How can I accurately simulate collisions between circles and inclined planes?

[StefenRoebke]

I am writing a 2D simulation that involves multiple circles colliding with inclined planes.

I want to collision response to be as true-to-life as possible but have hit a bit of a snag.

Depending on the velocity of the circles, when colliding with the inclined planes, they may either bounce off the plane, or they may not bounce and just begin rolling down the plane.

Is there a single mathematical model which would accomidate bothe of these instances. I am trying to make the calculations as straight forward as possible, without any conditionals.

 

[sir_manning]

Some ideas:

There are a number of factors to take into account if you want to improve the realism. Since you mentioned you want some of the circles to roll down the planes, I'm assuming this simulation involves gravity. Correct?

If so, the equations that describe the motion of the circles before and after contact with the planes will be the kinematics equations, with your acceleration as g.

For the collision with the plane, you can use the conservation of momentum. Are your circles going to bounce without loosing any energy (elastic collision) or bounce with loosing energy (inelastic collision)? If it's the latter, a frictional impulse force will act against the ball's momentum when colliding with the plane, which will reduce the outgoing momentum. I'm guessing this impulse force would be a function of the circle's collision speed (which you could control to get a wimpy squishy circle or a hard bouncy circle), but I'm not sure.

As for the rolling part, each circle will start spinning in the air right after its first bounce. I don't know how you'd implement this. However, if you ignored this, you could set a circle to start rolling down a plane after it has "stopped" bouncing. (You might say a ball has stopped after it no longer bounces off the plane with a certain magnitude of momentum.)

Modeling the rolling will require that you define a moment of inertia for your circles: are they going to roll easily, like a wooden ball, or are they going to roll slower, like a lead ball? An easier approach (although less realistic) would be to have the circles slide down the ramp as opposed to rolling... depending on how you render the circles in your simulation (can you tell if they're spinning?), it might not look that different. If you did want to model the rolling, hopefully someone else will point your in the right direction!

 

[StefenRoebke]

Well I have the rolling implemented just fine on flat surfaces. I basically take the distance traveled and divide it by pi*D to get the percentage of rotation, and multiply that by 2PI to get the radians of rotation.

The way I was going to implement it was how you stated above, it a condition to state it is no longer bouncing, but rolling. I was hoping I could just have a single calculation that would take into account rolling and bouncing.