## 4 Results  
Figure 4a,b aptly illustrate the recorded sound data, incorporating both sound level and sound intensity. Within these recordings, Peaks 1 (t\({}_{1}\)) and 2 (t\({}_{2}\)) correspond to discernible acoustic events, signifying the actuation of the switch and the subsequent impact of the steel ball with the ground, respectively. Subsequent peaks capture the acoustic signature of the restitution bounce exhibited by the steel ball, characterized by a gradual diminishment in amplitude. The time data for a steel ball falling from varying heights, acquired through both theoretical calculations and experimental measurements, are presented in Table 1. The time period obtained at each altitude consists of three values. All three values are used to calculate the average and standard deviation (S.D.) in order to analyze the distribution of the data around the average. The experiment's results revealed that the time intervals obtained from measuring the sound level and intensity between the pressing of the mechanical switch and the steel ball hitting the ground were consistent. Hence, we have the flexibility to analyze the results using data from either source.  
Figure 3: Recording sound level and intensity using the smartphone with an iPhone 8 Plus model.  
The theoretical time calculations presented in Table 1 are derived by substituting the altitude and gravitational acceleration values into Equation (6), using \(g\) = 9.81 m/s\({}^{2}\). In Figure 5a, researchers present the correlation between T\({}_{\text{theory}}\) and T\({}_{\text{experiment}}\). The empirical findings indicate that the time measurements obtained through the proposed research method exhibit a relative error of 3.27% when compared to theoretical values. In Figure 5b,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**H (m)**} & \multicolumn{2}{c}{**Theory**} & \multicolumn{2}{c}{**Experiments**} \\ \cline{2-5}  & **T (s)** & **T (s)** & **T\({}^{2}\) (s\({}^{2}\))** & **S.D.** \\ \hline
0.60 & 0.350 & 0.330 & 0.109 & 0.011 \\ \hline
0.80 & 0.404 & 0.392 & 0.153 & 0.005 \\ \hline
1.00 & 0.452 & 0.448 & 0.201 & 0.012 \\ \hline
1.20 & 0.495 & 0.482 & 0.232 & 0.002 \\ \hline
1.40 & 0.535 & 0.522 & 0.272 & 0.002 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Time measurements for the descent of a steel ball from different heights: theoretical calculations versus experimental observations.  
Figure 4: Sound signals corresponding to the pressing of a switch and the impact of a steel ball with a table at different time intervals, showcasing **(a)** sound level and **(b)** sound intensity.  
researchers illustrate the correlation between T\({}^{2}\) and H, revealing a slope of 4.9137 m/s\({}^{2}\) and an R\({}^{2}\) coefficient of 0.9952.  
This derived slope is then employed to calculate the \(g\) using Equation (7), yielding a value of 9.8274 m/s\({}^{2}\). Remarkably, this calculated \(g\) exhibits a mere 0.17737% relative error when juxtaposed with the theoretical value. Furthermore, the propagated uncertainty, denoted as \(\pm\)0.01, underscores the precision of using smartphones for the meticulous measurement of time intervals pertinent to gravitational experiments.