Ball Falling On Inclined Ramp

From a resting state, a small ball falls vertically on an inclined ramp of angle \(\theta\), in respect to the horizontal, resulting in several elastic collisions.

If \(d\) is the initial distance from the ball to the ramp, find the distance between the \(n^{\text{th}}\) point of collision and the first point of collision in a function of \(d\), \(n\) and \(\theta.\)

This question appeared on ITA's 2014 Physics Paper, and is inspired by a question from Saraeva's book "Problemas Selecionados de Fisica Elementar".

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