I'm sorry I did not understand. v What is the speed of the ball. vx1,vx2,vy1 or vy2.  complate code is;

-(void) tophareket {
ballspeedx=0; ballspeedy=4; balltwospeedx=0; balltwospeedy=-4;

// top top1'e de?erse
double  r1=19, r2=19, m1=1, m2=1, x1=ball.center.x,  y1=ball.center.y,  x2=balltwo.center.x,  y2=balltwo.center.y,
vx1=ballspeedx,  vy1=ballspeedy,  vx2=balltwospeedx,  vy2=balltwospeedy;

double dx = x2-x1, dy = y2-y1;
// where x1,y1 are center of ball1, and x2,y2 are center of ball2
double distance = sqrt(dx*dx+dy*dy);
// Unit vector in the direction of the collision
double ax=dx/distance, ay=dy/distance;
// 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+1)*(va2-va1)/(1+1);
double vaP2=va2 + (1+1)*(va1-va2)/(1+1);
// 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

ballspeedx =  vx1;
ballspeedy =  vy1;
balltwospeedx =  vx2;
balltwospeedy =  vx2;

dx=x2-x1, dy=y2-y1; d=sqrt(dx*dx+dy*dy);

double vp1= vx1 *dx/d+vy1*dy/d;
double vp2= vx2*dx+vy2*dy/d;
double dt =(r1+r2-d)/(vp1+vp2);

x1-= vx1*dt;
y1 -= vy1*dt;

ball.center = CGPointMake(x1, y1);
balltwo.center = CGPointMake(x2, y2);

x1+= vx1*dt;

ball.center = CGPointMake(x1, y1);
balltwo.center = CGPointMake(x2, y2);

ball.center = CGPointMake(ball.center.x + ballspeedx, ball.center.y + ballspeedy);
balltwo.center = CGPointMake(topbir.center.x + balltwospeedx, topbir.center.y + balltwospeedy);

and I tried the java programs you provide, tenth the collision happened. I wonder where I made a mistake?

-(void) tophareket {

// top top1'e de?erse
  double alpha,  m1=1,  m2=1,  r1=19,  r2=19,
x1=top.center.x,  y1=top.center.y,  x2=topbir.center.x,  y2=topbir.center.y,
vx1=topspeedx,  vy1=topspeedy,  vx2=topbirspeedx,  vy2=topbirspeedy,;
double r12=sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
double cs=(x2-x1)/r12; double sc=(y2-y1)/r12;
double vp1=vx1*cs+vy1*sc;
double vp2=vx2*cs+vy2*sc;
// back to collision time
double ddt=(r1+r2-r12)/(vp1-vp2);

x1-=vx1*ddt; y1-=vy1*ddt;
x2-=vx2*ddt; y2-=vy2*ddt;
r12= sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
//calculate component of velocity
cs=(x2-x1)/r12; sc=(y2-y1)/r12;
//  normal components do not change
double vn1=vx1*sc-vy1*cs;
double vn2=vx2*sc-vy2*cs;
//calculate component of velocity in original X,Y coordinate
double px1=vp1*cs; double py1=vp1*sc;
double px2=vp2*cs; double py2=vp2*sc;
double nx1=vn1*sc; double ny1=-vn1*cs;
double nx2=vn2*sc; double ny2=-vn2*cs;

topspeedx =  px1+nx1;
topspeedy =  py1+ny1;
topbirspeedx =  px2+nx2;
topbirspeedy =  py2+ny2;

int i=2;
top.center = CGPointMake(top.center.x + i*topspeedx, top.center.y + i*topspeedy);
topbir.center = CGPointMake(topbir.center.x + i*topbirspeedx, topbir.center.y + i*topbirspeedy);