Correct Choice for Shortest Uphill/Downhill Takeoff
John T. Lowry, PhD
[Note: This article was originally published in
the October 1996 issue of Mountain Pilot. Reprinted with their permission.
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.
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
"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
"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
"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.
"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
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
"Only way that could happen," he answered, "is
if you were dumb enough to come into a strip from which you couldn't
"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
"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,
"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.