# Is the coefficient of restitution a constant?(/ Validity over range)

We were taught about coefficient of restitution and it's definition and that it is treated as a constant. This was termed "Newton's law of restitution".

However, I don't seem to understand why that would seem to be. It seems pretty arbitrary why it would hold true.(No rigours or intuitive proof).

If there is one, I haven't seemed to find it online. (As there is for other laws like ohms law and Newton's law of cooling *)

Although I have accepted it to be a fact (to solve questions). I can not imagine it (COR) being anything more than a fancy way to give us an extra information which was previously missing.(As I have noticed with impulse-momentum equations).

I would like to gain some theoretical insight to why it is treated as a constant.(Or over a range of values) Thanks in advanced.

(*By the proofs of the laws, what I meant: Newton's law of cooling is an approximation of the Stephen-Boltzmann equation which is understood well. Also ohms law can be "derived" through knowledge of drift velocity/relaxation time/Electric field. Which is what I am looking for)

## 1 Answer

The Law of Restitution is usually stated as a constant ratio $e$ between relative velocities of separation and approach for a particular pair of colliding objects. A more intuitive formulation is that a constant fraction $1-e^2$ of the total kinetic energy lost in the collision.

Like Ohm's Law and Hooke's Law, the Law of Restitution is empirical. It is an approximation which is valid over a limited range of relative speeds of approach, typically 0.1 to 100 m/s. For some materials, notably rubber, which does not obey Hooke's law, it not a good approximation over this range. The law is not based on theory, but models of deformable materials can be used to "explain" how $e$ depends on the material, geometric and kinetic properties of the colliding bodies.

According to K L Thornton in Contact Mechanics, CUP 1987, chapter 11, p 363 :

The coefficient of restitution is not a material property, but depends on the severity of the impact. At sufficiently low velocities $V \lt V_Y$ the deformation is elastic and $e \approx 1$. The coefficient of restitution falls gradually with increasing velocity. When a fully plastic indentation is formed our theory suggests that $e \propto V^{-1/4}$.

For low impact speeds all colliding bodies behave like a spring and dashpot. The deformation is elastic, in the sense that the body returns to its original shape, but some energy is lost due to internal friction (elastic hysteresis). The amount of frictional loss relates to the speed of sound waves $c$ in the material. If the material is very hard (stiff), so that the deformation is small and $c \gg V$ then almost all of the stored energy is returned as kinetic energy ($e \approx 1$). For 'soft' materials ($V \approx c$) a significant portion of the KE is lost ($e \lt 1$).

Thornton's analysis suggests that collisions cease to be elastic ($e \approx 1$) when $p=\rho V^2/Y \lt 10^{-6}$, where $\rho$ is density, $V$ is impact speed and $Y$ is the yield stress of the softer material. For a hard steel sphere striking a medium hard steel floor the impact speed at which plastic deformation starts is $V_Y \approx 0.14 m/s$. Thereafter permanent deformation occurs in part of the contact region, and the size of this region increases as impact velocity increases. Fully plastic deformation occurs for $p \approx 10^{-3}$ when typically $V \approx 5m/s$, above which the $V^{-1/4}$ law dominates.

The shallow indentation theory breaks down at $p \approx 10^{-1}, V \approx 100m/s$. The onset of hydrodynamic flow at $p \approx 10, V \approx 1000m/s$ is marked by plastic deformation outside the contact region.