Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / Section B / 6 / g / Solution

For actual complete bounce distances $D_1=4\mathrm{m}$ and $D_{n+1}=r{D}_n$, the geometric series givesA natural intended estimate compares the observed first round trip with the loss-free round trip from the initially compressed position,If the reduction factor is taken to be $r=D_1/D_{\mathrm{ideal}}=0.77857$, the answer is $D_{\mathrm{total}}\simeq 18.1\mathrm{m}$. The first term is the actual 4 m, not the hypothetical loss-free distance.

This numerical inference needs the stated comparison assumption. Strictly, knowing the first actual bounce distance and saying subsequent distances share a ratio does not independently specify that ratio. A change in spring compression also means that an energy-loss fraction is not automatically the same as a vertical-distance ratio.