I am trying to animate an elastic collision and I don't how to code to describe the motion after the impact. Here's where I'm at right now:

S1i[t_, V1i_] := 0.2 + t*V1i;
S2i[t_, V2i_] := 0.8 + t*V2i;
S1f[t_, V1f_] := t*V1f;
S2f[t_, V2f_] := t*V2f;
V1f[m1_, m2_, V1i_, V2i_] := 2 (m1*V1i + m2*V2i)/(m1*m2) - V1i;
V2f[m1_, m2_, V1i_, V2i_] := 2 (m1*V1i + m2*V2i)/(m1*m2) - V2i;
ts := (0.6 - r1 - r2)/(V1ie - V2ie);
y = 0.4;
r1 = 0.06;
r2 = 0.06;
V1ie = 0.2;
V2ie = -0.1;
m1e = 0.03;
m2e = 0.03;
V1fe = 2 (m1e*V1ie + m2e*V2ie)/(m1e*m2e) - V1ie;
V2fe = 2 (m1e*V1ie + m2e*V2ie)/(m1e*m2e) - V2ie;
xy[t_] := {S1i[t, V1ie], y};
zy[t_] := {S2i[t, V2ie], y};
py[t_] := {S1f[t, V1fe], y};
qy[t_] := {S2f[t, V2fe], y};
traj1 = Table[xy, {t, 0, ts, 0.1}];
traj2 = Table[zy, {t, 0, ts, 0.1}];
traj3 = Table[py, {t, ts, 15, 0.1}];
traj4 = Table[qy, {t, ts, 15, 0.1}];
Manipulate[
 Show[ListPlot[traj1], ListPlot[traj2], ListPlot[traj3], 
  ListPlot[traj4],
  Graphics[{
    Black, Circle[xy[t], r1], Red, Circle[zy[t], r2]}],
  AspectRatio -> Automatic], {t, 0, 10}]

where S1f and S2f describe the motion after the impact of the Black Circle and the Red Circle, respectively. Also, ts is the time of the impact. I should also mention that whatever I'd tried didn't lead anywhere.

One more question: why doesn't the animation stop at the point of impact unless \I set the ruler to be {t,0,ts} in the Manipulate function. Shouldn't it stop because of this piece of code?:

traj1 = Table[xy, {t, 0, ts, 0.1}];
traj2 = Table[zy, {t, 0, ts, 0.1}];

Thank you in advance, and I am begging you to be somewhat thorough with your answer as I've just started using Mathematica. :)