Coefficient of restitution
The coefficient of restitution, denoted by e, is a dimensionless quantity in classical mechanics that quantifies the elasticity of a collision between two objects by measuring the degree to which kinetic energy is conserved.[1] It is defined as the ratio of the magnitude of the relative velocity of separation to the magnitude of the relative velocity of approach along the line of impact, mathematically expressed as e = \frac{|v_2 - v_1|}{|u_2 - u_1|}, where u_1, u_2 are the initial velocities and v_1, v_2 are the final velocities of the two bodies.[1] This parameter directly relates to the fraction of kinetic energy retained after the collision, with values typically ranging from 0 to 1: e = 1 indicates a perfectly elastic collision where no kinetic energy is lost, e = 0 represents a perfectly inelastic collision where the bodies stick together and all relative kinetic energy is dissipated, and intermediate values describe partially elastic collisions.[2] The concept was first introduced by Sir Isaac Newton in 1687 as part of his experimental law of impacts in the Philosophiæ Naturalis Principia Mathematica, predating the formal definition of kinetic energy by over a century and serving as a phenomenological description of collision outcomes based on empirical observations.[2] Newton's formulation emphasized the relative velocities, making it Galilean invariant and applicable to a wide range of material pairs, though the value of e depends on factors such as the materials involved, surface conditions, and impact velocity.[2] For practical measurement, such as in a drop test, e can be approximated as the square root of the ratio of rebound height to initial drop height, assuming constant gravitational acceleration.[3] In modern physics and engineering, the coefficient of restitution is essential for modeling and predicting the behavior of colliding systems, enabling the solution of momentum conservation equations alongside restitution to determine post-collision velocities.[1] It finds widespread applications in fields like sports science—for regulating equipment such as baseball bats via the batted ball coefficient of restitution (BBCOR) standard, which limits the effective coefficient of restitution to 0.50 to ensure performance comparable to wood bats—[4] automotive crash analysis to assess energy absorption in safety designs, and granular flow simulations in geophysics and materials processing. These uses highlight its role in bridging theoretical collision dynamics with real-world phenomena, where deviations from ideal values often reveal material deformation or frictional effects.Fundamentals
Definition
The coefficient of restitution, denoted by e, is a dimensionless empirical parameter in classical mechanics that characterizes the elasticity of a direct collision between two bodies, ranging from 0 for a perfectly inelastic collision to 1 for a perfectly elastic one. It is mathematically defined along the line of impact as e = -\frac{v_2' - v_1'}{v_2 - v_1}, where v_1 and v_2 are the pre-collision velocities of the two bodies, and v_1' and v_2' are their post-collision velocities, with the velocities taken as signed scalars in the direction of the line connecting their centers at impact. The negative sign ensures that e is positive for cases where the bodies rebound (i.e., the relative velocity reverses direction), as the separation velocity after collision opposes the approach velocity before collision.[5] This parameter serves as an empirical measure for direct central collisions, capturing the fraction of relative kinetic energy conserved without deriving from fundamental forces.[2] The concept was introduced by Isaac Newton in his 1687 work Philosophiæ Naturalis Principia Mathematica, where it was presented as a measure of the "resiliency" or relative velocity after impact in collisions of hard bodies.[6]Physical interpretation
The coefficient of restitution, denoted as e, serves as a dimensionless parameter that quantifies the elasticity of a collision between two bodies by measuring the degree to which they rebound relative to their approach speed. It classifies collisions along a spectrum: a value of e = 0 corresponds to a perfectly inelastic collision, where the bodies adhere upon impact and exhibit no rebound, resulting in maximum deformation and energy loss. In contrast, e = 1 indicates a perfectly elastic collision, in which the bodies separate with the same relative speed as their approach, conserving all kinetic energy without dissipation to deformation, heat, or other forms.[7]/7%3A_Linear_Momentum_and_Collisions/7.3%3A_Collisions)[8] Values between 0 and 1 describe partially elastic collisions, where some rebound occurs but kinetic energy is partially converted into other forms, such as internal deformation energy, thermal energy, or sound. A lower e signifies greater dissipation, reflecting more irreversible deformation during the brief contact phase, while higher values approach ideal elastic behavior with minimal energy loss. This interpretation links e directly to the efficiency of momentum transfer versus energy conservation in the collision process.[9][7] Intuitive examples illustrate these extremes: a ball made of soft clay or putty dropped onto a hard surface typically yields a low e (around 0.1), with the material deforming permanently and barely rebounding as kinetic energy is largely absorbed into plastic deformation. Conversely, polished steel spheres colliding can achieve a high e (up to 0.95), rebounding vigorously with most kinetic energy restored, due to the materials' rigidity and elastic recovery.[10][11] Unlike the coefficient of static friction, which measures the maximum frictional force opposing the onset of relative motion between surfaces before sliding, the coefficient of restitution specifically addresses the normal component of velocity reversal in impacts, independent of tangential friction effects./7%3A_Linear_Momentum_and_Collisions/7.3%3A_Collisions)Range of values
The coefficient of restitution e satisfies $0 \leq e \leq 1, where e = 0 indicates a perfectly inelastic collision with no relative rebound velocity, and e = 1 indicates a perfectly elastic collision with full recovery of relative velocity. Values exceeding 1 are theoretically impossible for isolated systems, as they would require an increase in kinetic energy, violating conservation of energy.[12][8][13] The value of e depends on the specific pair of colliding objects rather than being an intrinsic property of a single material, as deformation and energy dissipation occur at the interface between them. For example, steel colliding with steel typically yields e \approx 0.65 for standard surfaces, but up to 0.95 for polished spheres, whereas steel colliding with wood results in lower values due to greater absorption in the softer material.[11][10][14] Typical values for common material pairs, often measured in drop tests onto hard surfaces, illustrate this variability:| Material Pair | Approximate e | Notes/Source |
|---|---|---|
| Superball on hard surface | 0.9 | High elasticity from polymer composition.[15] |
| Glass marble on hard surface | 0.85 | Measured in controlled drop experiments.[16] |
| Tennis ball on hard court | 0.7–0.8 | Varies with ball pressure and surface; average from sports testing.[17][14] |
| Golf ball on hard surface | 0.8 | From impact studies on practice balls.[14][18] |
| Modeling clay on hard surface | ~0.1 | Near-inelastic due to plastic deformation. |
| Ice on ice | 0.5 | Varies with temperature and velocity; average from low-speed collisions.[19][20] |