Veit- I'm still not quite able to reconcile your posts (if I understand them correctly) with what I'm now observing. Velocities are very different depending on whether the balls are touching. I hadn't noticed this before. When the balls are initially separated the final velocities seem to be essentially 1/81, 4/81, 4/27, 4/9 and 4/3. When balls are initially touching the final velocities are clearly *not* 1/81, 4/81, 4/27, 4/9 and 4/3. The velocity of the final ball is somewhat greater than in the previous scenario. The velocities of the first four balls are very close to zero. Here's one approach that seems to predict what I'm observing: "Assumptions: At the moment of collision all the balls are scrunched up and become one entity as far as momentum and kinetic energy are concerned, because of serial compression from left to right. Then when the compression relaxes (at the speed of sound) the last ball is ejected, converting most of the stored potential energy of the compression to kinetic energy. The residue of the energy and momentum is shared between the four remaining balls. The equations for the velocities are as follows: Let the final velocity of balls 1-4 be x and of ball 5 be y. The combined mass of balls 1-4 is 5, and that of ball 5 is 1. Original momentum was 2 (2*1). ∴ Final momentum = 2 ∴ Equation A: 5*x + y =2 Original kinetic energy was 1 (0.5*2*1*1). ∴ Final kinetic energy = 1 ∴Equation B: 0.5*5*x^2 + 0.5*1*y^2 =1 Solving these equations gives one non-physical solution (where ball 5 has a negative velocity) and the other solution is as shown earlier: x = velocity of Balls 1-4 = 0.12251482 y = velocity of Ball 5 = 1.38742589 I think the key reason that the two-ball interactions don't work here is that the heavier ball cannot be completely stopped by any one ball, as opposed to the equal weight Newton's cradle. Hence it continues forward during the collision, inexorably compressing the other balls from left to right." On Mon, Dec 13, 2010 at 5:46 PM, Veit Elser <ve10@cornell.edu> wrote:
On Dec 13, 2010, at 7:33 PM, Gary Antonick wrote:
am not sure I'm following completely. I've tried this experiment now with several different model cradles (all with 1st ball mass 2m and remaining balls mass m) and get the same result each time: the last ball flies out and the remaining balls barely move. The balls clearly do *not* swing out in a fan pattern predicted by pairwise collisions.
then the balls come together and interact again. after this *second* set of collisions the the pairwise model seems accurate (because the balls started off slightly apart)— the balls fan out in the predicted pattern.
Experiments should be taken seriously. By "fan pattern predicted by pairwise collisions" I assume you mean this pattern of final velocities:
2 balls 1/3 4/3
3 balls 1/9 4/9 4/3
4 balls 1/27 4/27 4/9 4/3
etc.
Another thing to bear in mind. In the standard cradle the number of collisions for 2,3,4,... balls is 1,2,3,... But for your modification the number of collisions (pairwise model) goes like 1,3,6,10,... so a non-ideal coefficient of restitution degrades the kinetic energy more.
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