# Low Speed Impact of an Elastic Ball with Tapes and Clay Court
## 4 Results and Discussion
The mean absolute error \(e\) between the theory and the experiments is analyzed. The corresponding \(k\) and \(b\) values  
Figure 9c depicts the normal and tangential components of the displacement for both the theory and one of the experimental results with \(\beta=22.5^{\circ}\). The analysis of the experiment shows that the initial velocity is \(\textbf{V}_{\text{C}}(t_{0})=-0.774\)\(t-1.849\)\(j\) m/s and the final velocity is \(\textbf{V}_{\text{C}}(t_{f})=-0.228\)\(t-1.521\)\(j\) m/s. The simulation outcome is \(\textbf{V}_{\text{C}}(t_{f})=-0.232\)\(t-1.443\)\(j\) m/s. The errors between tangential and normal velocity components are 1.5% and 5.1%, respectively. The spring-damper model is a good fit with the experiments investigated above. In all cases, the model produced less than 10% error. Effect of SurfaceThe coefficient of restitution was statistically analyzed to gain a more profound understanding of how dynamic impact changes through clay courts and court tapes. The experiments consisted of three factors: a surface factor of three levels, a drop height factor of four levels, and an incline angle of four. The drop-height element is replaced with the pre-impact normal velocity covariate for the ANOVA analysis. Table 1 illustrates the ANOVA test results. The test consists of two factors and a covariate. The surface factor has two degrees of freedom, whereas the angle factor has three degrees of freedom. The total degree of freedom of the test is 143. Analysis results reveal that the surface factor is significant with a strong probability of \(p<0.001\) and \(F=41.8\). The angle factor is also essential, with a strong probability of \(p<0.001\) and \(F=8.39\). On the other hand, covariate \(\pi_{Cy}(t_{0})\) was not significant for this study's experiment range. Interactions between the factors are not included in the model. Figure 9: Comparison of the theoretical and experimental displacement on the local coordinate system, \(\left[\textbf{i}\;\textbf{j}\;\textbf{k}\right]\). (**a**) The normal and tangential component \(\beta=7.5^{\circ}\) for impact on the granular surface. (**b**) The normal and tangential component component \(\beta=15^{\circ}\) for impact on tape 1. (**c**) The normal and tangential component component \(\beta=22.5^{\circ}\) for impact on tape 2. Figure 10 shows a pairwise comparison of the mean coefficient of restitution for different surfaces. The mean coefficient of restitution on clay is 0.7265, whereas the mean coefficient of restitution on white and green tapes is 0.7617 and 0.7587, respectively. The procedure suggests a statistically significant difference between the impact on the granular surface and line tapes. Figure 11 shows fitted values vs. the studentized residuals plot, and Figure 11 illustrates the normal probability distribution of the residuals of the linear model. The diagnostics show slightly skewed data. The skewness of the coefficient of restitution is 0.545, and the kurtosis of the coefficient of restitution is 3.866. This study analyzed the impact of a tennis ball on all-weather green court clay (granular surface), white herringbone tape (tape 1), and green herringbone tape (tape 2). A viscoelastic contact force for compression and an elastic contact force for restitution were used in the mathematical model. Normal impact results were used to fit spring coefficients and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum Sq.** & **d.f** & **Mean Sq.** & **F** & **Prob \(>\) F** \\ \hline Surface & 0.03656 & 2 & 0.01828 & 41.8 & \(p<0.001\) \\ Angle & 0.011 & 3 & 0.00367 & 8.39 & \(p<0.001\) \\ \(v_{Coy}(t_{0})\) & 0.00046 & 1 & 0.00046 & 1.05 & 0.3066 \\ Error & 0.0599 & 137 & 0.00044 & & \\ Total & 0.10788 & 143 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: ANOVA test results. Figure 10: Multiple comparisons of surface effect. damping coefficients. The coefficients were calculated as follows. For clay, \(k\) = 25,650 N/m and \(b\) = 14 Ns/m. For tape 1, \(k\) = 30,800 N/m, and \(b\) = 15 Ns/m.