"Which way you boys heading out?" he asked as he approached. "Upstream or down?"  

"Just what we were trying to decide," I said. "Any ideas?"  

"Maybe." He looked around at the tattered wind sock and swaying leaves. "Looks like it's about 11 or 12 knots even 
down at wing level. It's a three degree slope. I suppose you boys have your no-slope no-wind lift off distances, for this 
elevation and temperature, and your aircraft weights. Got your lift-off speeds?"  

"We can get them."  

And we did. The wizened old duffer had one of those calculator watches. He went to work. In just a minute he pulled his 
knit sleeve down over his watch, cleared his throat, and spoke.  

"The Mooney should take off uphill. But the 172 should go downstream."  

"What about me?" asked Paul.  

"Oh yes, the 152...doesn't matter. Do whatever you want."  

"How do you figure?" we all said.  

"Little-known formula," he said. "Works every time. Got a scrap of paper?"  

Here's what he wrote down and explained:  

Breakeven is the "breakeven head wind," in knots. Angle theta is the runway slope up (in degrees); dLO(0,0) is the POH distance to lift off with no slope and no wind (in feet); and VLOT is the lift-off speed expressed in KTAS. The 1/5 is really an easy-to-remember approximation to 0.1971 = g*sine 1 degree*0.5924682. The 0.592468 factor comes from converting feet/sec to knots. Factor g is the acceleration due to gravity, 32.2 ft/sec2. He also used the fact that, for small angles theta, sin(theta) = theta in degrees times 0.0175 (the sine of one degree).  

"All you have to do," he said, "is use this formula to calculate the breakeven head wind for your airplane on this day. 
That's the wind which would give you the same distance to lift off whether you went upstream or down. For the M20K, 
for instance, it came out to 7.9 knots. Since the actual head wind is larger than that, the Mooney should take advantage of it."  

"And heavy as I am today," I offered, "my breakeven head wind was higher than the actual 11 or 12 knots."  

"Right," he said. "Breakeven came out 14.1 knots for your 172."  

"And my breakeven came out right on the actual!" Paul exclaimed. "So it doesn't really matter which way I go out."  

"Of course there might be other considerations," he mused. "Here you don't have terrain clearance to consider, but often 
that will tilt you towards downstream. But on the other hand, if you take off with a tail wind, you'll have a flatter climb out 
because of increased ground speed. And the usual wind shear, wind getting stronger as you leave the ground, will give 
you an increasingly stronger tail wind while you climb out. It's a balancing act."  

I had an objection. "But what about the distance to lift off? It wouldn't do us much good to take off in the best direction if 
that lift off distance is still longer than the air strip!"  

"Only way that could happen," he answered, "is if you were dumb enough to come into a strip from which you couldn't 
take off even if it were level and there were no wind. I doubt you'd do that. When the actual wind is the same as the 
computed breakeven wind, you get the longest take off you can have as long as you follow the take off direction given by 
the formula. And it turns out that the breakeven distance to lift off is the same as the distance to lift off under the same 
airplane and density altitude conditions but on a level runway in a calm."  

"Are these formulas and prescriptions exact?" I wanted to know. "Or only approximate?"  

He thought a minute. "Everything about these airplanes, in a practical sense, is approximate. Except maybe the number of 
wings and the number of cylinders..."  

"And around overhaul time," Larry piped up, "even my number of cylinders is a bit vague."  

"Naturally there's a fuller story," the old guy said. "Lemme see that piece of paper." On the back of it he scribbled:  

"But even that's an approximation. By the way the lift-off speed in the denominator is in feet per second. And to get the rule about the breakeven take off the same as the no-slope no-wind take off, you have to use the binomial expansion to simplify the power in the numerator. Gotta run. Fish to clean."  

And off he sauntered towards his cabin, laden with fishing gear and formulas. We finished loading the aircraft. Jake, the 
proprietor of the Resort, came out to see us off.  

"Did you ask the old fisherman about which way to take off?" Jake queried.  

"We didn't have to ask him; he volunteered," Larry said.  

"Yeh, who was that unasked man?" Paul put in.  

"Joe something. Teaches physics at the college. Good fisherman," Jake said.  

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