Low Speed Impact of an Elastic Ball with Tapes and Clay Court

Abstract

The study aims to investigate the normal and oblique impact of an elastic sphere (tennis ball) on a granular surface (clay) and two different plastic tape lines. In this research, we model the impact force with a mathematical elastoplastic force model, and a differential approach is used. The model is applied for an impact with granular material (green clay) and plastic surfaces (line tapes). We investigated the normal and oblique impact dynamics of a sphere (tennis ball). The impact duration is divided into two phases: compression with an elastoplastic force and restitution with an elastic force. The laboratory experiments in various configurations are recorded with a high frame-per-second camera and analyzed using image processing methods. The mathematical model for the impact with rebounds is verified with the experimental set-up for the considered surfaces. The viscoelastic and elastic forces agree well with the experimental data. The impact parameters of the granular surface and plastic tapes are compared. The ANOVA test suggests robust statistical significance in the coefficient of restitution between granular surfaces and plastic tapes. Our force model for impact performs well, and the impact responses of the sphere on the granular surface and the plastic line tapes are significantly different.

1. Introduction

The study of impact is of crucial interest for various applications, including sports. The first scientist to study impact was Galileo, who worked on falling objects, and the collision of spheres was analyzed by Newton in 1686 with a kinematic coefficient of restitution [1].

An important contribution to tennis analysis was made by Brody [2], who studied the oblique impact of a ball on a court, assuming the ball is rigid and the coefficient of restitution is constant [3]. Later, experimental investigations revealed that the coefficient of restitution decreases with the increase of the incident impact velocity [4,5]. The coefficient of restitution for oblique impact was higher than the normal impact coefficient of restitution [3,6,7,8].

The effects of the friction between the tennis ball and the racket string were analyzed by ref. [9]. The author experimentally measured sliding and rolling friction coefficients and developed an analytic model. The study revealed that a coefficient of sliding friction below 0.3 is critical, and slight chances of friction could cause a significant change in the ball’s rebound angle. According to refs. [10,11], the ball slides through the surface for low incline angles, yet a combination of sliding and gripping occurs for larger incline angles. Recently, Texier and Tadrist [12] studied the impact of the pressured membrane on the rigid ground. The authors concluded that energy dissipated in vibrations may be modeled with two mass models.

Chatterjee et al. [13] concluded that contact and impact forces can be estimated using a rigid-body impact model when deformations are small. Viscoelastic contact models are feasible to numerically simulate the ball impact problem as they do not include permanent deformation. A viscoelastic contact model consists of a damping (viscous) term combined with an elastic (conservative) force model. The Kelvin–Voigt analytic model was used by Dignall and Haake [14]. The authors experimentally measured the restitution and contact time coefficient while the stiffness and damping coefficients were calculated analytically. Later, a more complex viscoelastic model with a combination of a spring parameter, material damping, and momentum flux was developed by Goodwill and Haake [15]. Momani et al. [16] compared the performance of viscoelastic contact models with experimental data. The authors concluded that the nonlinear Hunt–Crossley model performs better regarding contact force and deflection than the Kevin–Vogit model but provides a relatively longer contact time prediction. The Kevin–Vogit model was found to perform better for contact time estimation. Recently, Chen et al. [17] studied rockfall problems based on a viscoelastic model assuming spherical geometry. The authors reported a good reliability of maximum impact force.

For the impact of a cricket ball with a rigid surface, a dynamic force deflection using a spring-damper model was developed [18]. The damping coefficient varied with the contact area, and the spring coefficient was described with a pre-impact velocity alone.

A nonlinear spring–damper model investigated tennis ball impact on the racket [19]. The contact force was given by a serially connected spring damper and a nonlinear damper. The string–bed effect was simulated with the nonlinear damper. The string tension significantly affects the coefficient of restitution and impact time duration, and the racket head size and string axial rigidity were revealed to be negligible. Ghaednia et al. [20] divided the viscoelastic contact model into compression and restitution phases and treated the fully elastic restitution phase with no damper. They reported a good agreement between the theoretical model and the experimental measures. The contact dynamics of multibodies is a highly nonlinear phenomenon, and a good selection of recently available force models can be seen in ref. [21].

Tennis games have three main courts: hard, grass, and clay. Clay courts are considered much slower than grass and hard courts. For clay courts, the lines are created by nailing synthetic tapes to the surface.

Kang et al. proposed a modified Archimedes’ law to analyze penetrations of solids into granular material [22]. Several authors have also investigated penetration depth and acting forces acting during the interaction with the granular medium [23,24]. The main focus of our paper is the rebound dynamics of the impact.

This study investigates the tennis ball’s impact on green all-weather clay [25], on white herringbone tape (tape 1) [26], and on green tape (tape 2) [25] experimentally and theoretically. The impact model is separated for compression and restitution. For the compression phase, a viscoelastic model is employed and the restitution phase is assumed to be perfectly elastic. The force coefficients are determined from the normal impact experiments and validated with the oblique impact cases. The force coefficients are compared for the different surfaces. We considered two fundamental properties of a tennis court surface: friction and restitution. An ANOVA test is conducted to compare the statistical significance of the coefficient of restitution for surfaces.

This is the first study of the impact on a granular medium with a rebound. Tennis balls and clay court materials were selected for the study. An analysis of impact video recordings showed that the impact at a relatively low velocity limits deformation at the contact region, and rigid body assumptions hold for the rest of the ball. Also, the effect of the court tape on the ball bounce has not been studied until now. It is very annoying for the players when the ball bounces differently in contact with the lines. It is advisable to have the same coefficients of friction and restitution for the whole playing surface. Our mathematical model for the ground impact with viscoelastic compression and elastic restitution is also unique.

The coefficient of restitution is the key parameter for characterizing impact dynamics. By investigating the coefficient of restitution for a clay court and court tapes, we can assess how similar or different their collision dynamics are. The study findings can help the industry improve the specifications of its products, and better court consistency increases game quality.

2. Impact Dynamics

Consider an impact between a tennis ball and a flat surface. The angle of the flat surface with a horizontal axis, $x_0$, is $\beta$. The tennis ball is considered a perfect sphere of radius R and mass center at C. The contact between the tennis ball and the flat surface occurs at point E.

The global reference frame of unit vectors is $[{\mathsf{𝚤}}_0,{\mathsf{𝚥}}_0,{\mathit{k}}_0]$. The local reference frame is placed at point E, with unit vectors $[\mathsf{𝚤},\mathsf{𝚥},\mathit{k}]$ as shown in Figure 1. The x-axis is tangential to the flat surface, and the y-axis is perpendicular to the flat surface.

To describe the instantaneous configuration of the ball, the generalized coordinates $q_1$, $q_2$, and $q_3$ are attached. The generalized coordinate $q_1$ denotes the contact point displacement along the x-axis. The generalized coordinate $q_2$ indicates the contact point displacement along the y-axis, and the $q_3$ coordinate defines the ball’s rotation.

The position vector of the impact point E is

$${\mathit{r}}_E=q_1\mathsf{𝚤}+q_2\mathsf{𝚥},$$

(1)

and the position vector of the center of the ball C is

$${\mathit{r}}_C=q_1\mathsf{𝚤}+(q_2+R)\mathsf{𝚥}.$$

(2)

The angular velocity and acceleration of the ball are

$$\mathit{\omega }={\dot{q}}_3\mathit{k}and\mathit{\alpha }={\ddot{q}}_3\mathit{k}.$$

(3)

The linear velocity and acceleration of the mass center C are

$${\mathit{v}}_c=\frac{d{\mathit{r}}_C}{dt}and{\mathit{a}}_c=\frac{d^2{\mathit{r}}_C}{d{t}^2}.$$

(4)

The linear velocity and acceleration of the impact point E are

$$\begin{array}{c}\hfill {\mathit{v}}_E={\mathit{v}}_C+\mathit{\omega }\times {\mathit{r}}_{CE},\\ \hfill {\mathit{a}}_E={\mathit{a}}_C+\mathit{\alpha }\times {\mathit{r}}_{CE}+\mathit{\omega }\times (\mathit{\omega }\times {\mathit{r}}_{CE}),\end{array}$$

(5)

where ${\mathit{r}}_{CE}={\mathit{r}}_E-{\mathit{r}}_C$.

2.1. Contact Force

Due to compression, the local contact region generates a reaction force. The contact force starts at the instant the ball touches the flat surface and disappears with the separation of the ball and the flat surface.

The contact force used in the study is a hybrid model. The compression model has an elastic force and viscous damping, whereas restitution is solely elastic. Figure 2 depicts an example of contact force.

2.1.1. Compression Phase

The impact starts with the compression phase. At the instant the ball and the surface touch, the normal contact force P is zero. The impact force is perpendicular to the flat surface and acts on the y-axis. The impact force is given by

$$\mathit{P}=(k\delta -b\dot{\delta })\mathsf{𝚥},$$

(6)

where $\delta =q_2\left(t\right)$ is the elastic deformation during the compression alongside the y-axis; k is the spring coefficient, and b is the damping coefficient.

The total force at point E is

$$\mathit{T}=F_f\mathsf{𝚤}+P\mathsf{𝚥},$$

(7)

where $F_f$ is the friction force that acts in the opposite direction to the tangential velocity of the ball at contact point E. The friction force is assumed to be

$${\mathit{F}}_f=\mu P\mathsf{𝚥},$$

(8)

where $\mu$ is the kinetic coefficient of friction.

Equations of motions for a rigid ball are given below.

$$\begin{array}{c}\hfill m{\mathit{a}}_c=\mathit{G}+\mathit{T}\\ \hfill I_C\alpha ={\mathit{r}}_{CE}\times \mathit{T},\end{array}$$

(9)

where G is the weight vector of the ball, described by $\mathit{G}=-mgsin\beta \mathsf{𝚤}-mgcos\beta \mathsf{𝚥}$. $I_C$ is the mass moment of inertia about the center of the mass C, and gravitational acceleration is g = 9.81 m/s².

The initial conditions for the restitution phase are

$$\left\{\begin{array}{cc}{q}_1\left(t_0\right)=q_{10},\hfill & {\dot{q}}_1\left(t_0\right)=-v_{0}sin\beta \hfill \\ q_2\left(t_0\right)=q_{20},\hfill & {\dot{q}}_2\left(t_0\right)=-v_{0}cos\beta \hfill \\ q_3\left(t_0\right)=q_{30},\hfill & {\dot{q}}_1\left(t_0\right)=0.\hfill \end{array}\right.$$

(10)

The final values of $q_i\left(t_m\right)$ and ${\dot{q}}_i\left(t_m\right)$ for the compression phase will be the initial conditions for the restitution phase.

Compression ends when the ball’s normal velocity of the impacting point vanishes $v_E\mathsf{𝚥}=0$. At this moment, the maximum deformation ${\delta }_m$ and the maximum normal contact force $P_m$ are observed.

2.1.2. Restitution Phase

The restitution phase starts at the moment of maximum compression. During the restitution phase, the contact force reduces from its maximum value, $P_m$, to zero. The normal force for the restitution phase is

$$\mathit{P}=k\delta \mathsf{𝚥}.$$

(11)

The total contact force for the restitution is $\mathit{T}=F_f\mathsf{𝚤}+P\mathsf{𝚥}$. Equations of motion are solved for the initial conditions at maximum compression.

$$\left\{\begin{array}{cc}{q}_1\left(t_m\right)=q_{1m},\hfill & {\dot{q}}_1\left(t_m\right)=-v_{xm}\hfill \\ q_2\left(t_m\right)=q_{2m},\hfill & {\dot{q}}_2\left(t_m\right)=-v_{ym}\hfill \\ q_3\left(t_m\right)=q_{3m},\hfill & {\dot{q}}_1\left(t_m\right)=w_m.\hfill \end{array}\right.$$

(12)

The equations of motion are given by Equation (9). The differential equations of motion are solved with MATLAB (2023b). The restitution phase ends when the contact force vanishes. The coefficient of restitution is calculated as

$$e=|\frac{v_E\left(t_f\right)\mathsf{𝚥}}{v_E\left(t_0\right)\mathsf{𝚥}}|,$$

(13)

where $v_E\left(t_0\right)$ is the initial velocity of the point E and $v_E\left(t_f\right)$ is the final velocity of the point E for the compression phase.

3. Experiments

An experiment-specific setup was built to provide consistent initial conditions. Figure 3 depicts a schematic of the setup layout.

A mechanical arm that can travel vertically was used to release the ball from different heights in the range of 0.2–0.4 m. Tennis balls with a diameter of 0.067 m and a mass of 0.0567 kg were used for the experiments. For each height, at least three consistent drops were conducted. The flat surface was attached to the tripod at one side, so the incline angle of the impacting surface could be adjusted by changing the tripod’s height. A high-speed digital camera was used to record every drop. The camera was mounted to a tripod, and the angle was always parallel to the dropping surface. The incline angle of the camera and the impact surface were controlled precisely with a digital angle gauge with a precision of ±0.5°.

Figure 4a shows a raw video image of the ball during impact. Figure 4b displays the center point tracking of the ball. Dashed lines covering the ball show the ball boundary for the current frame. The dotted scatter plot shows all the recorded positions for the center. Note that the camera and the surface are parallel in an oblique line.

The ball’s motion was recorded using an EDGETRONIC SC1 camera (Sanstreak Corp., Campbell, CA, USA) and Nikon AF Nikkor 50mm f/1.8D lens (Nikon, Tokyo, Japan). The high-speed camera was set to 10,000 frames per second (fps). Grayscale frames have a resolution of 512 × 512 pixels with an 8-bit depth where 0 is pure black and 255 is pure white. The ball’s radius and center were found using the Hough transform method for all frames. The ratio between the physical ball size and the average ball radius of the video determines a scale factor. Unit transformation with this method from pixel to meter eliminates the out-of-plane displacement errors. Figure 4b offers the center point tracking of the ball. Dashed lines covering the ball show the ball boundary for the current frame. The dotted scatter plot shows all the recorded positions for the center. Note that the camera and the surface are parallel in an oblique line.

The effective coefficient of friction is calculated using the tangential and the normal velocities before and after the impact. Calculated friction coefficients for individual tests are used in simulations of the proposed model. The ratio of the tangential and normal momentum change gives the effective coefficient of friction:

$$\mu ={\displaystyle \frac{m(v_{Cx}\left(t_f\right)-v_{Cx}\left(t_0\right))}{m(v_{Cy}\left(t_f\right)-v_{Cy}\left(t_0\right))}}.$$

(14)

Figure 5 shows the clay tennis surface, and tape 1 and 2 on the clay. Experiments are carried out on clay surfaces; later, tapes were positioned over the clay and carefully fixed with tape nails. Later, the investigations were repeated with the same dropping heights and impact angles.

4. Results and Discussion

The general behavior of the coefficient of restitution is graphically investigated in the following section. Each data point in Figure 6, Figure 7 and Figure 8 illustrates the mean of three experiments repeated at the same drop height, the same impact angle, and on the same surface.

The coefficient of restitution (e) of a tennis ball on a granular surface is shown in Figure 6. Figure 6a shows the chance of e with respect to pre-impact velocity, ${\mathit{V}}_C\left(t_0\right)$, and Figure 6b shows the chance of e with respect to impact angle, $\beta$. Figure 6a reveals no trend in the change of e with normal pre-impact velocity; moreover, there is no apparent separation between angle groups. Mean e’s and standard deviations are $\overline{e}=0.735$ and $\sigma =0.0097$ for normal impact, $\overline{e}=0.716$ and $\sigma =0.0062$ for oblique angle $\beta ={7.5}^{\circ }$, $\overline{e}=0.718$ and $\sigma =0.0219$ for oblique angle $\beta ={15}^{\circ }$, $\overline{e}=0.709$ and $\sigma =0.0302$ for oblique angle $\beta ={22.5}^{\circ }$. Figure 6b shows a slight decrease in e with increasing angle. The standard deviation in Figure 6b is $\sigma =0.0112$.

The coefficient of restitution (e) of tennis balls on tape 1 is shown in Figure 7. Figure 7a shows the chance of e concerning pre-impact velocity, ${\mathit{V}}_C\left(t_0\right)$, and Figure 7b shows the chance of COR with respect to impact angle, $\beta$. Figure 7a does not show any systematic behavior of COR with normal pre-impact velocity and impact angle. Mean es and standard deviations are $\overline{e}=0.777$ and $\sigma =0.0219$ for normal impact, $\overline{e}=0.754$ and $\sigma =0.0204$ for oblique angle $\beta ={7.5}^{\circ }$, $\overline{e}=0.751$ and $\sigma =0.0262$ for oblique angle $\beta ={15}^{\circ }$, and $\overline{e}=0.777$ and $\sigma =0.0500$ for oblique angle $\beta ={22.5}^{\circ }$. Figure 7b shows a slightly higher e for normal impact and $\beta ={22.5}^{\circ }$ degrees compared to $\beta ={7.5}^{\circ }$ and $\beta ={15}^{\circ }$. The standard deviation in Figure 7b is $\sigma =0.0317$.

The coefficient of restitution (e) of tennis balls on tape 2 is shown in Figure 8. Figure 8a shows the chance of e with respect to pre-impact velocity, ${\mathit{V}}_C\left(t_0\right)$, and Figure 8b shows the chance of e with respect to impact angle, $\beta$. Figure 8a shows no trend or separation of e with normal pre-impact velocity and impact angle. Mean es and standard deviations are $\overline{e}=0.772$ and $\sigma =0.0121$ for normal impact, $\overline{e}=0.745$ and $\sigma =0.0123$ for oblique angle $\beta ={7.5}^{\circ }$, $\overline{e}=0.761$ and $\sigma =0.0313$ for oblique angle $\beta ={15}^{\circ }$, and $\overline{e}=0.755$ and $\sigma =0.0084$ for oblique angle $\beta ={22.5}^{\circ }$. The standard deviation in Figure 8b is $\sigma =0.0116$.

Figure 6, Figure 7 and Figure 8 illustrate no apparent linear trend for the coefficient of restitution. However, the variation in the coefficient of restitution is small. Therefore, the coefficient of restitution can be almost constant in practical applications. Previous studies on tennis ball impact for different configurations [20,27] also show no linear trends with a small deviation.

The stiffness $\left(k\right)$ and damping $\left(b\right)$ coefficients must be determined to verify the mathematical model with the oblique impact. To find k and b, at least two independent experimental results are needed. The normal impact experimental data set consists of 12 experiments in three replications of four different initial drop heights. The mean absolute error $ϵ$ between the theory and the experiments is analyzed. The corresponding k and b values of minimum error are selected. The best fits that minimize $ϵ$ are k = 25,650 N/m and b = 14 Ns/m; the mean absolute error between theory and the experiment is $7.1\%$. The force coefficients for impact on tape 1 are k = 30,800 N/m and b = 15 Ns/m, respectively. Error, $ϵ$, is found to be $6.3\%$. The force coefficients for impact on tape 2 are k = 31,825 N/m, b = 15 Ns/m, and the corresponding error is $3.3\%$.

Figure 9 compares the displacement plots of the theory and the experiments on the local coordinate system. The first column of sub-figures illustrates the normal component, and the second column of figures represents tangential components of the same impact. The initial velocities are measured from the experiments and used for the simulations. The output velocities are compared.

Figure 9a illustrates the normal and tangential components of the displacement for both the theory and one of the experimental results with $\beta ={7.5}^{\circ }$. The analysis of the experiment shows that the initial velocity is ${\mathit{V}}_C\left(t_0\right)=-0.206\mathsf{𝚤}-1.787\mathsf{𝚥}\mathrm{m}/\mathrm{s}$ and the final velocity is ${\mathit{V}}_C\left(t_f\right)=-0.297\mathsf{𝚤}-1.421\mathsf{𝚥}\mathrm{m}/\mathrm{s}$. The simulation outcome is ${\mathit{V}}_C\left(t_f\right)=-0.309\mathsf{𝚤}-1.383\mathsf{𝚥}\mathrm{m}/\mathrm{s}$. The errors between tangential and normal velocity components are $1.1\%$ and $2.6\%$, respectively.

Figure 9b shows the normal and tangential components of the displacement for both the theory and one of the experimental results with $\beta ={15}^{\circ }$. The analysis of the experiment shows that the initial velocity is ${\mathit{V}}_C\left(t_0\right)=-0.468\mathsf{𝚤}-1.890\mathsf{𝚥}\mathrm{m}/\mathrm{s}$ and the final velocity is ${\mathit{V}}_C\left(t_f\right)=-0.246\mathsf{𝚤}-1.444\mathsf{𝚥}\mathrm{m}/\mathrm{s}$. The simulation outcome is ${\mathit{V}}_C\left(t_f\right)=-0.257\mathsf{𝚤}-1.470\mathsf{𝚥}\mathrm{m}/\mathrm{s}$. The errors between tangential and normal velocity components are $4.4\%$ and $1.8\%$, respectively.

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 ${\mathit{V}}_C\left(t_0\right)=-0.774\mathsf{𝚤}-1.849\mathsf{𝚥}\mathrm{m}/\mathrm{s}$ and the final velocity is ${\mathit{V}}_C\left(t_f\right)=-0.228\mathsf{𝚤}-1.521\mathsf{𝚥}\mathrm{m}/\mathrm{s}$. The simulation outcome is ${\mathit{V}}_C\left(t_f\right)=-0.232\mathsf{𝚤}-1.443\mathsf{𝚥}\mathrm{m}/\mathrm{s}$. 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 Surface

The 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. ANOVA test results.

Source	Sum Sq.	d.f	Mean Sq.	F	Prob > F
Surface	0.03656	2	0.01828	41.8	p < 0.001
Angle	0.011	3	0.00367	8.39	p < 0.001
$v_{Cy}\left(t_0\right)$	0.00046	1	0.00046	1.05	0.3066
Error	0.0599	137	0.00044
Total	0.10788	143

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 $v_{Cy}\left(t_0\right)$ was not significant for this study’s experiment range. Interactions between the factors are not included in the model.

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 11a shows fitted values vs. the studentized residuals plot, and Figure 11b 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 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. For tape 2, k = 31,825 N/m, b = 15 Ns/m.

The mathematical model and oblique impact experiments were compared using the evaluated force coefficients from a normal impact. The model and the experiments overlap with less than $10\%$ for every analyzed situation.

The model suggests that tape 1 has a $20\%$ more significant spring coefficient and a $7\%$ greater damping coefficient than the granular surface. Similarly, the analysis demonstrates a $24\%$ more significant spring coefficient and a $7\%$ greater damping coefficient for tape 2 than the granular surface.

Statistical analysis reveals a statistically robust significance between the coefficient of restitution on clay compared to white and green tapes. The mean coefficient of restitution on clay is $0.7265$, whereas the mean coefficient of restitution on white tape is $4.9\%$ greater than $0.7617$. Comparably, the mean coefficient of the distribution of experiments on green tape is $0.7587$ and $4.3\%$ larger than clay.

In our study, we have introduced a viscoelastic contact force model to analyze the impact of a tennis ball on a clay court and court tapes. The model is divided into two phases: compression and restitution. While existing studies focus on the impact of tennis balls on the tennis racket or the court surface, our research aims to explore the distinctions between the impact on clay tennis courts and court tapes, which has not been studied before.

Our force model aligns well with the experimental data, yet more studies are required with different impact angles and higher impact velocities. The study’s outcome shows a significant difference between the coefficient of restitution for a clay court and court tapes. A close coefficient of restitution on court and court tapes is required for better game quality. Further studies are required to assess how the significant difference in the coefficient of restitution between court and court tapes affects the game quality.

Moreover, more investigations are needed of the coefficient of friction and impact dynamics on different clay court materials (e.g., American red clay, European red clay, etc.).

5. Conclusions

The study’s outcome shows that, despite the remarkable difference in the theoretical spring constants ($20\%$ and $24\%$, respectively), the difference between the experimental restitution coefficient remains significant but low ($4.9\%$ and $4.3\%$, respectively). This study and future studies of higher velocities will help us better understand the impact of tennis balls on granular court surfaces and line tapes.

We analyzed the rebound for different surfaces. The coefficient of restitution and the coefficient of friction play an essential role in the collision problem. We can improve the tape quality if we manufacture lines with property (coefficient of restitution and friction) close to the court property. In this way, the bounce when the ball impacts the line will be similar to the court rebound. We can design better court surfaces if we better understand the characteristics of the ball’s impact.