Up and down and up and down and...

Not every oscillation in nature is a harmonic oscillation - in this problem, we will examine a non-harmonic oscillation. Suppose we had a rubber ball with a perfect coefficient of restitution so that, when dropped, it would always return to the same height. The period of the bouncing is the time \(T\) between successive bounces and the amplitude \(A\) of this non-linear oscillation is the maximum height of the ball above the floor. For a harmonic oscillator, the period and amplitude are independent, but not so here. Instead, there's a relation between them: \(T\) proportional to \(A^w\). Find \(w\).

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