hello professor
i read your equations about 2d collision but did not understand these steps

// Projection of the velocities in these axes
double va1=(vx1*ax+vy1*ay), vb1=(-vx1*ay+vy1*ax);
double va2=(vx2*ax+vy2*ay), vb2=(-vx2*ay+vy2*ax);
// New velocities in these axes (after collision): ed<=1,  for elastic collision ed=1
double vaP1=va1 + (1+ed)*(va2-va1)/(1+mass1/mass2);
double vaP2=va2 + (1+ed)*(va1-va2)/(1+mass2/mass1);
// Undo the projections
vx1=vaP1*ax-vb1*ay;  vy1=vaP1*ay+vb1*ax;// new vx,vy for ball 1 after collision
vx2=vaP2*ax-vb2*ay;  vy2=vaP2*ay+vb2*ax;// new vx,vy for ball 2 after collision

why we need vb1 and vb2 ?? and also what is your means by undo projection..
actually sir i am doing work on molecular dynamics in gas and when particals are collide we use your model of 2d collision is this correct for us to use these equations at the time of collision ??