Air Resistance and the Origin of Vertebrate Flight
C. A. LONG, G. P. ZHANG, T. F. GEORGE, AND C. F. LONG
Departments of Biology; Physics; Chemistry & Biochemistry/Physics & Astronomy, Biology
University of Wisconsin-Stevens Point; Indiana State University; University of Missouri-St. Louis
Stevens Point, Wisconsin 54481; Terre Haute, Indiana 47809; St. Louis, Missouri 4499
U.S.A.
Abstract:-- Flutter-gliding is a synthesis of the ground-up and tree-down theories of flight, using wingflapping and gravity for air speed, extended flight (in time and space) and softening of impacts. Air resistance
works with ground friction against running take-offs, but counters gravity for gliding.
Key-Words: -- Air resistance, Origin of flight, Archaeopteryx, proto-birds
glide
distances related to the height. The
biological fitness improved as better locomotion,
lengthened escape distance, or attack distance.
This scenario fits the principles of physics, and for
pterosaurs and certainly bats has become accepted
explanation. Flutter-gliding applied to ledgenesting, piscivorous, bipedal feathered reptiles or
semi-arboreal proto-birds. Fitness was increased
by the height climbed (increasing potential energy)
and the use of air resistance to slow the
momentum of vertical fall. Fluttering led to
important expansion of the wing-tips[1-4].
The “flutter-glide” explanation is a
synthesis in that potential energy of gravity was
available to feathered bipedal clamberers, and the
air resistance that limits and inhibits leaping or
running was useful for several advantages by
vigorously flapping feathered wings. Furthermore,
the synthesis agrees with the general
developmental progression in fledglings or birds
with poor flight (Long et al. [5]). The flutter-glide
synthesis avoids friction of running and energy
costs for hind limbs [2].
1 Introduction
Of the physical forces thrust, drag, ground friction,
gravity, and lift, drag and lift are air resistance.
Air resistance works against thrust; without it
there would be no lift (to counter gravity). What
seems a negative force in flight, and inhibits takeoffs, supports a moving projectile and acts as a
kind of lift for parachutists. If Newton’s physics
underlies evolution of flight, then air resistance
favors parachuting and gliding. Flutter-gliding
seems general in birds. Fixed-wing gliding is
reasonable for bats. Both used potential energy
from gravity and avoided ground friction. Also,
flutter-gliding by bilateral wing thrusts softened
impacts, and inevitably led to powered flight.
The two opposite theories explaining the
origin of vertebrate flight are: (1) running and
leaping into flight (“ground up theory”) and (2)
gliding to flight (“tree down theory”). The ground
up model features a biologically powered, ballistic,
bipedal form that ran or leaped into its trajectory.
The longer the flight the more dangerous the
impact. The bipedal form might have leaped from
the ground after running and lifting itself by
rudimentary wings, with maneuverability and
thrust consequently evolving. Physically, this
evolution seems unlikely, and for a poorly adapted
proto-flyer absurd. Some arboreal animals (flying
squirrels, probably pterosaurs) became parachutistgliders after mutating patagia, using potential
energy from gravity and deflecting vertical fall to
2. Problem Formulation and Solution
2.1 Physical Basis
In running for lift-off or in flight itself, a
profound retardation of speed and force by air
resistance is created by the product of the crosssectional area of the projectile and speed divided
by mass. Forelimbs of an erect bipedal reptile and
1
its light body both would physically retard the
speed and cost extra energy [6]. Quadrupedal
runners usually swing the hind limb counter to the
opposite forelimb, and even in evolution of bipedal
runners, forelimbs counterbalance the hind limbs.
Birds and gliders leap and spread their
wings or patagia with bilaterally synchronous
thrusts. Instead of increasing wing thrust [9],
which was lessened anyhow [6], a powerful
downstroke of both wings together might facilitate
a running or leaping take-off.
Running to obtain lift and air speed
expends a tremendous output of energy, especially
by the hard-working hind limbs [7,8]. Accelerating
significantly increases both air drag and ground
friction. In archaic flyers, how long might
thrusting with the hind limbs and wings continue
before all metabolic energy was spent?
Considering problems of air resistance (frictional
and induced drag against a bipedal strider),
gravity, and ground friction, a running take-off
seems impossible without highly developed wings
contributing lift and forward thrust [6].
Flapping would be adaptive for the
fluttering glider, because every downthrust
increases lift. There is no sudden need for extra
energy costs for a take-off. Propulsion by
specialized wings, said to resemble propeller drive,
integrates lift and thrust forces while minimizing
drag. Even for the fossil Archaeopteryx it was
likely. It may have leaped into flight, as Burgers
and Chiappe [9] suggested, but not by running and
thrusting with wings. Specialized flight feathers of
this bird, with two rows overlapping at the bend of
the wing, suggest separate inner and outer wing
functions and an extensible wing. Yalden [10]
discussed wing flexing in Archaeopteryx, the
primary and secondary feathers, and sufficient
wing area.
There is a likelihood of powered flight
gradually evolving in gliders by preadaptive stages
[2,3,11,12]. Wing thrusting might evolve (over
many generations) increasing momentum or
softening impacts. With wings creating lift in
descending flight, how relatively simple it would
have been to evolutionarily lengthen the tips
providing greater lift and even thrust.
Air resistance,  V, also called total drag
( Dt 0 ) , is a key force in a ballistics analysis. It is
the sum of induced drag ( Di ) and frictional drag
( D f ) . A projectile shows an increase of drag
proportional to its velocity squared (V 2 ) . A
doubled speed equals a dramatic fourfold increase
of drag with remarkable deceleration. Flapping
projectiles have resistance [2,3,4,13] called
“induced drag”. Wings pushing downward upon
an air mass create a flapping force ( K  wq) .
When air density is d , and the wing length is b ,
the air mass (defined by the dimension b as the
radius of a cross-section) becomes q = d CdVb2.
Such a product of air density and a given volume
of air allows the weight (W = Mg) to be W =
dwVb2 and the downward push w = W/dVb2. An
"induced power" sufficient to raise W results from
flapping with flight muscles. The power becomes
Ww = (W)W/dVb2. If the projectile's speed is that
of a flapping bird, Di from such flapping becomes
W 2 / dV 2 b 2  because power is reduced by V 2 .
Gravity is helpful to obtain air speed to
overcome horizontal drag. A squared velocity and
evolutionarily increased wingspan at slow speeds
decreases the induced drag and thereby creates
more lift. This is a reason for flyers, gliders and
flutter-gliders (but certainly not runners) to
lengthen the wings [2-4].
While running on the ground the projectile
suffers both induced and frictional drag. The sum
of the drags inhibits take-off. Any slowing
dramatically increases Di. Halving the speed
increases induced drag fourfold [4]. Drag works
against a leaping and flapping projectile that
decelerates at every bound, loses purchase with the
ground, and struggles against gravity. Bipedal
runners, having little or no contact at all with the
ground, suffer the attendant problems retarding
slow flight before lift-off, as well as energy and
frictional problems. But drag retardation is helpful
for a descending bird fluttering to avoid impact.
Archaic birds with poorly formed wings,
perhaps long tails, and heavy hind limbs doubtless
2
were slow either in the air or on the ground, with
or without wing flapping. These proto-birds would
suffer induced drag by flapping [2]. If speed
increases on the ground, lift-off is not assured; the
air (Df) and ground friction would increase. A leap
upward would retard the forward velocity and
increase gravity. Thrust, if it were regularly
increased by flapping [9], might help to lift a
winged bird from the ground. However, in
acceleration for take-off, thrust wanes (see Von
Mises [14] and Long et al. [6]).
The proto-bird or the early Archaeopteryx
running on the ground before take-off would suffer
even more resistance and greater energy cost than
induced drag predicts. Von Mises [14] described
the ground to air take-off in terms of Dt 0 and V 2 ,
involving also negative pressure forces (F )
between the projectile and the ground. Ground
friction and drag increase exponentially. At the
slow speed of bipedal running, the induced drag
plays a significantly adverse role. Another relevant
problem is the hindrance by an appendage. The
resultant drag of a running body with two
appendages extended into the wind exceeds (by
30-50%) the sum of the drag forces of all three
parts. Such “interference” [14] of minor
appendages may more than double the total drag of
a streamlined body. Induced drag, total drag, and
drag from extended appendages all provide fitness
for the fluttering parachutist, and for the fluttergliding synthesis.
The physicist separates drag into vertical
and horizontal directions. Air resistance ( V), also
called total drag ( Dt 0 ) , is a force that combines
with gravity against upward velocity. It may be
regarded as a type of lift. A downward glide or
flutter gliding may soften dangerous landings.
Putting aside for the moment the ground friction
that retards running and leaping, we will analyze
horizontal and vertical velocities with regard to air
resistance that retards running or flapping flight.
With the air drag  V, the velocity has
components Vx and Vy. The drag force has two
separable components (  Vx and  Vy), proving
that the drag along the x axis is proportional to Vx
and the drag along the y axis is proportional to Vy .
The following equations make physical
comparison easy for leaping and gliding
downward against air resistance. The initial values
are V x 0 and V y 0 . The simplest example is   0 ,
and y (t )  V x 0t  1 gt 2 , where t is time in units
2
of seconds and g is gravity (9.8 m/sec2). For
leaping, the maximal horizontal distance is
xmax  V02 / g , where V02  Vx20  V y20 ) and the
maximal height is ymax V02 / 2 g . The initial
value of the ascent, then, based on Pythagoras’
theorem, is V0 . When   0 , the distance x and
height y become
~
(1)
x (t ) V M /  (1  et / M )
x0
~
y (t )   Mgt /   ( M 2 g  MV y0 ) / 2 (1  e t / M )( 2)
~
(3)
x (t )  x(t )  V t 2 / 2 M
x0
~
y (t )  y(t )  V y 0t 2 / 2M .
(4)
Obviously the air resistance significantly reduces
not only the horizontal distance but the height. For
a larger initial velocity, the reduction becomes
even larger. The speed of the jumping or running
animal is retarded.
An upper limit is placed on the animal's
weight. Even the modern goose can hardly leap
into the air [10,13,15]. In hypothetical proto-birds,
reportedly resembling the dinosaur Deinonychus
(Ostrom [16]), powerful hind limbs evolved and
the forelegs were reduced. It must have been
cantilevered by a heavy tail [15]. No known flyers
or gliders have heavy tails. If more ground speed
was necessary, it would be more economical for
energy E , to gradually lengthen the legs than to
increase the rate of striding for bipedal runners
[5,6,17].
Reduction of leaping by air resistance is
~
y
 ( M 2 g / 2 ) ln( 1  V / Mg )  MV / 
max

y0
3
2
ymax  V y 0 / 3Mg .
y0
(5)
Suppose V y 0 = 10 m/sec, M = 0.2 kg (an
estimated value for Archaeopteryx) and  = 0.1
3
or
1
2
~
~
Tdown  [2 y / g  Tdown / 3Mg ] .
kg/sec; then ymax = 5.10 m becomes ymax =
3.36 m. A reduction of 1.74 m is made in the
height from the ground. Flight requires that
ymax > 0, which follows from the constraint on
the ratio  between the initial force f 0 and
weight,
3
3
f 0  V y 0  Mg (6a );   f 0 / Mg 
(6b)
2
2
With air resistance, increasing the velocity
may not yield take-off. Two options for the animal
are: (1) to reduce the surface area A of the wing,
and (2) to reduce the angle of attack,  . The first
choice is counterproductive as the flapping animal
needs large wings. Lift to drag might be increased
by lengthening the wings. For the second choice,
a finite angle is needed to generate sufficient
aerodynamic lift, but increasing the angle  would
decrease the time in flight and shorten distance.
Although Burgers and Chiappe [9] suggest
Archaeopteryx attained lift and increased velocity
of several m/sec by running and sculling the
wings, they did not consider the deceleration of
flapping-caused air drag. Perhaps they presumed
that the described thrust [9] overcame the force of
drag. Their hypothesized counterclockwise
rotation of the power resultant indicates
graphically that thrust wanes toward zero.
Without air resistance, the time for upward
(9)
~
2
If   0 , then t down  [2 y / g ] . Thus, Tdown is
always larger than t down . This extra time would be
an advantage in gliding evolution, and lead toward
induced flight. The y cannot be high due to the air
resistance, not to mention the limitations due to
ground friction and the leaper's limited muscular
power [2, 3] [5, 6]. The  cannot provide a larger
value, to satisfy Equation (6). These factors limit
the time in the air. If we use y  3.36 m and the
same resistance as before, then t down  0.828 sec,
~
and Tdown  0.834 sec, an insignificant difference.
For a horizontal escaping distance x, when
the variable   0 , the maximal distance x max is
1
1
xmax, glide  Vx0 / g (V y 0  [V y 02  2 gy ] 2 , (10)
where y is the animal's vertical distance measured
upward from the ground or between two different
heights. If y = 0, we then go to the leaping process
xmax,leap  2Vx0V y 0 / g . With the same initial
velocity, x max, glide  x max, leap .
Interestingly,
becomes
even
greater with larger y . The y has no limit since it
is not determined by the running velocity, but by
how high the animal climbed above the ground.
Gliding is energetically "cheap" unless a great deal
of energy were expended in climbing, which is, at
least
in
modern
animals,
metabolically
replenished. For leaping, the minimal energy is
and downward paths is tup  tdown  V y 0 / g . The
total time is the sum, t to  2 V y 0 / g . For the initial
velocity V y 0 = 10 m/sec, tt 0 = 2.04 sec. If   0 ,
we should treat them separately. For the upward
leap against  ,
~
Tup  M /  ln (1  V y 0 / Mg )
 tup  V y 0 2 / Mg 2 .
x max, glide
MV y 0 2 / 2 , and it would cost 10 J of energy for an
animal of M = 0.2 kg to reach the instantaneous
velocity V0 = 10 m/sec.
Norberg [2,3] and Long et al. [5,6]
mentioned the remarkable output of power needed
by a bipedal proto-bird running on the ground and
concomitantly flapping to increase speed. She and
Tennekes [4] added that gravity also enters into the
problem at lift-off. Results on flight energetics of
small parrots (Melopsittacus) in wind tunnels
show that the minimal energy used in flapping
(7)
Thus, due to air resistance, the time aloft is
shortened, and for the above parameters reduced to
~
0.52 sec. After Tup , the vertical velocity is zero
and the animal descends. The falling time from a
~
y  (Mg /  ) Tdown  M 2 g / 2 (1  e  Tdown / M )
~
~
y  g (Tdown2 / 2   / 6 M (Tdown3 )
(8)
4
flight is in a fluttering descent, and the most
expensive flight is flapping upward [18,19]. The
evolutionary economy of energy (upward of 33%)
likely parallels that observed in the wind tunnel, if
both phenomena relate to underlying physics.
For a flying animal, the horizontal plane
has a small resistance  , while in the vertical
plane,  is large. This disparity works against
deep (high) and narrow bodies and has helped
streamline body forms during evolution. A broad
and shallow form suggests tree-down evolution. A
bipedal runner with vertical stance presents its
entire frontal plane to the wind [6]. The glider
avoids much resistance by its flattened shape.
From Equation (1), we know the flight distance,
inversely proportional to x , becomes longer.
Secondly, a long duration time, Tdown , is needed,
which is determined by the height y and resistance
y [see Equation (9)]. The prolongation helps the
animal attain more opportunity to manipulate its
rudimentary wings. A leaping or flying animal
functionally and evolutionarily may decrease air
resistance by assuming or evolving a flattened
projectile shape (e.g., leaping spread-eagled,
extending the limbs outward, and pushing against
the air). A deep breast (keeled sternum and
massive pectoral muscles) was absent in
Archaeopterx [15,16,19].
falling forcibly is counter to either flutter-gliding
or especially gliding with fixed wings, but fitness
compensation may overrule in evolution by
increased survival from predation, better mobility,
or obtaining a new food source. Climbing tree
trunks in Archaeopteryx may have been impossible
(a negative attribute for gliders?),but flitting from
branch to branch [21] and clambering with clawed
wings seem plausible. Leaping, which seems not
feasible leading to take-off, does increase the
initial velocity and extended trajectory of gliding
descents. It also might facilitate leaping from the
ground into low branches of trees, preadaptive to
climbing and gliding.
Two dendrograms based on character
matrices [ 22] using biological and paleontological
information (scaled 0-2) compare the models on
flight. Such graphs depict parsimonious and
objective analyses. By disparaging the two running
models, the two gliding explanations are greatly
strengthened. The most parsimonious tree,
containing only 19 steps, supports our fluttergliding synthesis. Both kinds of tree-down models
pair as a branch, and running take-offs seems
most disparate. If one combines the gliding with
either of the running models, it adds 9 steps. It
adds ten steps to combine fluttering with running.
A second matrix included data published on the
reptilian, bird-like fossils from Texas and China,
including such evolutionary-morphological traits
as supracoracoideus, feathers, and keeled sternum
in Protoavis (but not in Archaeopteryx). The same
dendrogram was created. The poorest results arose
from double splitting of the two kinds of gliding
and the two kinds of running. The most feasible
pairing seems to be of the two kinds of gliding.
The data for either tree down model reflects
evolutionary fitness. A third matrix based on
ballistic features created a similar tree. Eight
features were negative for either running take-off
or running on water, and there was a net gain of
five (for gliding) and six for flutter gliding. The
ballistics dendrogram resembled the others.
Flutter-gliding might
multiply fitness
owing to the multiple wing strokes, which may
contribute to the extended length of the maximal
2.2 Biological and Physical Comparisons
In comparing four possible modes of evolution of
flight, a tabulation (Long et al. [5]) was made of
fitness attributes and their negatives for
Archaeopteryx, admittedly arbitrary but those
found had been published as facts. They include a
preponderance of positive values for fluttergliding. The totals tell only part of the comparison,
because both of the ground-up hypotheses had
numerous negatives (some of which falsify the
running hypotheses). The negatives relate to both
function and evolutionary history. For example,
the competition of body energy E for both running
and flapping is a serious physiological problem,
not only for the ascending proto-bird itself, but
likely it shaped the history of evolutionary
modification. A possible negative attribute of
5
glide (against gravity and air resistance).
Biological or ballistics information similarly
suggest that the flutter gliding synthesis is, at the
least, as robust as the fixed-wing gliding model.
The ballistics analysis appraises immediate effects
on a projectile (mostly of the important force air
resistance). One may expect physical principles to
underlie both functional performance and longterm evolution.
runs of bipedal winged vertebrates. Archaeopteryx.
20, 2002, pp. 63-71.
7. Pritchard, W. & Pritchard, J. Math models of
running. Amer. Scientist 82, 1994, pp. 546-53.
8. Cavagna, G. A., Saibene, F., & R. Margaria,
Mechanical work in running. J. Applied Physiol.
19, 1964, pp. 249-56.
9. Burgers, P. & L. M. Chiappe, L. M. The wing of
Archaeopteryx as a primary thrust generator.
Nature (London) 399, 1999, pp. 60-62.
10. Yalden, D. W. The flying ability of
Archaeopteryx. Ibis 113, 1971, pp. 349-56.
11. Spurway, H. Shadow elimination and origin of
flight. Symposium on Organic Evolution. Bull. Nat.
Inst. Sci. India. 7, 1955, pp. 110-11.
12. Bock, W. J. The role of adaptive mechanisms
in the origin of higher levels of organization.
Systematic Zool. 14, 1965, pp. 272-287.
13. Pennycuick, C. J. Mechanics of flight. In:
Avian Biology, Vol. 5, (Farner, D. S. & King, J. R.,
eds.) pp. 1-75. Academic Press, London. 1975.
14. Von Mises, R. Theory of flight. Dover, New
York. 1959.
15. Thompson, D'Arcy On growth and form.
Cambridge University Press, England. 1961.
16. Ostrom, J. H. Archaeopteryx and the origin of
flight. Quarterly Rev. Biol. 49, 1974, pp. 27-47.
17. Taylor, C. R. The energetics of terrestrial
locomotion and body size in vertebrates., In: Scale
effects in animal locomotion (Pedley, T. J., ed) pp.
127-141. Academic Press, London. 1977.
18. Tucker, V. A. Respiratory exchange and
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Experimental Zool. 48, 1968, pp. 67-87.
19. Tucker, V. A. Gliding birds: The effect of
variable wing span. J. Experimental Zool. 133,
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20. Ostrom, J. H. The ancestry of birds. Nature
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21. Chatterjee, S. & R. J. Templin. The flight of
Archaeopteryx. Naturwissenschaften, 90, 2003, pp.
27-32.
22. Feduccia, A. The origin and evolution of
birds. Yale University Press, 1996.
3. Conclusions
The running and leaping (ground-up)
origin of flight in vertebrates is not feasible
physically in space or time, considering air
resistance or forces against bipedal running. Air
and mechanical resistance and energy allocation
from hind limbs to wings favor the tree-down
models. Gliding [3,22] or flutter-gliding [5] makes
use of the potential energy from height, and the air
resistance in falling seems useful to soften the fall
(especially in flutter-gliding). Energetics, body
form, behavior, and ontogeny of modern birds and
the relation of Archaeopteryx and other fossils
support the synthesis [5]. Physical theory based on
Newton’s principles also supports this synthesis.
References:
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2. Norberg, U. Evolution of vertebrate flight: An
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3. Norberg, U. Vertebrate flight. Mechanics,
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4. Tennekes, H. The simple science of flight from
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6. Long, C. A., G. P. Zhang, G. P., & T. F. George.
Physical and evolutionary problems in take-off
6