Interesting Quadratic Models

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Vorpals
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Interesting Quadratic Models

Postby Vorpals » Fri Sep 30, 2011 1:49 am UTC

For a presentation to my class in pre-calculus, I have to find a quadratic model and detail its usage in real life.

Bonus points are given for an interesting and unique one. Do any of you guys know of a cool quadratic model with interesting uses in the real world?

Note: it cannot be a general equation; it has to have constants and be defined in terms of x and not much else.

Thanks in advance!

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jmorgan3
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Re: Interesting Quadratic Models

Postby jmorgan3 » Fri Sep 30, 2011 2:33 am UTC

Aerodynamic drag is quadratic in velocity.
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Re: Interesting Quadratic Models

Postby skeptical scientist » Fri Sep 30, 2011 2:35 am UTC

When you say a quadratic model, you mean a physical situation where you have two quantities x and y which both vary, but maintain the relationship y=ax2+bx+c, for some constants a,b,c?
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Vorpals
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Re: Interesting Quadratic Models

Postby Vorpals » Fri Sep 30, 2011 2:38 am UTC

skeptical scientist wrote:When you say a quadratic model, you mean a physical situation where you have two quantities x and y which both vary, but maintain the relationship y=ax2+bx+c, for some constants a,b,c?


Yeah exactly! It can't be a general equation where a, b, and c also vary.

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Talith
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Re: Interesting Quadratic Models

Postby Talith » Fri Sep 30, 2011 11:47 am UTC

Parabolic mirrors and reflectors (any satelitte dish is a parabolic reflector) have that the height of the surface above the centre (when layed flat) is a quadratic function of the horizontal distance from the centre of the surface.

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Yakk
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Re: Interesting Quadratic Models

Postby Yakk » Fri Sep 30, 2011 2:25 pm UTC

This is a problem which much be approached with the greatest gravity and care. Who knows what direction (other than down) such a search could end up going? We will need to bend space and time to find the answer -- or at least the low density/velocity approximation thereof. When finished, what fruit will fall from the tree of knowledge, possibly hitting someone on the head, at least mythologically? But hark -- what is this? Someone born on the 25th of December may have the answer, as well as answers to many questions that have plagued humanity from ancient days.

My friends, we need not accelerate into the solution too rashly. There is a level of proper distance we must maintain -- and, if I may be so brash, we should moderate our velocity before it is too late. Why, I've half a mind to not even let some random square know the answer! If only we had a sign, cos that would let us know where to place our target.

Spoiler:
Hopefully that wasn't too derivative.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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z4lis
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Re: Interesting Quadratic Models

Postby z4lis » Fri Sep 30, 2011 3:24 pm UTC

Morse theory is totally a real-life application of quadratic models. :twisted: (I was going to suggest it as a possibility, and then I reread your original post!)


Anyway, lots of physical systems have quadratic terms. For instance, the kinetic energy of classical mechanics has a quadratic term. It also represents the potential (stored) and kinetic energy of the frictionless harmonic oscillator, which is a really important simple system for physics. Drag was mentioned above. The trajectory of falling bodies in a gravitational (or any inverse square field) trace out parabolas. In probability and statistics, we use quadratic functions for standard deviation because it's convenient algebraically. The standard deviation of a set of data basically tells you how "spread out" it is. This link has some other examples, along with some nifty pictures.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

gorcee
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Re: Interesting Quadratic Models

Postby gorcee » Fri Sep 30, 2011 4:40 pm UTC

Ricatti control theory is quadratic.


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