first of all, we have a serious problem in that one ball appears to be kicking out only one ball. so pretty much everyone might be wrong.. or at least a new group is a bit more right. and now completely perplexed. am epoxying together a larger cradle (1" and 1.25" balls) this moment. second... in defense of Fred (from very much the layperson view on this) it does seem that given three colliding bodies all you know is that the ending velocities (along with the beginning velocities) will lie on a circle at the intersection of a sphere (kinetic energy) and a plane (momentum). I don't see how this endpoint could ever be determinate without another constraint. On Sat, Dec 11, 2010 at 10:45 AM, Robert Munafo <mrob27@gmail.com> wrote:
How ill-defined?
For example, take the case where all three have equal mass, the central body starts out motionless and the outer two are approaching at the same speed (from opposite directions). The midpoint between the positions of the outer two is X, and isn't changing until the first impact because of the initial conditions I just stated.
If the central body starts out at position "X+epsilon", I get two non-simultaneous collisions after which the central body ends up at position "X-epsilon". In the limit as epsilon goes to zero, I get two simultaneous collisions, the central body remains motionless and the outer two simultaneously bounce off it.
If I decompose in the other order (starting with a negative "epsilon" so the other impact happens first) the limit still gives me the same answer.
Why is the limit ill-defined? Because the velocity of either particle is not defined (and discontinuous) at the moment of impact? Wouldn't that make all individual collision events ill-defined?
(Of course this is all assuming the bodies are ideal perfectly rigid, i.e. the speed of sound in their material is infinite and similar idealizations)
- Robert
On Sat, Dec 11, 2010 at 07:55, Fred lunnon <fred.lunnon@gmail.com> wrote:
[...] I've never been convinced that these problems are even well-defined, for the same reason that a simultaneous collision between three bodies in the plane is ill-defined. In that case, it's immediately obvious that you get completely different results according to how you decompose into a sequence of 2-body collisions.
The resemblance to the preceding discussion is striking! WFL
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com
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