# Balls in a semicircle!

$$N$$ identical balls lie equally spaced in a semicircle on a frictionless horizontal table, as shown. The total mass of these balls is $$M$$. Another ball of mass $$m$$ approaches the semicircle from the left, with the proper initial conditions so that it bounces (elastically) off all $$N$$ balls and finally leaves the semicircle, heading directly to the left.

In the limit $$N → \infty$$ (so the mass of each ball in the semicircle, $$\frac{M}{N}$$, goes to zero), find the minimum value of $$\frac{M}{m}$$ that allows the incoming ball to come out heading directly to the left.

Hint - Use the result obtained in Maximum deflection angle!

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