# 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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