The coefficient of restitution for collisions of happy balls, unhappy balls, and tennis balls

Abstract

A perfectly happy ball is one that bounces to its original height when dropped on a massive, rigid surface. A completely unhappy ball does not bounce at all. In the former case, the coefficient of restitution ͑COR͒ is unity. In the latter case, the COR is zero. It is shown that when an unhappy ball collides with a happy ball, the COR increases from zero to unity as the stiffness of the happy ball decreases from infinity to zero. The COR is independent of the mass of each ball. The implication of reducing the COR of a tennis ball, as a possible means of slowing the serve in tennis, is also considered. It is shown that ͑a͒ the COR for a collision with a racket varies with the impact point and is a maximum at the vibration node near the center of the strings, and ͑b͒ the serve speed is reduced by only about 20% if the COR for a bounce on the court is reduced to zero.

Ritualized fighting and biological armor: the impact mechanics of the mantis shrimp's telson

“…However, it is important to note that the coefficient of restitution is a characteristic of a collision; it is not a material property. The coefficient of restitution depends on the size, shape, structural and material properties, and relative velocities of the two impacting objects (Cross, 2000;Goldsmith, 1960;Stronge, 2000). Under certain conditions, such as one of the colliding objects being significantly more rigid than the other, the coefficient of restitution can be referred to as a property of the less rigid object (Cross, 2000).…”

“…where e is the coefficient of restitution (Cross, 2000;Nathan, 2000). Additionally, the duration of impact, defined as the time between first contact and separation, was recorded.…”

“…The stiffness of the colliding objects is an important factor determining impact behavior. Thus, impact can also be modeled as a collision between two springs (Bayman, 1976;Cross, 1999;Cross, 2000). We calculated the spring stiffness, k, of the telson and abdomen using two different equations:…”

Viscoelastic impact characterisation of solid sports balls used in the Irish sport of Hurling

“…Over the course of testing the sliotar cores, a difference was observed between these two measurements for ball types that exhibited lateral expansion. Previous publications' statements of equivalence [30][31][32] between COM displacement and diameter compression methods were shown in the present study as being acceptable, not for the reasons given in the original papers but rather due to the absence of lateral expansion in the ball impacts examined. As the lateral expansion occurred parallel to the impact plate and hence was imperceptible to the axial load-cell, it augmented the diameter compression due to volumetric spreading of ball material without affecting the COM displacement as derived from the load-cell force data.…”

“…Numerous methods for quantifying deformation are featured in the literature: diameter compression, the reduction of diameter of the ball normal to the impact plate; lateral expansion, the increase in ball diameter parallel to the impact plate; and centre of mass (COM) dis-placement, calculated from the double time integral of the force divided by ball mass. Many publications assume, either explicitly or implicitly, that diameter compression and COM displacement are equivalent [30][31][32], neglecting the effect (if any) of lateral expansion. The studies that have considered lateral expansion do not refer to COM displacement [33,34].…”