# Is there a way to theoretically estimate the restitution coefficient of a rubber ball?

Let's consider a rubber ball like the ones shown in the picture:

When dropped onto the floor, the ball will rebound, but it won't regain its initial height.

The restitution coefficient is defined as the ratio of the speed immediately after the collision to the speed immediately before the collision:

$$R = \frac{v_{\rm after}}{v_{\rm before}}.$$

Is there a way to theoretically estimate the restitution coefficient of such a rubber ball? How close can it be to unity?

I am looking for a straightforward estimate based on an analysis of the collision dynamics. The estimate may depend on the diameter of the ball, constants related to the rubber's properties, and the incident speed.

However, it doesn't need to be overly precise or rely on complex mathematical equations. Rather, I'm interested in simple equations and reasoning that provide an order-of-magnitude estimate of the difference between the restitution coefficient and unity, similar to how physicists make rough estimates.

Here are the assumptions:

1. The rubber should be assumed to be of the highest quality (among realistically available rubbers) in terms of achieving the highest restitution coefficient.

2. The incident speed should be assumed to correspond to a drop height of about one meter.

3. The ball should be assumed to be more or less of standard size, 3-8 cm in diameter.

I understand that the restitution coefficient may be much easier to measure experimentally by measuring the drop height and the rebound height, but I'm interested whether a rough theoretical estimate can be made based on an analysis of the collision dynamics - and, if so, how.

• I don't know about how the properties you mentioned would affect the coefficient of restitution. But I feel that you can simply calculate the coefficient of restitution by measuring the height reached, like drop it from certain height, calculate the velocity when it touches the ground, earth is massive so we can assume no velocity is imparted to it. had it been $e$=1, the velocity of the ball after impact would have been same. Now calculate the height reached, you can calculate the velocity just after the impact from this data. The ratio of $v_{final}/ v_{initial}$ should give you $e$ (or $R$) Commented Mar 15 at 11:46
• @PinkAura No, I want an estimate that doesn't use any measurements, including measurements of heights Commented Mar 15 at 11:48
• @Mitsuko I'm afraid, there's no simple theoretical estimates (without sort of measurements needed), because modelling restitution coefficient depends on many parameters, like young modulus of material, radius, homogeneity, temperature, inside structure (hollow or not, etc). And still it's not enough because it will also depend on properties on the surface of ground where ball will scatter upon, like also ground elasticity, etc. So to say coefficient of restitution is coupled property of ball-ground, that's why only "convoluted" modelling theories exists. Commented Mar 15 at 13:59
• @Mitsuko Why don’t you cite the elasticity literature you’ve researched to clarify the gap that you think should be filled. What is the simplest model that you consider still too “convoluted”? You don’t want to do experiments, you don’t want to use existing tables, you don’t want to conduct a comprehensive analysis. I know you have the capacity to research and apply—perhaps simplify—a model of mechanical hysteresis in elastomers, which is a key phenomenon here preventing perfect rebound. Commented Mar 15 at 15:43
• Unfortunately, there is no absolute coefficient of restitution. COR is a value between two objects. A ball bouncing against concrete will have a high COR while the same ball bouncing against compact dirt will have a lower value. Commented Mar 15 at 17:11

If you are interested in a rough estimate, the kinetic energy of the rubber ball will be converted into potential elastic energy but for some dissipation. In solid materials with little dissipation one way describe this using the Zener model through a loss angle $$\phi$$ that modifies Hooke's law like this: the elastic constant $$k \rightarrow k(1 + j \phi)$$: an imaginary part $$j k \phi$$ is introduced that results in dissipation. Same happens for Young's modulus E.
With $$\phi \ll 1, 2 \pi \phi$$ indicates the fraction of stored energy that is lost in a compression-expansion cycle of the ball. $$\phi \pi$$ measures the relative energy loss during the bounce half cycle and so the relative height loss between two bounces. This neglects further tangential friction between ball and ground. As said by Agnius Vasiliauskas $$\phi$$ depends on the material, but also on thermal treatment and deformation history. For glass $$\phi$$ can be as low as $$10^{-7}$$, for harmonic steel $$10^{-4} - 10^{-3}$$, for rubber 0.05-0.1. There, little can be done starting from theory only. The restitution coefficient, which relates speeds, is $$R=\sqrt{1-\phi}$$.