# Calculating the Coefficient of Restitution Based on two Objects

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.

• This involves solving the partial differential equations describing the internal motion. When the two objects touch, at the point they touch the velocity (the derivative of the displacement vector) along the contact boundary is the same and the displacement vectors are zero. You impose a "no penetration condition", this gives you the initial conditions for the equation of motion of the elastic body. From the solution you deduce the moment the bodies move away, this is when the normal component of the stress across the boundary changes sign (the objects are not glued together). – Count Iblis Mar 23 '16 at 20:03
• I would think toughness and surface hardness would be important. Maybe even intrinsic structural damping. Pure elasticity, not so much. Although elasticity drives the impact time. – ja72 Mar 23 '16 at 21:18
• This coefficient is empirically derived for the specific situation and range of impact speed. It cannot be derived from first principles (at least not readily). Anyone else have a different opinion? – ja72 Mar 28 '16 at 18:05

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:

1. A body can deform in such a way that it returns all of it's gained energy - this is against the 2nd law of thermodynamics since work is done in deforming the body, or;
2. A body can instantly reverse direction with no deformation. This implies an infinite force that acts over zero time

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.

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):

1. Strain-rate independent loss in the material: any material that exhibits hysteresis during a deformation cycle ("internal friction") will exhibit loss of kinetic energy during collision.
2. Strain-rate dependent losses: many materials have different visco-elastic properties as a function of strain rate. This is seen in materials like Silly Putty, but there is a broad class of materials with such properties. This means that you need to determine the local strain rate (not just strain) during the collision to determine the loss.
3. Non-adiabatic losses in inflated balls: when the air in a ball is compressed during a collision, it is heated. Some of this heat will be given off to the material of the ball so the internal energy of the air is lowered and it is unable to "give back" all the energy
4. Interfacial friction: during a collision, the shape of the surface changes (like the flat spot on a tire touching the ground). The size of the load and the elasticity of the surfaces affects the size of the contact patch and the corresponding surface deformation. The deformation can lead to materials sliding over each other and dissipating energy (in essence not all motion perpendicular to the impact velocity is converted to elastic potential energy).

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).