# Low Speed Impact of an Elastic Ball with Tapes and Clay Court
## 1 Introduction  
The study of impact is of crucial interest for various applications, including sports. The first scientist to study impact was [PERSON], who worked on falling objects, and the collision of spheres was analyzed by [PERSON] in 1686 with a kinematic coefficient of restitution [1]. An important contribution to tennis analysis was made by [PERSON] [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, [PERSON] and [PERSON] [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. [PERSON] 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 usedby [PERSON] and [PERSON] [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 [PERSON] and [PERSON] [15]. [PERSON] 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, [PERSON] 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. [PERSON] 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.