Editing Parabola (section)

== In the physical world ==
In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of [[physics]] is the [[trajectory]] of a particle or body in motion under the influence of a uniform [[gravitational field]] without [[air resistance]] (for instance, a ball flying through the air, neglecting air [[friction]]).

The [[Projectile motion|parabolic trajectory of projectiles]] was discovered experimentally in the early 17th century by [[Galileo]], who performed experiments with balls rolling on inclined planes. He also later proved this [[mathematical]]ly in his book ''Dialogue Concerning Two New Sciences''.<ref>Dialogue Concerning Two New Sciences (1638) (The Motion of Projectiles: Theorem 1).</ref>{{efn|However, this parabolic shape, as Newton recognized, is only an approximation of the actual elliptical shape of the trajectory and is obtained by assuming that the gravitational force is constant (not pointing toward the center of the Earth) in the area of interest. Often, this difference is negligible and leads to a simpler formula for tracking motion.}} For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the [[center of mass]] of the object nevertheless moves along a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.

Another [[hypothetical]] situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by [[Sir Isaac Newton]], is in [[two-body orbit]]s, for example, the path of a small planetoid or other object under the influence of the gravitation of the [[Sun]]. [[Parabolic orbit]]s do not occur in nature; simple orbits most commonly resemble [[hyperbola]]s or [[ellipse]]s. The parabolic orbit is the [[degeneracy (math)|degenerate]] intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exact [[escape velocity]] of the object it orbits; objects in [[elliptical orbit|elliptical]] or [[hyperbolic orbit|hyperbolic]] orbits travel at less or greater than escape velocity, respectively. Long-period [[comet]]s travel close to the Sun's escape velocity while they are moving through the inner Solar system, so their paths are nearly parabolic.

Approximations of parabolas are also found in the shape of the main cables on a simple [[suspension bridge]]. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a [[catenary]], but in practice the curve is generally nearer to a parabola due to the weight of the load (i.e. the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used.<ref name="Troyano">{{cite book
|title=Bridge engineering: a global perspective
|first1=Leonardo Fernández
|last1=Troyano
|publisher=Thomas Telford
|year=2003
|isbn=0-7277-3215-3
|page=536
|url=https://books.google.com/books?id=0u5G8E3uPUAC&pg=PA536}}
</ref><ref>{{cite book
|title=A memoir of suspension bridges
|first1=Charles Stewart
|last1=Drewry
|publisher=Oxford University
|year=1832
|page=[https://archive.org/details/amemoironsuspen00drewgoog/page/n183 159]
|url=https://archive.org/details/amemoironsuspen00drewgoog}}
</ref> Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola (see {{slink|Catenary#Suspension bridge curve}}). Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola. Suspension-bridge cables are, ideally, purely in tension, without having to carry other forces, for example, bending. Similarly, the structures of parabolic arches are purely in compression.

Paraboloids arise in several physical situations as well. The best-known instance is the [[parabolic reflector]], which is a mirror or similar reflective device that concentrates light or other forms of [[electromagnetic radiation]] to a common [[Focus (optics)|focal point]], or conversely, collimates light from a point source at the focus into a parallel beam. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer [[Archimedes]], who, according to a dubious legend,<ref>{{cite journal
|last      = Middleton
|first     = W. E. Knowles
|date      = December 1961
|title     = Archimedes, Kircher, Buffon, and the Burning-Mirrors
|journal   = Isis
|volume    = 52
|issue     = 4
|publisher = Published by: The University of Chicago Press on behalf of The History of Science Society
|pages     = 533–543
|doi = 10.1086/349498
|jstor = 228646|s2cid = 145385010
}}</ref> constructed parabolic mirrors to defend [[Syracuse, Italy|Syracuse]] against the [[Roman Empire|Roman]] fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to [[telescope]]s in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in [[microwave]] and satellite-dish receiving and transmitting antennas.

In [[parabolic microphone]]s, a parabolic reflector is used to focus sound onto a microphone, giving it highly directional performance.

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the [[centrifugal force]] causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the [[liquid-mirror telescope]].

[[Aircraft]] used to create a [[Weightlessness|weightless state]] for purposes of experimentation, such as [[NASA]]'s "[[Vomit Comet]]", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in [[free fall]], which produces the same effect as zero gravity for most purposes.

=== Gallery ===

<gallery mode="packed" heights="200px" style="text-align:left">
File:Bouncing ball strobe edit.jpg|A [[bouncing ball]] captured with a stroboscopic flash at 25 images per second. The ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and [[air resistance]], causes the curve swept out to deviate slightly from the expected perfect parabola.
File:ParabolicWaterTrajectory.jpg|Parabolic trajectories of water in a fountain.
File:Comet Kohoutek orbit p391.svg|The path (in red) of [[Comet Kohoutek]] as it passed through the inner Solar system, showing its nearly parabolic shape. The blue orbit is the Earth's.
File:Laxmanjhula.jpg|The supporting cables of [[suspension bridge]]s follow a curve that is intermediate between a parabola and a [[catenary]].
File:Rainbow Bridge(2).jpg|The [[Rainbow Bridge (Niagara Falls)|Rainbow Bridge]] across the [[Niagara River]], connecting [[Canada]] (left) to the [[United States]] (right). The parabolic arch is in compression and carries the weight of the road.
File:Celler de Sant Cugat lateral.JPG|Parabolic arches used in architecture
File:Parabola shape in rotating layers of fluid.jpg|Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See [[Rotating furnace]])
File:ALSOL.jpg|[[Solar cooker]] with [[parabolic reflector]]
File:Antenna 03.JPG|[[Parabolic antenna]]
File:ParabolicMicrophone.jpg|[[Parabolic microphone]] with optically transparent plastic reflector used at an American college football game.
File:Solar Array.jpg|Array of [[parabolic trough]]s to collect [[solar energy]]
File:Ed d21m.jpg|[[Thomas Edison|Edison]]'s searchlight, mounted on a cart. The light had a parabolic reflector.
File:Physicist Stephen Hawking in Zero Gravity NASA.jpg|Physicist [[Stephen Hawking]] in an aircraft flying a parabolic trajectory to simulate zero gravity
</gallery>

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