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Correct Choice for Shortest Uphill/Downhill Takeoff  

John T. Lowry, PhD  

Flight Physics  

November 1993  

[Note: This article was originally published in the October 1996 issue of Mountain Pilot. Reprinted with their permission. 
For subscription information, call them at 303-397-7600 or fax them at 303-397-7619.]  

We didn't exactly have a problem, but we were making one up. It was morning at Pleasant Meadow Resort, the air was still sliding down the valley, as the three of us reluctantly contemplated jumping into our respective airplanes and heading back to so-called civilization. The problem was: uphill or downhill? We had a headwind helping us on the uphill take off. 
On the other hand, gravity would help us on a downhill take off. Terrain clearance wasn't a problem. In fact there wasn't much of a problem at all: the Resort strip is sloped a good three degrees, true, but it's paved and 2000 feet long. And the breeze was only about 11 or 12 knots. So none of us--Larry in his Mooney, Paul with his Cessna 152, or me in the 
Cessna 172--would have any real problem no matter which way we went.

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But it was the principle of the thing. We got out our Pilots Operating Handbooks. They gave us lift-off distances and
 
indicated air speeds for different weights, different pressure altitudes, and different temperatures. They even gave us rules 
of thumb to adjust for head wind or tail wind. But none of the POHs even mentioned runway slope. I know it's flat in 
Kansas, but Kerrville? Supposed to be in the Texas "hill country."  
We were standing around, our handbooks on our horizontal tails, wondering how to figure the effect of runway slope on 
distance to lift off, when out of the willows towards Slippery Slough a shadowy figure materialized. He was an old guy, 
decked out for fishing with rod and creel and a vest full of trout flies. His face was obscured by the tight weave of his 
mosquito headnet.  

"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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