• Why does the Zero not turn right as quickly as the Corsair? Zero Mystery
• What angle of bank should you choose? Pilot's Nightmare
• Will you be glad you did that? Sloping in the Dark
• For winning an Air-Race, what is the best cruise altitude? Air Race
  

rC = rC0 + j h, where...

rC0 = rate-of-climb at h = 0 (sea level) = 700 fpm
j = climb rate coefficient = 0 - rC0 / 14,000 = - 0.05 fpm/ft

vKTGS = vMD + k h, where k = 0.0023 kt/ft
vKTGS = vMX + k h, where k = 0.0013 kt/ft
vKTGS = vNE + k h, where k = 0.0023 kt/ft

Nota bene, at that moment, t = 33 min, Red has flown 84 miles from fix f1 and is located 25 nm ahead of Blue, which has flown only 59 nm from fix f1.  Let's call that ∆xMAX = 25 nm.

We see that during cruise, Blue flies faster than Red; however, even at an altitude higher by 6,000 ft, the speed advantage is only 8 knots.  It would take 188 minutes -- more than three hours -- for Blue to catch up with Red.  Flying at 180 KTAS, Blue will be then wind up 'down-course' by 628 nm from fix f1.  Hey, that's 333 nm beyond the end of the Air-Race at fix f2.

Instead, at t = 86 min into the flight, Blue begins descending for its arrival at f2.  The plane commences a high-speed 'shallow dive' at full power from the cruise altitude hC = 11,500 ft to hPA = 1,000 ft MSL.  As shown in the diagram above, the descent uses an assumed rate-of-descent rD = 350 fpm at all altitudes while adjusting pitch to maintain the 'never-exceed speed' vNE = 200 KIAS.

Surprise!  At t = 105 min, while descending through 5,500 ft MSL, Blue catches up with Red, which is just beginning its own shallow dive.  Both planes will then reach Fix f2 11 minutes later, as one sees in the graph above at t = 116 min.  Thus, the Air-Race ends in a tie. 

Reminder: Our model for the Air-Race postulated no wind.  Of course, as a practical matter there is no such thing as 'no wind'.  Whatever its intensity, from a breeze to a gale, 'winds aloft' have pronounced effects on flights in general and an Air-Race in particular.  The graph below has been adapted from the solution to the Wind Circle puzzle. 

Consider the effect on the Air-Race of a hypothetical wind having a steady intensity vW and direction aW at all altitudes throughout the entire area.  For cases of direct 'tailwind', aW = 180º, and we see in the graph that aircraft groundspeeds can be derived by adding vW in full measure to true airspeeds. 

For example, let vW = 18 kts, which happens to be 10% of Blue's cruise speed of 180 KTAS.  If aW = 180º, then vG = 180 + 18 = 198 KTGS and vG = 172 + 18 = 190 KTGS.  Likewise the average ground speeds calculated above will become:  vAVG = 107 + 18 = 125 KTGS (+ 17%) and vAVG = 153 +18 = 171 KTGS (+ 12%).  However, ∆xMAX = 25 nm does not change.

Our hypothetical wind in the model is hardly representative of winds aloft.  Indeed, vW typically increases with altitude, and, in the northern hemisphere, aW typically rotates clockwise with altitude.  Accordingly, the decision about best cruise altitude in an Air-Race seems to be dominated by winds aloft...

Headwind: Use lowest altitude clear of obstacles, whatever the length of the flight segment. Tailwind: Higher the wind, higher the altitude, especially with longer flight segments.

In the table below are the performance results, using 'minimum drag speed'  vMD = 90 KIAS for climbing at the beginning of the hypothetical Air-Race.  In ordinary flying it is customary to trim for a recommended 'cruise climb' speed vCC, optimized for engine cooling and fuel economy.  To test the effects of a climb speed only slightly faster than vMD, the table shows that for a vCC = 95 KIAS, the rate of climb decreases from 700 ft/min to 670 ft/min.  Thus, the time to climb, say, 10,000 ft, increases from 31 min to 33 min.  We observe also that ∆xMAX increases from 25 nm to 27 nm.

Air-Race Performance Model Results
Climb Selected:	Minimum Drag			Cruise Climb
S/L Rate-of-Climb	700 fpm			670 fpm
Average Rate-of-Climb *	534 fpm	319 fpm		510 fpm	305 fpm
Time to Climb 10,000 ft	31 minutes			33 minutes
S/L Climb Speed	90 KIAS			95 KIAS
Average Climb Speed **	97 KTAS		107 KTAS	102 KTAS	112 KTAS
Maximum Lead ∆xMAX	25 nm			27 nm

*   measured from 1,000 ft MSL to the respective cruise altitudes 5,500 ft and 11,500 ft MSL.
** measured from the beginning of the Air-Race to the point of maximum lead distance ∆xMAX (see text).

During the climb, if a tailwind is detected, the solution to the Air-Race puzzle calls for taking advantage of increases in wind speed with altitude.  The table shows that climbing at vMD will hasten that benefit.