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Math PPT. PARABOLAS

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Chesler Thomas

on 29 May 2013

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Transcript of Math PPT. PARABOLAS

Applications of Parabolas in Real Life Parabolic Reflectors If a reflector is axially symmetrical and shaped so its cross-section is a parabola, it has the property of bringing light that has come from a very distant source such as the sun, with rays of light are effectively parallel, to a point focus. If the axis of symmetry is aimed at the sun, any object that is located at the focus receives highly concentrated sunlight, and therefore becomes very hot. This is the basis for the use of this kind of reflector for solar cooking. Main Concept The parabolic reflector functions due to the geometric properties of the paraboloidal shape: any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or "focus". Solar Cooker Flashlights A flashlight projects light in one general direction because of the reflecting surface behind the lightbulb. This redirection of light is called collimation. It turns out that the paraboloid is the only surface that collimates perfectly. Parabolic Microphone A parabolic microphone is a microphone that uses a parabolic reflector to collect and focus sound waves onto a receiver.Typical uses of this microphone, which has unusually focused front sensitivity and can pick up sounds from many metres away, include nature recording, field audio for sports broadcasting, eavesdropping (for example, espionage), and law enforcement.
Parabolas only focus waves with a wavelength much smaller than the diameter of the parabola. Since sound waves travel at 342 m/s through the air (speed of sound), obtaining hi-fidelity sound (down to 20 Hz, the lower limit of human hearing) would require a parabola with a diameter greater than 17 metres (= 342 m/s / 20 Hz). Most parabolic microphones sacrifice low-end fidelity to get a more manageable size. Suspension Bridges Up close, the suspension bridge is an amazing and beautiful structure that can span rivers and connect cities hundreds of miles apart. From a distance they look fragile, hanging from almost transparent threads. Despite their seeming fragility, suspension bridges are very, very strong thanks to their design and the materials used to build them. These awe-inspiring bridges alone balance the forces of tension and compression, managing to stay up through hurricanes, storms, and earth-quakes. Suspension bridges are capable of spanning long distances and actually are the only type of bridge to span the longest distances possible for a bridge. This is because the shape of the suspension bridge is actually one of the most stable structures there is. The cable of the bridge is inherently stable against any disturbance if it is thick enough to withstand any tension. But what sets the suspension bridge apart from most conventional bridges is that all the forces do not "internally-cancel." Instead, the forces are directed in a way that the tensions are resisted by the ground, which is in compression. The Most Famous Examples are the BROOKLYN BRIDGE and GOLDEN GATE BRIDGE. Galileo Galilei found that all objects thrown form a parabolic path, no matter what. He deduced this by the simple observation of watching objects being thrown. Galileo is responsible for the modern concepts of velocity and acceleration to explain projectile motion that is studied today:
A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola. Path of Projectile Motion The ideal case of motion of a projectile in a uniform gravitational field, in the absence of other forces (such as air drag), was first investigated by Galileo Galilei. To neglect the action of the atmosphere, in shaping a trajectory, would have been considered a futile hypothesis by practical minded investigators, all through the Middle Ages in Europe. Nevertheless, by anticipating the existence of the vacuum, later to be demonstrated on Earth by his collaborator Evangelista Torricelli[citation needed], Galileo was able to initiate the future science of mechanics.[citation needed] And in a near vacuum, as it turns out for instance on the Moon, his simplified parabolic trajectory proves essentially correct.
In the analysis that follows we derive the equation of motion of a projectile as measured from an inertial frame, at rest with respect to the ground, to which frame is associated a right-hand co-ordinate system - the origin of which coincides with the point of launch of the projectile. The x-axis is parallel to the ground and the y axis perpendicular to it ( parallel to the gravitational field lines ). Note: The maximum range, for a given initial speed , is obtained when the initial angle is 45 degrees. Examples of Projectiles To Be Continued..............
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