I like David's problem a lot. A simpler scenario that seems interesting (and is probably simpler to analyze) involves a gun that shoots a bullet to the right at a random speed that is either 1 or 2 in an iid (independent, identically distributed) fashion, with respective probabilities p and q, with p+q=1. (If you like, we can focus on the case p=q=1/2.) Here the speed of the furthest bullet has trivial asymptotics but other features of the evolution of this system may not be so obvious. There will be a whole bunch of bullets moving at speed 2 that are to the right of all the bullets moving at speed 1 and hence can never be destroyed (call them the immortal bullets, at least until someone comes up with a better name); how does this bunch of bullets grow over time, and how far apart are they typically spaced? It's not obvious to me that the spacing between the successive immortal bullets should be iid. I'd be glad if someone wrote up some code and generated some pictures for us to see, for both my dynamics and David's. I'd also like to see what happens with a non-random gun that simply alternates between emitting speed-1 bullets and speed-2 bullets; probably it gives something with a simple fractal structure, but I'm too tired right now to do by-hand simulations or to code. Jim Propp On Tue, Jun 5, 2012 at 3:16 PM, David Wilson <davidwwilson@comcast.net>wrote:
A gun sits on a line.
Every second, the gun shoots a bullet to the right at a random constant speed between 0 and 1.
If two bullets collide they annihilate.
It's probability 0, but if more than two bullets collide, the slowest bullets annihilate in pairs.
Can we expect the speed of the furthest bullet to approach 1 over time?
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