8 junio, 2024

9 applications of the parable in everyday life

The applications of the parable in everyday life they are multiple. From satellite dishes and radio telescopes to concentrate signals, to car headlights sending parallel light beams.

A parabola, in simple terms, can be defined as a curve in which the points are equidistant from a fixed point and a line. The fixed point is called the focus and the line is called the directrix.

The parabola is a conic that is traced in different phenomena, such as the movement of a ball driven by a basketball player or the fall of water from a fountain.

The parabola is especially important in various areas of physics, in resistance of materials or in mechanics. At the base of mechanics and physics the properties of the parabola are used.

Applications of the parable in everyday life

satellite dishes

The parabola can be defined as a curve that arises when making a cut to a cone. If this definition were applied to a three-dimensional object, we would obtain a surface called a paraboloid.

This figure is very useful because of a property that parabolas have, where a point inside the parabola is moving in a line parallel to the axis, it will «bounce» off the parabola and be sent towards the focus.

A paraboloid with a signal receiver at the focus can cause all signals that bounce off the paraboloid to be sent to the receiver, without pointing directly at it. A great signal reception is obtained using the entire paraboloid.

This type of antennas are characterized by having a parabolic reflector. Its surface is a paraboloid of revolution.

Its shape is due to a property of mathematical parabolas. They can be transmitting, receiving or full duplex. They are called that way when they are capable of transmitting and receiving at the same time. Usually, they are used at high frequencies.

satellites

A satellite sends information to Earth. These rays are perpendicular to the directrix for the distance at which the satellite is.

When reflected from the antenna dish, which is generally white, the rays converge at the focus, where a receiver is located that decodes the information.

the jets of water

The jets of water that come out of a fountain have a parabolic shape.

When numerous jets leave a point with the same speed, but with different inclination, another parabola called «safety parabola» is above the others and it is not possible for any other of the remaining parabolas to pass above it.

solar cookers

The property that characterizes parabolas allows them to be used to create devices such as solar cookers.

With a paraboloid that reflects the sun’s rays, what is going to be cooked would easily be placed in its focus, causing it to heat up quickly.

Other uses are the accumulation of solar energy using an accumulator on the focus.

Vehicle headlights and parabolic microphones

The previously explained property of parabolas can be used inversely. By placing a signal emitter located towards its surface at the focus of a paraboloid, all signals will bounce off it. In this way, its axis will be reflected outward in parallel, obtaining a higher level of signal emission.

In vehicle headlights this occurs when a bulb is placed in the bulb to emit more light.

In parabolic microphones, it occurs when a microphone is placed at the focus of a paraboloid to emit more sound.

Hanging bridges

Suspension bridge cables adopt the parabolic shape. These form the envelope of a parabola.

In the analysis of the equilibrium curve of the cables, it is assumed that there are many stays and it can be considered that the load is distributed uniformly horizontally.

With this description, it is shown that the equilibrium curve of each cable is a simple equation parabola and its use is frequent in the art.

As examples of real life are the San Francisco bridge (United States) or the Barqueta bridge (Seville), which use parabolic structures to give the bridge greater stability.

trajectory of celestial objects

There are periodic comets that have elongated trajectories or ellipses. When the return that comets make around the solar system is not proven, it seems that they describe a parabola.

Sports

In every sport in which a pitch is made, we find parables. These can be described by balls or thrown artifacts as in soccer, basketball, or javelin throwing.

This launch is known as a «parabolic launch», and consists of throwing an object upwards (not vertically). The path that the object makes when going up (with the force applied to it) and going down (by gravity) forms a parabola.

A more concrete example are the plays made by Michael Jordan, an NBA basketball player.

This player has become famous, among other things, for his «flights» to the basket, where at first glance he seemed to be suspended in the air much longer than other players.

Michael’s secret was that he knew how to use proper body movements and a great initial speed that allowed him to form an elongated parabola, making his trajectory close to the height of the vertex.

Lightning

When a cone-shaped light beam is projected onto a wall, parabolic shapes are obtained, as long as the wall is parallel to the generatrix of the cone.

References

Arnheim, C. (2015). Mathematical Surfaces. Germany: BoD
Boyer, C. (2012). History of Analytic Geometry. USA: Courier Corporation.
Frante, Ronald L. (1980). A Parabolic Antenna with Very Low Sidelobes. IEEE Transactions on Antennas and Propagation.
Kletenik, D. (2002). Problems in Analytic Geometry. Hawaii: The Minerva Group.
Kraus, J.D. (1988). Antennas, USA: McGraw‐Hill.

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