\(N\) spheres are placed in a semicircle on a frictionless surface. The mass of all these spheres is \(m_1\).

A sphere of mass \(m_2\) is then projected so that it collides with all \(N\) balls and finally leaves the semicircle, moving towards the right.

If \(N\to \infty\), the ratio between the final and initial velocities of \(m_2\) is \[-\frac{a}{b}\]

Find \(a+b\).

You might wish to make small approximations.

This is part of Ordered Disorder.

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