# Parabolas everywhere!

There a books about the Pythagorean theorem, about the exponential function and even about the gamma constant. I haven't seen any decent book about parabolas yet...

• they form the equations of rocket science (literally) because they describe the movement of rockets
• they have a focal point which leads to their usage as parabola antennas and headlights
• they are at the heart of parabolic partial differential equations (e.g. the heat equation or Black Scholes equation in finance)
• they even have strong connections to stochastic calculus (as an intuition: whereas with standard calculus one has straight lines at the infinitly small, with stochastic calculus one has parabolas)

• ...and you could build a nifty analog multiplier with it (see here) - I did this with my kids, they loved it and were fascinated by it! :-)

Do you know of other interesting connections, books, articles, links...?

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Air resistance seriously causes problems with imagining the motion of a rocket as a parabola. – KConrad Jul 30 '10 at 15:16
Just to follow up on KConrad's remark: a parabola is unstable as a conic section (an arbitrarily small shift in eccentricity gives either an ellipse or a hyperbola), so even in the absence of air resistance, it seems unlikely that one will actually see parabolas in the arcs traced out by falling bodies. (I'm mentioning this because when it was first pointed out to me, it was very shocking to realize that all those cannon balls moving in parabolas were truly an idealization of an idealization!) – Emerton Jul 30 '10 at 15:23
P.S. I had not seen this multiplier before, and really liked it! – Emerton Jul 30 '10 at 15:25
Actually, parabola - just a one example of conic section, and there a lot of books about them, starting with the classical book of Apollonius of Perga. Perhaps it would be moore useful to learn about all the conic sections simultaneously. – Nurdin Takenov Jul 30 '10 at 15:34
In light of Emerton's comment concerning perturbation, "parabolas almost nowhere" might be more accurate. – S. Carnahan Jul 30 '10 at 15:38

It seems that this is already community wiki, so posting something that by no means answers the stated question will work out.  This is just about projectile flight in air, experiencing drag http://en.wikipedia.org/wiki/Drag_(physics) which I remembered to be proportional to the square of the velocity, so I had that part right.  Anyway, in baseball, I have often heard that the optimal angle to hit the ball is 35 degrees above horizontal, not 45 which would apply to a genuine parabola. This may originate with a book by Roger Adair called "The Physics of Baseball." I was able to find some pages including the 35 degree idea, http://uw.physics.wisc.edu/~himpsel/107/Lectures/BaseballRotated.pdf  That being said, some very famous mathematicians worked on classified projects in World War II, some of which were about artillery trajectories. I thought this odd when I first heard of it, but over large distances a trajectory will indeed differ notably from the idealized parabola for the initial speed and direction.  from http://www.science20.com/science_20/science_baseball_what_farthest_home_run_and_did_mickey_mantle_hit_it "Before there were measurements, '500 foot home runs' were commonly stated but that's because people did not understand trajectories; people assume they are even so its distance when it reaches its apex will be mirrored in distance on the far side. But gravity and drag do not work that way. Using some calculations you can also duplicate (2) you can see the trajectories drop a lot even for a ball hit at an optimum 35 degree angle."