Multiple collisions

A block of mass \(m\) sits on a horizontal, frictionless surface some distance \(\ell\) from a wall. Another block of mass \(M\) collides from the right with velocity \(V.\) Depending on that ratio \(x =\frac Mm,\) the blocks will collide a total of \(n\) times.

Find \[\displaystyle \lim_{x\rightarrow \infty} \frac{n}{\sqrt{x}}.\]

Assume that all collisions are elastic and that the motion is in 1-dimension.

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