Scratch

I thought you just do conservation of momentum? And use the coefficient of restitution for the material to work out the "bounciness" for elastic collisions? I only say this because we did it in mechanics just this morning... 

Last edited by blob8108 (2012-10-18 11:33:56)

Does this work? It's only for scalar velocities, but hopefully the maths is equally valid for vectors.

First object:
m_1     — mass
u_1     — velocity before collision
v_1     — velocity after collision (find)

Similarly for second object...
m_2     — mass
u_2     — velocity before collision
v_2     — velocity after collision (find)

e       — coefficient of restitution
When e=0:       inelastic collision, v_1 = v_2
When 0 < e < 1: elastic
When e=1:       "perfectly elastic"



Momentum is conserved:
m_1*u_1 + m_2*u_2  =  m_1*v_1  +  m_2*v_2

m_2*v_2  =  m_1*u_1 + m_2*u_2 - m_1*v_1

v_2  = (m_1*u_1 + m_2*u_2 - m_1*v_1) / m_2



Coefficient of restitution:

e  =  (v_2 - v_1) / (u_1 - u_2)

v_2 - v_1  =  e*(u_1 - u_2)

v_2  =  e*(u_1 - u_2) + v_1


Combining the equations to find v_1:

e*(u_1 - u_2) + v_1  =  (m_1*u_1 + m_2*u_2 - m_1*v_1) / m_2

e*(u_1 - u_2) + v_1  =  (m_1*u_1 + m_2*u_2) / m_2  -  m_1*v_1/m_2

v_1  =  (m_1*u_1 + m_2*u_2) / m_2  -  m_1*v_1/m_2  - e*(u_1 - u_2)

v_1  +  m_1*v_1/m_2   =  (m_1*u_1 + m_2*u_2) / m_2 - e*(u_1 - u_2)

v_1(1  +  m_1/m_2)   =  (m_1*u_1 + m_2*u_2) / m_2 - e*(u_1 - u_2)

v_1  =  ((m_1*u_1 + m_2*u_2) / m_2 - e*(u_1 - u_2)) / (1  +  m_1/m_2)
~

then, find v_2:

v_2 - v_1  =  e*(u_1 - u_2)

so

v_1  =  v_2 - e*(u_1 - u_2)

Hope that works...

Note to self: do maths on *paper* next time. 

Last edited by blob8108 (2012-10-18 14:51:58)

Thanks for the help; I wouldn't mind seeing what you got, if it's not too much trouble (I assume you don't sit next to scanners 24/7, so I may be asking for a lot).

I tried the Wolfram|Alpha equation, it didn't seem to work right. Here's the JS which does the force calculations, if you want to take a look. Note that I messed up big time: the exertForce function actually takes a velocity, not a force, as input, so basically you should divide by the mass of the body before passing forces to that function. I don't know why that happened, could have been due to the fact that I was coding at 10PM. 

var effectiveForceLine = new KMVector(body.x-otherBody.x, body.y-otherBody.y); // Effective vector along which forces of collision act
var effectiveForceBody = body.velocity.projectOn(effectiveForceLine).total();  // Effective force exerted by body
var effectiveForceOther= otherBody.velocity.projectOn(effectiveForceLine).total(); // Effective force exerted by other body
if (otherBody.fixed) {// Ignore this, this is basically for universally fixed objects (the ground)
    // ???
} else {
    // Wolfram formula:
    var final1 = ((body.mass*effectiveForceBody)-(otherBody.mass*effectiveForceBody)+otherBody.mass*(effectiveForceOther-effectiveForceBody))/(body.mass+otherBody.mass);
    
    body.exertForce(effectiveForceLine.resizeTo(effectiveForceBody)); // Stop the body. I don't particularly like doing this, it's ugly. I should be applying just one force, right?
    body.exertForce(effectiveForceLine.resizeTo(final1)); // Apply the relevant force to this body
}

blob8108 wrote:

Hardmath123 wrote:

Thanks for the help; I wouldn't mind seeing what you got, if it's not too much trouble (I assume you don't sit next to scanners 24/7, so I may be asking for a lot).

Sadly not    But here, I tried it again for you. Although I got a wrong sign this time... 

Cool, that's exactly what Wolfram says. So my coding is wrong.

Did you use Penultimate or something for that?

I tried the Wolfram|Alpha equation, it didn't seem to work right.

Hmm. It should work, as far as I know... How didn't it work? Maybe the conversion to vectors breaks it. 

The end velocities didn't match up to what they should be in obvious cases (i.e. one static body vs. one moving one, etc.)

...the exertForce function actually takes a velocity, not a force, as input, so basically you should divide by the mass of the body before passing forces to that function. I don't know why that happened, could have been due to the fact that I was coding at 10PM. 

I thought force / mass = acceleration, not velocity?

Right, acceleration changes the velocity of the body. So basically the exertForce funciton should be renamed accelerateByVector or something.

Hardmath123 wrote:

blob8108 wrote:

Hardmath123 wrote:

Thanks for the help; I wouldn't mind seeing what you got, if it's not too much trouble (I assume you don't sit next to scanners 24/7, so I may be asking for a lot).

Sadly not    But here, I tried it again for you. Although I got a wrong sign this time... 

Cool, that's exactly what Wolfram says. So my coding is wrong.

I'm not sure I understand your code... 

Did you use Penultimate or something for that?

I used InkBook on OS X — I have one of those Wacom Bamboo Pen tablet thingies 

Thought (sorry if this is obvious, don't mean to offend!): are you handling the collision in one go for both objects, or are you iterating through all the objects and handling collisions for each object? If the latter, have you already updated the velocity by the time you handle the collision for the second object? I remember thinking my physics library would have been much better if I'd thought about handling each collision in one go... I imagine you've already thought of this, however 

... very stupid mistake ... now it works.