Planes Flight how Planes Fly Physics Bernoulli Algebra Formula

How an Airplane Flies: A Question of Physics

Many people look to the skies and see a streak of white smoke through the clouds, smiling as they realize that they are seeing the trail of a jet or some other type of airplane. With a sparkle in their eyes, they wonder how such a huge piece of metal is able to traverse the atmosphere with such ease.  There is a wide variety of mathematical reasons to explain why airplanes can fly above our heads, even though they are such massive pieces of architecture.  Planes of all sorts can lift into the sky due to their velocity, foil-wing lift, the Coanda Effect, and even the viscosity of atmospheric liquids.

Magic through Mathematics

Although Swiss Mathematician Daniel Bernoulli began the explanation of flight by examining the flight patterns of insects and how their wings are able to keep them aloft, in modern air-travel studies it is not enough to think in those terms. As Robert Kunzig explains in his article The Physics of Airplanes’, Bernoulli’s explanation that “Air flows faster over the top surface of the wing than underneath the bottom” (Kunzig, 2001) is incorrect. He continues to explain that the air that travels over the top of the wing, in contrast to the earlier beliefs, actually reaches the trailing edge faster than the under-side air currents. The air over the top follows the curve of the wing, creating what is known as the Coanda Effect. Kunzig further explains that the Coanda Effect “is just the tendency of air or any even slightly viscous fluid to stick to a surface it is flowing over, and thus to follow the surface as it bends.” (Kunzig, 2001) He continues to explain that, between the low-pressure zone that is created when the bent air pulls and distends the air above it and the angle of attack (the angle that is created by the curved surface of the wing and its tilt) as well as the velocity of the aircraft in question, an airplane can lift off the ground.

Formula for Successful Flight

Lift, equivalent to the aircraft’s weight in pounds, equals one half of the current altitudinal air density, multiplied by the square of the velocity of the aircraft in feet per second, times the square footage of the airplane’s wing area, multiplied by the lift coefficient, which can be determined through the angle of attack and the type of airfoil being used. Thus you have the equation: L = (1/2) * d * (v^2) * s * CL {where L is lift, d is air density, v is velocity, s is wing area, and CL is lift coefficient}. (Hodanbosi, 1996) With this algebraic formula, one can determine several factors about the aircraft in question, depending on the information that is acquirable or on-hand. Given the fact that airplanes are smothered with gauges that analyze and display this information for the pilots to utilize, many of these factors can be determined by looking at these gauges or paying attention to weather patterns. If you were building an airplane, therefore, you could begin with the weight of the aircraft in question in order to determine the chances that the aircraft will achieve flight.


Through Bernoulli’s explanation and the thoughts of modern physicists, people can have a better understanding of flight than ever before. As complex as these concepts might sound at first, you can easily determine the functional parameters of flight by analyzing chunks of data over time spans. This data can easily be turned over, twisted, and looked at in such a way that even novice mathematicians and physics enthusiasts will be able to figure out just how air flight happens with such large objects.  Flight can be an awe-inspiring event to watch, but is much more interesting when you understand the nuances that provide the means of this flight.


Hodanbosi, C. (1996, August). Lift Formula. Retrieved September 4, 2008, from NASA – NASA Glenn Home:

Kunzig, R. (2001, April 1). The Physics of … Airplanes. Retrieved September 4, 2008, from DISCOVER: Science, Technology, and the Future: