yes - the n lines intersect with the 2 main lines at equal intervals. and I made a mistake. The structure should then be stretched until the angle between the 0th line (not 1st line) and the n/2+1th line is the same as the angle at which the ball was tossed. -------- Original message -------- Subject: Re: [math-fun] parabola puzzle From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> CC: It's looking a lot better now, anyway. Presumably the "n lines" are spaced at equal intervals along the "base" lines? WFL On 11/28/11, Gary Antonick <gantonick@post.harvard.edu> wrote:
Hi Fred,
Thank you for checking this out. The problem I'm having is being precise with rotating and scaling main parabola. I'm looking for an underlying structure that indicates all bounce patterns at once.
Here's a concrete example I just ran. The bounce pattern and necessary reflector angle for 20 bounces. All steps can be done quickly and with a lot of precision except for
- creating the 45 degree angle between the 1st and n/2+1 (in this case the 11th) added tangent line and - rotating the structure until the n/2+1 tangent line is vertical.
http://www.flickr.com/photos/70917169@N02/6417153069/in/photostream
- Gary
On Sun, Nov 27, 2011 at 10:38 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
I followed your pictures well enough to understand the problem now, but still can't grasp what the (solution? or) property that you have in mind might be.
WFL
On 11/28/11, Gary Antonick <gantonick@post.harvard.edu> wrote:
Phooey. I don't think my description was very enticing.
Here's a more specific question. In the linked image below, what is the relationship between lengths a, b, c and d?
http://www.flickr.com/photos/70917169@N02/6416169329/in/photostream
I've googled around a bit.. haven't found anything on using a parabola on its side to trace out the pattern of multiple bounces. Any suggestions would be very much appreciated.
Best,
Gary
On Sun, Nov 27, 2011 at 3:06 PM, Gary Antonick <gantonick@post.harvard.edu>wrote:
Hi all,
I happened to notice something interesting about parabolas several months ago. Am wondering if it would make a good puzzle.
It's about a ball bouncing up an incline.
Say you point a cannon at 45° and fire. A cannonball flies out and hits a perfect reflector at the same height as the mouth of the cannon. If the cannonball bounces once on the reflector and then returns along its original path to the mouth of the cannon you'd know the reflector was at 45 °.
But what if the ball bounces 5 times? What's the angle of the reflector? And - a related question - what trajectory does the bouncing ball take?
All frictionless, of course, and w/o air resistance. And no calculus (or formulas of any kind) required.
- Gary
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