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(22d) Airplane Flight                             
          -- How High? How Fast?

(Optional unit on the principles of airplane flight)


22.Reference Frames

22a.Starlight Aberration

22b. Relativity

22c. Flight (1)

22d. Flight (2)

23. Inertial Forces

23a. The Centrifugal Force

  23b. Loop-the-Loop

  24a.The Rotating Earth

24b. Rotating Frames

The Sun

S-1. Sunlight & Earth

S-1A. Weather

S-1B. Global Climate

S-2.Solar Layers

S-3.The Magnetic Sun

S-3A. Interplanetary
        Magnetic Fields

S-4. Colors of Sunlight
    An airplane stays aloft because of the flow of air above and below its wing. That flow creates an upward "lift" force opposing gravity, which keeps the airplane from falling.

Streamlining and Drag

    The cross-section ("profile") of an airplane wing must meet two demands. First, its rear must taper down to a thin edge, like a wedge. That is where two air flows join together, from above and below the wing, and such "streamlining" assures that they meet smoothly, without swirling flows which increase air resistance. In contrast, an open parachute, whose rear is a half-sphere, creates a great deal of swirling behind it and has a large resistance; trucks which end abruptly in a high cargo door similarly encounter relatively high air resistance.

    [Contrary to intuition, the shape of the front is less critical. Perhaps our intuition owes too much to the bows of ships, which need a sharp edge to slice through surface waves. Deep-sea nuclear submarines have blunt spherical fronts, just as]

Forces on airplane

   Streamlining reduces air resistance ("drag" in aviation). Experiments have shown that the drag force D (bold face is not used here to distinguish vectors) increases with velocity v--in fact, it increases like v2. It is also proportional to the air density d; all other factors we lump here into a coefficient A which is proportional to the wing area and depends on the shape of its cross section (that is where streamlining comes in) and and on the "angle of attack" with which is faces the air flow (angle=zero when the wing is lined up with the air flow).

D   =   A d  v2

    The proportionality of the drag of a streamlined wing to v2 was found from observations, but it can also be supported by the rough ("hand-waving") argument below. Follow it slowly!.

    A wing of an airplace facing an airstream loses energy mostly by shoving aside air whose space it proceeds to occupy. If the pattern of flow lines around the wing at low and high speeds is the same (reasonably true), when v doubles, the velocity of air shoved aside is also doubled, and from this process alone, the (kinetic) energy imparted to it, proportional to mv2, should increase 4 times.

   How about m? With the doubled velocity, the wing advances twice the distance as before each second, so the mass of air m shoved out of the way also doubles. The overall rate at which energy is imparted by the wing to the surrounding air, therefore, increases 8 times.

    That rate should match the mechanical work done each second by the force D (i.e. the power required): it, too, should increase 8-fold. Since the distance covered per second is v, the work done each second is Dv. If v is doubled and Dv increases 8-fold, then D must grow 4 times --a growth proportional to v2.


   The second requirement is that the wing produces lift, an upward force holding the airplane aloft. To produce lift, the wing must be non-symmetrical--flat on the bottom but rising in a curve on top. That shape speeds up the air flow over the top, which reduces the air pressure there, and when the pressure on the bottom of the wing is larger than the one on the top, the net result is an upward force. [A wing symmetric on top and bottom, but meeting the air stream with a moderate angle of attack, also satisfies this non-symmetry condition.]

The Wright brothers used a wind tunnel to test models of different wing profiles, identifying cross sections suited for various types of flying. They also found by observations that the lift force produced by a wing was roughly proportional to the density d of the air and to the square of the velocity v of the airflow over it:

L   =   B d  v2

    Here L is the lift in (say) Newtons, d the air density (about 1.3 kg/meter3 at sea level) and v may be in meter/sec, mph or km per hour--whatever one prefers. The factor B depends on the profile of the wing, the wing's length and its width: a larger wing obviously gives a larger lift. Obviously, the lift is proportional to drag:airplanes usually fly at the "angle of attack" (defined earlier) which gives the most economical operation, when the lift/drag ratio is at its largest. That "best" value depends on a wing's design and can range from 10 (even less in warplanes) to 50 (in high-performance gliders).

    One can increase the lift by increasing the angle of attack (as one does with a kite), but at the price of a much greater drag. Furthermore, if the angle is too steep, the orderly flow above the wind is disrupted and the wing "stalls, " suddenly losing much of its lift. Many airplane crashes have been traced to sudden stalls.

How high, how fast?

    Suppose you design an airliner weighing W kilograms (about 10W Newtons). In level flight, of course, lift must balance the airplane's weight

L   =   W
B d v2   =   W

    The value of W is set by B--in other words, the wing must be long enough, wide enough and efficient enough to keep the weight W of the loaded airliner in the air.

    How high and how fast should the airliner fly? Passengers want to reach their destination quickly, so designers aim at a high "cruising speed." However, passengers also value safe landings, and hence the landing speed should be slow.

    Speed is the main reason why airliners fly at an altitude above 30,000 feet or close to 10 kilometers. Air density decreases to about 1/2 with each added 5 km of altitude, so at 10 km, d is about 1/4 its sea-level value and an airplane can double its speed to generate the same lift, with the same drag D (which, as shown, also grows like dv2). The main reason airliners have pressurized cabins is to allow them to fly higher, in order to fly faster.

    How fast? The practical limit seems to be around 600 mph (960 km/hour). Any closer to the speed of sound (1200 km/h = 746 mph, varies with temperature) and the air flow above the wing creates shock fronts which increase drag and reduce lift. Even to get that far, swept-back wings are needed.

Landing Safely

    A speed of 600 mph at an altitude of 10 km seems to imply a sea-level landing speed of 300 mph (d is 4 times larger, so v can drop to 1/2). That is still too fast--even the space shuttle is said to land at 270 mph. One could fly at 70,000 feet (about 20 km) the way the U-2 reconnaissance plane does, and land (even without changing the angle of attack) at 150 mph. However, to create the necessary lift in the thin air at that height, B must be much bigger--that is, the wing must be much larger--or else the weight W must be reduced (or both). That was done for the U-2, a light-weight airplane with a very long and efficient wing, but such a design would not work on the scale of an airliner.

    The practical solution is to increase the angle of attack during landings, and to temporarily increase the size of the wing. If you ever sat near the window of a landing airliner, you may have noted extra wing surfaces sliding out during the final approach, to provide more lift--and if you watched such an airliner from the ground, you would see that these surfaces extend downwards from the wing, at a steeper angle. All that of course makes drag much larger, but a landing airplane must anyway get rid of its extra speed.

    All these allow the airliner to land at about 150 mph. Landing is actually a precision maneuver, in which the airliner (ideally) runs out of airspeed just before its wheels touch the ground. Glide slope radars and other navigation aids make it possible, thousands of times each day.

Non-stop around the world--How high? How fast?

    One of the more memorable feats in aviation was the non-stop non-refuel flight of the Voyager airplane around the world, in December 1986. Designed by Burt Rutan and piloted by his brother Dick and by Jeana Yaeger, the plane now hangs above the lobby of the National Air and Space Museum in Washington.

    Initially the hope was to have a pressurized cabin and to fly at 25,000 ft., but weight limitations precluded that, so that a slower, lower flight was undertaken, lasting 9 days. The take-off weight of "Voyager" was 9700 pounds, and to lift such a heavy airplane, it used two engines--one pushing and one pulling--to reach the required airspeed of 138 mph.

    Halfway through the flight, as fuel was used up, less lift was needed. Therefore one engine was shut off, airspeed was allowed to drop to 79 mph, and to avoid reducing it even more, the flight altitude was raised to 11-12,000 feet. As a result the second half of the flight was much slower than the first, and much harder on the sleep-deprived pilots. They had, however, no other choice--flying faster would have required a less efficient wing angle, and would have wasted too much fuel.

Note:     The story of the non-stop flight around the world is told in "Voyager" by Jeana Yaeger, Dick Rutan and Phil Patton, viii + 337 pp, Alfred A Knopf, New York, 1987. Note: in 2005 Steve Fossett flew solo around the world, nonstop, on the jet-propelled Virgin Atlantic Global Flyer. The flight took less than 3 days because his cabin was pressurized and the airplane could reach around 340 mph (plus tailwind) at altitudes near 40,000 ft.

Questions from Users:
***     The landing speed of airplanes

Next Stop: #23  Accelerated Frames of Reference: Inertial Forces

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Author and Curator:   Dr. David P. Stern
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Updated 9-22-2004  ;  reformatted 24 March 2006  ;  updates 13 October 2016