# How can I derive a relationship between temperature and coefficient of restitution for a rubber bouncy ball theoretically?

For a high school research project, I am exploring the relationship between temperature and coefficient of restitution for a rubber bouncy ball. I have already carried out the experiment and have formed a relationship practically. However, I would also like to derive a theoretical formula for my hypothesis which I can then use to calculate the theoretical value of coefficient of restitution for the rubber ball at different temperatures, post which I can compare this with the values I obtained experimentally.

I carried out the experiment in terms of height and therefore used the below formula to calculate COR at each temperature:

$$\text{Coef. of Rest}=\sqrt{\frac{\text{Height }R}{\text{Height }D}}$$

where HeightR is the rebound height upon the first bounce and HeightD is the height the ball was dropped from.

I would appreciate any guidance on how I should derive an equation to relate coefficient of restitution to temperature for a rubber bouncy ball theoretically so that I progress in the correct direction. Just to clarify, I want to derive an equation for coefficient of restitution in terms of the temperature which the ball is at.

I want to derive an equation for coefficient of restitution in terms of the temperature which the ball is at.

The CoR is related to the concept of Elastic Hysteresis:

Elastic Hysteresis is the difference between the strain energy required to generate a given stress in a material, and the material's elastic energy at that stress. This energy is dissipated as internal friction (heat) in a material during one cycle of testing (loading and unloading).

When mechanical test data is plotted on a stress/strain curve, a material exhibiting elastic hysteresis will display one path during the loading phase of the test, and a different path during the unloading phase. The two paths will clearly diverge due to hysteresis loss (energy loss in the form of heat), with the area between the curves representing the energy dissipated.

Graphically:

Now you face two main problems.

1. Relation between simple 'lab' hysteresis and complex 'real life' hysteresis:

The Elastic Hysteresis of a visco-elastic material (like rubber) can be established using simple geometries like standardised dumbbells, where very simple deformations are in play but to try and predict the hysteresis of something as complex as a bouncing ball from these simple lab data is a different matter altogether.

1. Influence of temperature on Elastic Hysteresis:

A theoretical model linking these two variables is not available as far as I know.

Some relevant research has almost certainly been carried out in the field of (automotive) timing belts, as the energy losses due to hysteresis is an area of concern in that field of application. So it might be an idea to trawl the specialised trade literature for some insights.

So rather than try and develop a complex and potentially quite faulty theoretical relation between the CoR and temperature it might be much preferable to develop an empirical model.

From the coefficient of restitution you can calculate the loss of the macroscopic kinetic energy of the ball, macroscopic meaning the loss of external kinetic energy of the ball as a whole ($$1/2mv^2$$).

Then, neglecting sound and any possible inelastic deformation of the surface on which the ball bounces, you can assume all of the loss macroscopic kinetic energy equals the increase in microscopic kinetic energy of the molecules of the ball, its internal kinetic energy, which results in an increase in the temperature of the ball.

Finally, knowing the mass of the ball and the specific heat $$C$$ of its material, you can calculate the theoretical increase in temperature from

$$\frac{1}{2}mv^{2}=mC\Delta T$$

Where the left side of the equation is the loss of KE of the ball.

The theoretical temperature increase is likely to be small and not uniform throughout the ball. Moreover, the increase in temperature relative to the environment will result in heat transfer to the lower temperature environment. In short, don't expect to be able to actually measure the temperature increase, or at least don't expect to be able to measure it accurately.

Hope this helps.

• Thank you very much for your response but I think I wasn't clear enough with my question, apologies. I want to derive an equation for coefficient of restitution in terms of the temperature which the rubber ball is at. For example, if we were to take a temperature of 40 degrees Celsius, I would like to derive an equation to calculate coefficient of restitution of the rubber ball at this temperature. – Harsh Mishra May 2 at 16:29
• As far as I know there is no equation relating the absolute temperature of the ball to the coefficient of restitution (COR) You can only relate the COR to the CHANGE in temperature – Bob D May 2 at 16:36
• I see. Thank you very much for all your help! – Harsh Mishra May 2 at 17:04
• @BobD I was a rubber technologist for about 10 years and I can assure you the CoR depends strongly on the temperature it is being measured at. – Gert May 2 at 17:38
• @Gert Interesting, but I'm not sure what that has to do with it as long as you don't presume what the COR is. You do a test at a specified temperature and whatever the COR is you can determine the theoretical temperature rise based on that value of the COR. Don't you agree? – Bob D May 2 at 18:02