined up with the runway.  Finally, that distance beyond the end of runway available for climbing x is given by...
x = 2 (r sin + ycos ) + y3   -  y1
Taking into account the effects of balancing adverse yaw, then, we see in this tabulation...

 Angle of Bank b (degrees) Angle (radians) Rate of Turn w (radians / sec) Loss in Turn + 2 Loss due to Adverse Yaw Total Altitude  Loss h (feet) 15 1.494 0.077 874 42 916 30 1.266 0.157 419 89 507 45 0.935 0.246 261 144 405 60 0.610 0.358 186 221 427 75 0.346 0.555 146 366 512

...that the solution to the Pilot's Nightmare puzzle is b = 45o -- decidedly not b = 75o.  ut wait.  Have a look at Figure 6 on the right.  The elevation view depicts your loss of h feet in altitude while reversing course and lining up with the runway.  Figure 6 shows another reality -- that you still have to get back to the airport while gliding at Descent Angle a (the dotted red line).  The longer the runway the better -- net of your take-off roll (the unused runway is, so to speak, "reaching out" to receive your crippled craft).

Your altitude at any given point during the climb-out will depend on your Climb Angle c along with the distance flown before engine-failure (the green line).  As shown in Figure 6, c better be greater than a or you'll never make it. To determine c, you will need to know your airspeed v and rate of climb u.  By the way, wind helps your cause by making your angle of climb steeper and your angle of descent shallower -- and by shortening your take-off roll.  However, wind also increases your groundspeed at touch-down.

Accordingly, there are plenty of technical factors beyond those addresed in solving the Pilot's Nightmare puzzle -- factors that may convince you not to execute the course-reversing maneuver. 