Physics Asked by Limne on December 22, 2020
The coefficient of restitution is calculated based on the velocities of objects before and after a collision:
$$C_R = -frac{v_{2f} – v_{1f}}{v_{2i} – v_{1i}}$$
The coefficient of restitution tells us about the elasticity of a collision. What equation can calculate the coefficient of restitution based on the elasticity of the objects themselves?
An object’s elasticity describes it’s ability to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. These factors are related to the objects ductility and stiffness.
The coefficient of restitution tells you about the energy lost in the collision. Specifically e^2 is the ratio of the kinetic energy after to before the collision in the zero momentum frame. This depends not only on the elastic properties of the material, but also the structure of the body.
If you take a very simple example and have 2 springs hit each other head on then they will compress up to a point then begin to separate. At the point of separation the springs will still be compressed and therefore hold energy kx^2/2 each. This is the energy lost in the collision. If the springs were completely undamped then they would go on oscillating forever never losing this energy. In reality the spring will lose it's energy over a few cycles (as heating of the material) and in fact loses some during the initial compression phase also.
This is essentially what happens in a real collision of 2 objects. Perfect elasticity would imply one so 2 things:
All of the above is constrained to classical mechanics since the quantum reality is somewhat different.
So the coeff of rest tells you about the combination of material and structure. As an experiment you can drop a tennis ball from a height, then cut out a small patch from it and drop that from the same height.
Answered by Alan Swindells on December 22, 2020
Coefficient of restitution is closely related to energy loss mechanisms. Unfortunately those mechanisms don't just include the properties of the individual materials, but also their interface.
There are broadly four terms to consider (plus drag but that happens without the need for a collision):
I am not even mentioning processes that cause irreversible microstructure damage - they are somewhat implied in the first two points but often considered separate.
Given the above list, it is quite hard to determine the coefficient of restitution from first principles. You would have to do a numerical mode of the local deformation and combine it with stress-strain information obtained over a wide range of strain rates; then integrate over space and time. In many cases, the friction component is not negligible either (especially when one object is much softer than the other, like in the case of squash balls).
Answered by Floris on December 22, 2020
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