"Andrew Coppin" <orp### [at] btinternetcom> wrote in message
news:3fb53d4c$1@news.povray.org...

> ....
> And the biggy:
>
> #declare SA = sphere {<0, 0, 0>, 1}
> #declare SB = sphere {<+8, 0, 0>, 1}
>
> SA is stationary. SB is travelling along the X-axis. Since the two have
> equal mass (no, they do, cos it's my universe and I SAID SO :-), when SB
> hits SA, SB will come to a complete stop, and SA will continue on with
> (roughly?) the same velocity as SB had.
>
> How suppose we change the start conditions:
>
> #declare SA = sphere {<0, 0, 0>, 1}
> #declare SB = sphere {<+8, +0.2, 0>, 1}
>
> Unless I'm horribly mistaken, SB should bounce off SA at an angle (WHICH
> angle??), transferring only some (HOW MUCH?) of its momentum to SA.
>
> Anyone care to add details?
>
> Thanks.
> Andrew.
>

Andrew, you need to build two macro's out off the following theory:

1.    Detect collision.
**********************************************************
 The movement of a particle is given by:
        s =   s0 +     v.t
 The vector notation for particles A and B:
        vecSa = vecA + vecVa.t
 , with vecSa = position at time t,
        vecA  = position at time zero,
        vecVa = velocity, so:

        vecSb = vecB + vecVb.t

 The distance between A and B is:
        vecS  = vecSa - vecSb
                = (vecA + vecVa.t) - (vecB + vecVb.t)
                = (vecA - vecB) + (vecVa - vecVb).t
                =      vecD     +       vecV     .t
        vecS  = vecD + vecV.t

 When there is collision then:
        R = rA + rB = |vecS|
        R^2 = vdot(vecS,vecS)
        R^2 = vdot((vecD + vecV.t),(vecD + vecV.t))

 this gives:
        vdot(vecV,vecV).t^2 + 2.vdot(vecD,vecV).t
         + (vdot(vecD,vecD) - R^2) = 0

 The Discriminant of this equation is:
        Disc = 4.vdot(vecD,vecV)^2 - 4.vdot(vecV,vecV).(vdot(vecD,vecD)-R^2)

 CollisionTimes:

        t = _______________________________
              2.vdot(vecV,vecV)

 Eliminating the factor 2:

         t = _______________________________
                  vdot(vecV,vecV)

 The smallest time is:
                -sqrt(D) - vdot(vecD,vecV)
        t = _____________________________
                  vdot(vecV,vecV)
with:
        D = vdot(vecD,vecV)^2 - vdot(vecV,vecV).(vdot(vecD,vecD)-R^2)

Condition for collision:  D > 0      (if D = 0 they only touch each other)

 If t < 0 then this was a collision in the past and you have to try the
other solution as well.
*********************************************************

2. Calculate new velocities if collision takes place.
*********************************************************
  When 2 particles collide, they bounce from a plane perpendicular to their
  distance vector vecD.  The normal of this plane is the distance vector:
        vecN = vecD.

  The working lines of their velocities are given by vecVa_1 and vecVb_1
  and will normally cross each other.  We need to split these vectors in a
  component along the normal and a component in the colliding plane.
  (Make a 3D-drawing if you can't imagine this.)
  Along the normal we need the speed of the particles (Speed is velocity
  without the direction, so it is a scalar).  This is given by:
        Va_n_1 = vdot(vnormalize(vecN),vecVa_1)
        Vb_n_1 = vdot(vnormalize(vecN),vecVb_1)

  After the collision we get new speeds:
        Va_n_2 and Vb_n_2

  The new velocities are given by:
        vecVa_2 = vecVa_1 - (Va_n_1 - Va_n_2).vnormalize(vecN)
        vecVb_2 = vecVb_1 - (Vb_n_1 - Vb_n_2).vnormalize(vecN)

  Collision :
         Particle speeds:
                Va_1 and  Vb_1 (before)
                Va_2 and Vb_2 (after)
         Particle masses:
                Ma   and   Mb
         Reduced masses:



        Equations to solve:
                Ma.Va_1 + Mb.Vb_1 = Ma.Va_2 + Mb.Vb_2
                             Va_1 - Vb_1 = -e.(Va_2 - Vb_2)
         For molecules:          e = 1             (e = elasticity)

         Answers for elasticity is 1:



  In order to get a correct answer, we have to determine the relative
  signs of Va_1 and Vb_1, that will say:
        are they moving in the same or in the opposite direction along
        the normal of collision. This is done automatically by
            Vx_n_1 = vdot(vnormalize(vecN),vecVx_1), with x = a or b.
**********************************************************

You can count on the correctness of this theory, for I've used it in an
animation that I will place in p.b.a.

Succes with it,

Jaap Frank

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