Take parametric equations for the objects' motions and plug them into the distance formula (squaring both sides): distance formula squared = (a + txvel1 - (c + t(xvel2))^2 + (b + tyvel1 - (d + t(yvel2))^2. Remember, t is going to be negative. To find the time of collision for two circular objects we set the left side equal to (r+s)^2. Solving for t using the quadratic equation (and a great deal of very tedious algebra), we get the above "t=..." equation.

Take parametric equations (for example, newxpos1 = a + tw) for the objects' motions and plug them into the distance formula (squaring both sides): distance formula squared = (a + tw - (c + tx))^2 + (b + ty - (d + t*z))^2. Remember, t is going to be negative. To find the time of collision for two circular objects we set the left side equal to (r+s)^2. Solving for t using the quadratic equation (and a great deal of very tedious algebra), we get the above "t=..." equation.

FINAL EDIT/UPDATE: Answered, see my answer below.

EDIT/UPDATE: My biggest question right now is whether step 3's "t=..." equation is a good idea or there a better way to do it. Most other issues have been partially or fully addressed, but no comments or answers have really touched on this issue. Again, an analytic solution is probably required, the velocities and distances are too large, and the objects are too small, for any iterative/recursive solution (a few are suggested below in the comments) that I can think of (although if there is a special iterative/recursive solution that will handle these kinds of situations fine then I'm definitely open to it). Thank you very much for your help so far, you all are awesome and I really appreciate your thoughts and help!

EDIT/UPDATE: My biggest question right now is whether step 3's "t=..." equation is a good idea or there a better way to do it. Most other issues have been partially or fully addressed, but no comments or answers have really touched on this issue. Again, an analytic solution is probably required, the velocities and distances are too large, and the objects are too small, for any iterative/recursive solution (a few are suggested below in the comments) that I can think of (although if there is a special iterative/recursive solution that will handle these kinds of situations fine then I'm definitely open to it). Thank you very much for your help so far, you all are awesome and I really appreciate your thoughts and help!

EDIT/UPDATE: My biggest question right now is whether step 3's "t=..." equation is a good idea or there a better way to do it. Most other issues have been partially or fully addressed, but no comments or answers have really touched on this issue. Again, an analytic solution is probably required, the velocities and distances are too large, and the objects are too small, for any iterative/recursive solution (a few are suggested below in the comments) that I can think of (although if there is a special iterative/recursive solution that will handle these kinds of situations fine then I'm definitely open to it). Thank you very much for your help so far, you all are awesome and I really appreciate your thoughts and help!

FINAL EDIT/UPDATE: Answered, see my answer below.

EDIT/UPDATE: My biggest question right now is whether step 3's "t=..." equation is a good idea or there a better way to do it. Most other issues have been partially or fully addressed, but no comments or answers have really touched on this issue. Again, an analytic solution is probably required, the velocities and distances are too large, and the objects are too small, for any iterative/recursive solution (a few are suggested below in the comments) that I can think of (although if there is a special iterative/recursive solution that will handle these kinds of situations fine then I'm definitely open to it). Thank you very much for your help so far, you all are awesome and I really appreciate your thoughts and help!