You say your airplane's POH doesn't have some performance numbers you need? Or, because of airframe or powerplant modifications, your factory-original POH performance section is out of date? Or perhaps you need numbers for your one-of-a-kind homebuilt? Don't despair ... and don't guess! Now there's an easy way to calculate accurate light aircraft V-speeds, rates and angles of climb, thrust, drag, and much more. Aviation physicist and private pilot John T. Lowry shows you how. (This first installment deals strictly with simple fixed-pitch-prop airplanes; a follow-up article deals with constant-speed props and other complications.)
December 8, 1999
The Bootstrap Approach is a method which lets Joe (or Jo)
Pilot calculate his (or her) airplane's performance numbers performance for any gross
weight at any density altitude without a master's degree in aeronautical engineering.
You simply fly some sawtooth climbs and glides (less than an hour), calculate four
numbers, pick up five more numbers from your airplane's POH, then plug them into formulas.
Those formulas are taken care of automatically in the downloadable Excel spreadsheet which
accompanies this article. The nine basic numbers you'll need in order to use the
spreadsheet make up what we'll call the airplane's Bootstrap Data Plate, or BDP for short.
The Fine Print
We all learned to run before we learned to dance, and this subject is no exception. So
in this article we will restrict ourselves to steady flight performance of fixed-pitch
propeller-driven airplanes with a single normally-aspirated engine (properly leaned for
maximum power) running either at full open throttle or gliding with closed throttle, at
one flaps/gear configuration, with wings level in calm air. Follow-up articles will take
up the performance of constant-speed propeller aircraft, maneuvering, partial-throttle
operations, light twins, turbocharged engines, wind and updraft/downdraft effects, and
takeoff and landing operations.
Like many of us, the Bootstrap Approach has some built-in limitations. We assume the
thrust line is directed along the flight path (certainly no Ospreys or other powered-lift
aircraft need apply!) and that the flight path is not too steep, inclined less than 15
degrees from the horizontal. The airplane's movement is assumed 'steady' in the sense that
it is not being accelerated along the flight path. (Later on in the series, steady turns
will be allowed.) So the bootstrap approach spreadsheet can't tell you what your father's
J3 Cub would do halfway through a tail slide. (Not recommended!) But it can come up with
almost all the performance numbers most of us need.
Our current aim is to come up with two kinds of performance figures. First, we'd like
our airplane's interesting V-speeds, and performance numbers associated with those speeds,
for any desired altitude and gross weight. See Table 1 for specifics and examples. Notice
that the bootstrap approach does not produce the airplane's stall speeds, or any
structural-limitation V-speed such as maneuvering speed Va; you can get those
from the POH or by flying separate tests. Secondly, we'd like to know the airplane's
thrust and drag and its detailed flight path its angle or rate of climb or descent
for any desired speed, weight, and altitude. See Table 2.
||Maximum level flight speed
||Minimum level flight speed
||Speed for best angle of climb
||Best angle of climb
||Speed for best rate of climb
||Best rate of climb
||Speed for best glide
||Best glide angle
||Speed for minimum descent rate
||Minimum rate of sink
Table 1. Bootstrap V-speeds, and corresponding performance
numbers, for a particular Cessna 172 weighing 2200 lb. at an altitude of 7000 feet.
||Full-throttle rate of climb
||Full-throttle angle of climb
||Gliding rate of sink
Table 2. Bootstrap performance numbers for a particular Cessna 172
weighing 2200 lb. at altitude 7000 feet at 80 KCAS.
Since implementing the bootstrap approach is a detailed formula-based numerical
procedure, and because spreadsheets are an easy way to do repetitive lengthy calculations,
at this point you should consider downloading a copy of the companion spreadsheet which
can do the calculations for you.
|NOTE: You may want to download the spreadsheet now,
because it'll make it easier for you to follow the rest of this article. The spreadsheet
is available for download in two formats:
There are no macros or menus or anything fancy just input, output, and formulas in
The next three sections of this article will correspond to the spreadsheet's three
major sections Parts A, B, and C.
Part A: Obtaining Your Airplane's Bootstrap Data Plate
You've seen those little aluminum data plates riveted onto the fuselage beneath the
horizontal tail, giving the airplane model, gross weight, serial number, etc. Think of the
Bootstrap Data Plate as something similar, but invisible. It's a data plate which
once you decode it tells you a lot about your airplane's performance. But first we
must "develop" the airplane's Bootstrap Data Plate, rendering it visible. Table
3, from cell block A39:C47 of the spreadsheet, is a typical Bootstrap Data Plate.
|Wing area, S
|Wing span, B
|Rated engine power, P0
|Rated engine speed, N0
|Propeller diameter, d
|Parasite drag coefficient, CD0
|Airplane efficiency factor, e
|Slope of propeller polar, m
|Intercept of propeller polar, b
Table 3. Bootstrap Data Plate for a particular Cessna 172, flaps
The first five numbers in Table 3, the "easy" ones, come straight from your
POH. The last four "harder-to-get" numbers come from tough work, but
somebody's got to do it flying the airplane! See Figure 1.
Figure 1. The five "easy" BDP items come from the POH or
common knowledge; the four "harder-to-get" items come from the sawtooth climb
and glide tests.
Now let's focus on the sawtooth climbs and glides. See Figure 2. The object is to find,
by trial and error, the airplane's speed for best angle of climb Vx with
corresponding AOCmax (best or maximum angle of climb) and its speed for best
glide Vbg with corresponding BGA (best glide angle). POH values should give you
first good guesses, but if you're lighter than the standard weight used in the POH both
true Vx and true Vbg will be lower than their book counterparts.
Figure 2. Sawtooth climbs and glides (perhaps five or ten of
each), give you sufficient test data to calculate the four "harder-to-get"
Bootstrap Data Plate items.
You will need stabilized airspeed (hopefully within a knot) during each run. So that
means starting high on the glides, and low on the climbs, to give yourself time to
stabilize at your chosen target airspeed. The measured sawteeth need not be large;
altitude excursions of about 500 feet are usually sufficient. Table 4 shows the data
you'll jot down during each climb or glide. Figure 3 shows the kinds of graphs which might
result. As you can see, Vx here is approximately 61 KCAS and Vbg is
about 69 KCAS.
Table 4. Data collection form with entries for two climbs and two
glides. We've assumed the ASI has no calibration error. Gross weight can be filled in
later. The next-to-last column is calculated, and plotted against the airspeed (as in
Figure 3) to help pinpoint the minimum of the climb values and the maximum of the glide
Note that when doing these climb/glide tests, each climb and glide for which you record
KCAS and delta-t must be conducted between the exact same upper and lower altitude limits.
It doesn't really matter what the altitude change is (300 feet or 800 feet) so long as
it's the same for all the data points. The climb/glide tests should also be made during a
relatively short elapsed time so that the atmospheric temperature doesn't change
significantly between the first test and the last one.
For the best results, you should use a different (stabilized) airspeed for each climb
and each glide, covering a range from well under the expected Vx and Vbg
to well above it. Your final test should be made at your provisional best guess at Vx
Either during the test flight (assuming you have an assistant along) or when you're
done recording your climb and glide test results and are safely back on the ground,
calculate the seventh column of your data collection form (Table 4 above) by multiplying
the recorded KCAS by the recorded delta-T for each test. Then plot these values of (KCAS *
delta-t) against KCAS on a piece of graph paper, and try to draw a smooth curve through
all your data points. I prefer to do this inflight (using an asistant) so that I'm sure
we've honed in on Vx and Vbg. When you've graphed the values, you
should come up with something that looks like Figure 4 below:
Figure 3. Graphs of (KCAS * delta-t) vs. KCAS for both the climb
and glide series of points. The climb minimum shows Vx = 61 KCAS and the glide
maximum shows Vbg = 69 KCAS.
What equipment does it take to collect and process this flight data? Besides the
airplane, paper, graph paper, and a pencil, you will need:
- Calibrated airspeed indicator (and the calibration curve needed to get from KIAS to
- Calibrated altimeter (set to 29.92 in. Hg, so that it reads pressure altitude);
- Stop watch (and a calibrated thumb to operate it);
- Calculator (to get the seventh column in Table 4;
- Ordinary watch (to later figure fuel burn and hence gross weight); and
- OAT thermometer (to later help the spreadsheet correct pressure altitude to density
Part A of the downloadable spreadsheet will hold the experiment's identifiers, the five
"easy" BDP numbers, and will also process your collected flight test data to
produce the four "harder-to-get" BDP items. All the numbers you need to input go
into green cells in the spreadsheet. A millisecond after finishing your input you will
have the corresponding Bootstrap Data Plate output. Phase I of the Bootstrap process a
one-time job is now complete.
That last, about the one-time job, is important. To some, even to a few
engineers who've heard about the bootstrap approach, the process seems circular. They
think "Get Vx to later find out what Vx is? That's a big
advance? What's wrong with this picture?" What they fail to realize is that (still
focusing on Vx) it's more like: Get Vx once, for one known
weight and altitude, to be able to get Vx (and much more) later, for any
weight and for any altitude. It's somewhat like your pilot training. You first take
lessons, say for forty hours, to then later be able to fly around the countryside for 400
hours. Or for 4000 hours.
You've got your Bootstrap Data Plate. The training wheels are off.
Part B: Calculating Your Airplane's V-Speeds and V-Speed Performance
Think for a minute about your flight controls. Since we've restricted ourselves in this
first article to fully open or fully closed throttle, you have no choices there. Wings
stay level, and flight stays coordinated, so there's nothing creative to do with either
ailerons or rudder. Flaps stay fixed, probably up. All you have left in your control bag
is the ability to change airspeed, using the elevator control, and the freedom to move to
a different altitude. Only airspeed V and altitude h are at your disposal. And those only
within limits. In a sense you can also choose the airplane's gross weight, W, but you'd
best do that on the ground, while fueling. Passengers strongly protest being ejected from
the airplane in midflight. So in our current scenario the "operating" variables,
those somewhat under the pilot's control, consist of only airspeed V, density altitude h,
and gross weight W.
In the spreadsheet's V-speed section, Part B, speeds are outputs. Here you're down to
only the two operating variables h and W. Further, since a set of altitudes h has already
been provisionally chosen for you (see Table 5, though you are perfectly free to
substitute your own altitude choices), the only free choice left is gross weight W. You
may want printouts of Part B for three or four different gross weights. Stash them away in
your airplane's glove box or in your map case.
|Gross Weight, W: 2000 lbs.
Table 5. Part B of the accompanying spreadsheet Bootstp1.xls for a
Cessna 172 weighing 2000 pounds.
Part C: Calculating Thrust, Drag, and Rate or Angle of Climb/Descent
In this "any speed" section, Part C, you must choose a (density) altitude h
and a gross weight W. Then thrust and drag and rate of climb, etc. all the data items
of Table 2 are given for a range of airspeeds V. Again, feel free to tailor those
suggested speeds to your faster or slower airplane. If you want readily-available printed
output for say three altitudes and three gross weights, you'll end up with nine different
Part C performance sheets.
|Gross Weight, W: 2200 lb
|Density Altitude, h: 5000 ft.
Table 6. Part C of the accompanying spreadsheet Bootstp1.xls for a
Cessna 172 weighing 2200 pounds at 5000 feet.
AVweb readers may prefer to turn this kind of tabular material (their own
versions of Tables 5 and 6) into graphs. Figures 4, 5, and 6 below are examples. Using
various spreadsheet techniques (or by simply recalculating and jotting down changed
results) one can also put together graphs which do not come directly from the spreadsheet
cells. For example see Figure 7 for how maximum rate of climb depends on altitude for
three different gross weights.
Figure 4. Major powered V-speeds vs. altitude for a Cessna 172
weighing 2400 pounds.
Figure 5. ROC vs. V for a Cessna 172 weighing 2400 pounds at three
Figure 6. Thrust and drag vs. airspeed for the Cessna 172, flaps
up, at MSL weighing 2400 pounds.
Figure 7. Maximum rate of climb vs. altitude for the Cessna 172 at
three gross weights.
Since no one can (or should!) do these calculations in the cockpit, not even after
they've had time to absorb what they mean and how to do them, you'll want to sit down with
a cup of coffee and think through which graphs or charts you really want. That includes
selecting density altitudes at which you normally (and perhaps abnormally!) fly, at which
weights, and what the most appropriate ranges of featured airspeeds should be. Figure out
the least amount of most important information before you construct the final output
spreadsheets and graphs. Less is more, so start small. You can always add another graph or
table later on.
Next Bootstrap Steps
In the next article of this series,
we'll shift gears to the constant-speed propeller aircraft, the type flown by most AVweb
readers. In the meantime, those of you wanting more background on the bootstrap approach
can find it in books and articles, at selected Web sites, or in feedback comments here on AVweb.
The following references were all written by the author:
- Performance of Light Aircraft, AIAA 1999, available for purchase from AIAA at
www.aiaa.org, from Amazon at
www.amazon.com, or from Barnes and Noble at
www.bn.com. The mathematics level is mostly that of early
college with an occasional (but optional) bout of heavy lifting.
- Computing Airplane Performance with The Bootstrap Approach: A Field Guide, M
Press 1995, available from the author at firstname.lastname@example.org.
Lots of bootstrap formulas but almost no derivations. There is a companion disk of
- Aircraft Performance at High Density Altitude, USAF 1999, is a short book written
for Civil Air Patrol mountain search and rescue teams. Graphics instead of algebra. I'm
uncertain whether the Air Force version will be available other than to CAP members.
- Several professional engineering bootstrap articles have appeared in such publications
as Journal of Aircraft, Journal of Aviation/Aerospace Education and Research,
and Journal of Propulsion and Power. Contact the author, by email, for specific
- The author's Web site, www.mcn.net/~jlowry will
have rotating articles on aircraft performance (some bootstrap, some not), and has a link
to four bootstrap articles on the AllStar Network (supported by NASA and hosted by Florida
The best and simplest way to get your bootstrap questions answered is to ask them right
here on AVweb, through the article feedback facility (see buttons below). And to
stay tuned for future articles.