[3rd repost] What is maximum distance between the balls?

Question

Infinite number of frictionless snooker balls of radius R
are initially at rest on level surface along straight line
at equal intervals.

After the first ball is cued, it hits the second ball,
the second ball hits the third, and so on...

cue=====< O O O O O O O O O O .....

What is maximum distance D between the centers of
adjacent balls, such that the sequence of collisions
will continue for infinitely long time?

If the distance is too large, small initial error in direction will grow exponetially, and the sequence of collisions will be interrupted.

As eyesonthescreen correctly points out, small misalignments may also be either grow or vanish exponentially.

As far as I understand, critical distance for both misdirection and misalignment is the same.

Answer 1

If P is the initial position (i.e. center) of the incident ball, P' is its position when it's touching the target ball and Q is the position of the target ball, then P,P',Q form a triangle with PQ = D , P'Q = 2R and a =
sin(a)/2R = sin(pi-a-b)/D = sin(a+b)/D

For a,b small, that's:

(D/2R - 1)a = b

So the angle does grow or shrink exponentially depending on whether D is greater or less than 4R.

**************

The energy lost due to momentum in the perpendicular direction (a mv) is about:

a^2 (1/2)mv^2

The total energy lost in an infinite number of collisions would be about (sum of geometric series) : (1/2)mv^2 a^2/(1-(D/2R-1)^2). If D is below 4R there's always a sufficiently small value of the initial angle a.

Answer 2

The distance between centres would be 2*R so that all the balls are touching. Otherwise even the smallest initial error will stop the sequence of collisions eventually.

Answer 3

I agree with the first answer, but I'd also add that, even when touching (d = 2R), balls could be misaligned so the momenta would not be transferred direct on and mV > mv. In which case, eventually the energy would peater out and a ball would not budge because the change in momentum (force) on that unfortunate ball would be too low to accelerate it.