AOPA Pilot Magazine - November 2022 - 91

RUDDER & WRENCH
FLY ING SMART /
The power of two
Why math matters
BY CATHERINE CAVAGNARO
doubling B quadruples A, and cutting B
in half reduces A by a factor of four. For
example, since lift is proportional to the
square of airspeed, changes in airspeed
have a profound effect on lift-more so
than either air density or angle of attack.
When you see the power of two in a formula,
that should get your attention.
Let's consider four flight situations for
which the power of two can show us what
matters most.
Knowing what changes
to make in a pressure
situation can mean
WHETHER I T 'S PASSING a checkride or
getting out of a sticky flight situation,
pilots need to perform their best, and that
involves knowing key changes we can
make to ensure a successful outcome. For
example, if we are departing from a short
runway, increases in atmospheric pressure
can shorten takeoff distance. But decreasing
payload, or waiting for cooler temperatures
or better winds makes an even bigger
difference in aircraft performance. (see
" What Matters Most: If a Performance
Chart Could Talk, " July 2021 AOPA Pilot)
Knowing what changes to make in a pressure
situation can mean earning a new
commercial certificate or even walking
away from a landing after an engine failure.
Mathematical models can shed light
on ways to optimize performance. Saying
that one quantity is equal to some combination
of others tells a story about the
relative importance of those ingredients,
and it can betray what matters most.
Yeah, I get it. Although my degrees are
in mathematics, like most people my eyes
can glaze over when I see lots of symbols.
Like any story, these equations take time
to unpack to distill their messages. I hope
you'll agree that the relationships below are
compelling enough to be worth that effort.
Let's start by defining what it means
for a quantity A to be proportional to
B. In that case, A is a multiple of B, so
doubling B means that A is doubled too.
Cutting B in half has the same effect on
A. For example, the lift generated by our
airplanes is proportional to air density
so, all other factors equal, airplanes generate
about half the lift at 18,000 feet msl
than they do at sea level. Lift is also proportional
to (absolute) angle of attack,
so doubling angle of attack increases lift
by a factor of two provided the latter is
below the critical angle of attack.
Now, we should pay special attention
when A is proportional to B2
. In this case,
earning a new commercial
certificate or even walking
away from a landing after
an engine failure.
Canyon turn
When I give commercial or flight instructor
practical exams, candidates often report
that the chandelle is a great save-the-day
maneuver in a box canyon situation where
reversing course in a confined area is critical
to averting disaster. But considering
what's important for a tight turn shows the
chandelle is not a technique for our mountain-flying
toolbox.
Turn radius is proportional to
the square of airspeed (in a no-wind
situation), so any time spent at a higherthan-necessary
airspeed will result in a
much greater turn radius. The typical
chandelle starts from the maneuvering
speed of the aircraft which is roughly
twice its stalling speed. If reversing
course in a confined area is important,
the chandelle just isn't your answer.
AOPA PILOT / November 2022
91
MARCIN WOLSKI
AOPA Pilot Magazine - November 2022

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