Short Ride in a Fast Machine

A particle of \(5 \ \text{kg}\) is launched from the ground at an angle \(1\) radian above the horizontal with a speed \(10 \ \text{ms}^{-1}\). When it lands, it bounces off, losing \(\frac{1}{3}\) of the total energy as sound and heat. It bounces and lands many times before eventually coming to a halt.

What is the ratio of the total range of the particle to the displacement between launch and the first bounce?

Take gravitational acceleration to be \(g = 9.81 \ \text{ms}^{-1}\).

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