Joshua, Dan, Veit— thanks. All very helpful. Tiny thetas are now a lot more clear. But I still don't understand what's going on with *big* thetas. What happens when M/m=1, for example, or M/m=10. A line of slope of -1 should cut out about 1/3 of the circle. A line of slope of -10 should cut out about 1/31. But what do these lines *actually* cut? That's what I'm not quite clear about. A slope of -1 appears to cut the circle into four equal slices. But pi doesn't begin with a 4. How about slope -10? Cuts the circle into about 31 slices. But how many exactly? Fewer than 30.9? More than 31.1? I actually did make some progress on this since yesterday. One thing I realized is that the sum of these differences will never be more than the first slice. Which means the digits of pi are jotted out exactly. I also realized what this sum means: that the two balls M and m will always be traveling at different velocities after the final collision. The circle will always be divided into some 10^n pi equal wedges plus one extra (smaller) wedge. In pondering I came up with the following scenario which I think is kind of a fun intuition test. As M/m approaches infinity the size of the extra small wedge approaches zero. Does this mean at the limit the two balls will have the same final velocity? If so, what is this final velocity? (And how is this scenario reversible?) On Thu, Nov 25, 2010 at 11:58 AM, Joshua Zucker <joshua.zucker@gmail.com> wrote:
The angle in radians = arctan of the slope.
Maybe it's easier if you draw the line to the center of the circle and see that the slope is about 1/10 or 1/100 or so on, so the arctan is about 1/10 or 1/100 radians, and of course there are pi radians in a semicircle.
--Joshua
On Wed, Nov 24, 2010 at 9:08 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Thanks, Joshua. But I still don't get it. Why does a line of slope -10 carve out exactly 31 arcs and not, say, 30?
I'll try to think about some more.
On Wed, Nov 24, 2010 at 2:01 AM, Joshua Zucker <joshua.zucker@gmail.com> wrote:
tan theta = theta for small theta.
--Joshua
On Tue, Nov 23, 2010 at 9:55 PM, Gary Antonick <gantonick@post.harvard.edu> wrote:
Just noticed this yesterday and can't figure out why it works.
Draw a circle with radius 1 around 0,0. Draw a line with slope -10 that hits the circle at 1,0. This line will also hit the circle again and make a little arc. The length of the arc will be about 1/31 the perimeter of the circle.
Now try again with a line of slope -100. The line will create an arc 1/314 of the circle.
Starting to look a bit like pi. Am wondering why this is.
In case I made a mistake here's what I'm trying to describe.. if the circle were a clock the first line would cut an arc from 15 minutes (3:00) to almost the 13 minute mark.
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