Conservation of Linear Momentum+ Elastic and Inelastic collision

[Nanako]

hi everyone. I'm working on a physics engine as part of a game I'm making, a hobby project. Up until this point I've implemented inter object collisions in a rather quick and dirty manner. Whenever something moves into something else, it is simply teleported back out to the edge. This implementation worked for a while, but amongst other problems, it causes anything not currently moving to be an immovable object, which is not good.

Since i already have support for forces, impulses, and friction, I've decided it's time to re-examine my collisions and make a better model for it.

I believe the formula i need for elastic collision is here: http://en.wikipedia.org/wiki/Momentum#Conservation_of_linear_momentum

the big pair of formulae, just under where it says "In one dimension" in bold letters. however, i need a little bit of help rearranging it. Those formulae give the final velocities of the object as an output. But i'd like to not go that far. The output I'd like is the Impulse to apply to both objects, in Ns (Newton-seconds). As i understand, both objects in a collision will receive the same impulse. I want this so that i can do other (game related) things with it before applying it. If anyone could help with adjusting the formula to give me that, it'd be helpful.


The second issue i have though, is that the formula only covers perfectly elastic collisions. And there's a separate formula for perfectly inelastic collisions. This seems inherently wrong to me, as nothing in nature is completely one or the other.

What i'd like to do is to figure out a way to incorporate the coefficient of restitution into things (the measure of elasticity between the objects). Considering the problem, i believe it will work similar to the coefficient of friction, in that either component having low elasticity will drive the coefficient towards zero. For example, a snowball thrown at a window will smush against it rather than bouncing off, even though one of the two is highly elastic, because the other is not. This is mostly just logical thought on my part.

I already have a simple formula for calculating the friction coefficient that i would probably reuse (sqrt(ab)). Assuming i keep within the bounds of a and b always being between 0 and 1, that would ensure my coefficient is always likewise.

Once i have a coefficient of restitution, how to incorporate it is the next step. I really don't know here, but logic tells me that something like multiplying the impulse by the coefficient, would work well.

what i ideally want to do is to be able to assign individual elasticity values to objects within my engine, work out a coefficient between them. Work out a collision impulse for the collision based on the object's masses and velocities, and figure out a way to combine the impulse and coefficient, before applying the impulse to the objects. I believe this should give me a more well rounded collision simulation, but i need some assistance to fill in the blanks.

I'm interested in any thoughts and assistance anyone can provide on these two subjects

 

[Nanako]

bump. can anyone assist ?

 

[A.T.]

You have two things:
- Total momentum vector : always conserved 100%
- Total kinetic energy : conserved 100% only in perfectly elastic collisions. Otherwise reduced depending on the coefficient of restitution the mass ratio and the frame of reference.

 

[Nanako]

Hello everyone ^^ I got sidetracked from this for quite a while working out other collision related things. Now it's time to pick up and actually start it. Unfortunately i still need lots of help.

So then, i'll start with a few questions that's on my mind.


1.
If momentum equals mass*velocity, and momentum in a closed system is always conserved, then how is it possible for a collision to result in reducing the velocity of both objects? If one object is slowed by a collision, isn't it necessary that the other gains velocity to keep the total momentum the same?



2.
I'm also looking for some clarification of this quote:

In multiple dimensions

In the case of objects colliding in more than one dimension, as in oblique collisions, the velocity is resolved into orthogonal components with one component perpendicular to the plane of collision and the other component or components in the plane of collision. The velocity components in the plane of collision remain unchanged, while the velocity perpendicular to the plane of collision is calculated in the same way as the one-dimensional case.

For example, in a two-dimensional collision, the momenta can be resolved into x and y components. We can then calculate each component separately, and combine them to produce a vector result. The magnitude of this vector is the final momentum of the isolated system.

I already have my velocities stored as a two-part vector (this is 2D. x and y) but I'm having trouble understanding how this relates. Would i calculate the x momentum from the x velocity, and do all the calculations on that value, then repeat it for the y? or is it more complex than that ?


3.
I'm now unsure if i really understand impulse very well. From a quick glance here: http://en.wikipedia.org/wiki/Newton-second
It appears that the Newton-second is the unit of momentum, as well as impulse. This is a link didn't quite make mentally. Does that then mean that i can apply the total momentum of the system, as an impulse, to achieve the desired change in velocity? (of course, one object will receive the impulse*-1). Or would it perhaps be half of the total system momentum ?

although even then, I'm sure this would only give me the magnitude of the impulse value. I still need to directionalise it. This I'm not sure about, any thoughts are appreciated.

An impulse value is my ultimate aim for this exercise, as i want to be able to do game-y things to it before actually applying it.

 

[PJC01]

.

I appreciate your dedication to improving your physics engine and incorporating more realistic models for collisions. Conservation of linear momentum is a fundamental law in physics, stating that the total momentum of a system remains constant unless acted upon by an external force. This law is crucial in understanding and predicting the motion of objects in collisions.

In elastic collisions, the total kinetic energy of the system is conserved, meaning that the objects involved do not lose any energy during the collision. This is reflected in the equations you have referenced, where the final velocities of the objects are equal to their initial velocities. However, as you mentioned, perfectly elastic collisions are rare in nature and it is important to incorporate a coefficient of restitution to account for the loss of energy in a collision.

The coefficient of restitution is a measure of the elasticity between two objects, and as you correctly stated, it varies between 0 and 1. A value of 0 represents a perfectly inelastic collision, where the objects stick together after colliding, while a value of 1 represents a perfectly elastic collision. To incorporate this into your collision model, you can use the equation you mentioned, where the impulse is multiplied by the coefficient of restitution.

I also appreciate your idea of assigning individual elasticity values to objects within your engine. This would allow for more realistic and varied collisions between objects. I would suggest using a coefficient of restitution calculator to determine the coefficient between two objects based on their material properties. This will give you a more accurate value to use in your calculations.

Overall, I think your approach to improving your collision model is sound and I encourage you to continue exploring and experimenting with different methods. As a scientist, it is important to constantly seek improvement and refinement in our models and theories. Good luck with your project!