Predicting Coefficients of Restitution

[MCGenco]

Hello everyone, this is my first post here, so please try to be patient with me. Thanks! :)

I am trying to build a very basic physics engine, and seek a method to predict the coefficient of restitution for a collision between two objects by measuring those for the collisions of each object separately with a third object. I've tried to find a list of coefficients of restitution for multiple collisions with which I could compare any results, but was unable to do so. I would have empirically gathered the data myself, but unfortunately I do not have the means to perform any experiments.

I am particularly interested in the specific scenario of predicting the coefficient of restitution for a collision between two circular discs knowing those of the theoretical collisions of each disc separately with an immovable surface. However the solution to the more generic scenario would be preferred.

My Question:
Given only the coefficients of restitution for collision 1, between objects A and C, and collision 2, between objects B and C, is it possible to determine the coefficient of restitution for collision 3, between objects A and B, mathematically? Would there be any important restrictions to consider? How would this compare with empirical data?

Thanks in advance to everyone who reads this!

- MCGenco

 

[eljose79]

Hello MCGenco, welcome to the forum!

Your question is a great example of how physics principles can be applied to real-world scenarios. To answer your question, yes, it is possible to mathematically determine the coefficient of restitution for collision 3 using the coefficients of restitution for collisions 1 and 2. This can be done by using the principle of conservation of energy and momentum.

First, we need to understand the concept of coefficient of restitution. It is a measure of the elasticity of a collision, and is defined as the ratio of the relative velocity of separation to the relative velocity of approach. In simpler terms, it tells us how much energy is lost or retained during a collision.

Now, let's apply this to your scenario. We have two objects, A and B, colliding with each other, and we know the coefficients of restitution for their individual collisions with an immovable surface (object C). We can use the following equation to determine the coefficient of restitution for collision 3:

e3 = (v3f - v3i) / (v3i - v3c)

Where:
e3 = coefficient of restitution for collision 3
v3f = final relative velocity of separation between A and B
v3i = initial relative velocity of approach between A and B
v3c = relative velocity of approach between A and B and C

To find v3f, we can use the conservation of momentum principle, which states that the total momentum of a closed system remains constant. Therefore, we can say:

mAv3i + mBv3i = mAv3f + mBv3f

Similarly, to find v3c, we can use the conservation of energy principle, which states that the total energy of a closed system remains constant. Therefore, we can say:

1/2mAv3i^2 + 1/2mBv3i^2 = 1/2mAv3c^2 + 1/2mBv3c^2

Now, we have all the necessary values to plug into the equation for e3. This will give us the coefficient of restitution for the collision between A and B.

It is important to note that this mathematical solution assumes that the collisions are perfectly elastic and there are no external forces acting on the system. In reality, there may be other factors that can affect the coefficient of restitution, such