Hopefully this is not a fail. But, before I explain the math, I think it will be easier if I give a background of what I'm trying to do...
Imagine 2 line segments randomly defined in a $2D$ room. The line segments are 2D... Both are given a speed $v$ and $\omega$ (angular velocity). The point of the math, is to find the exact future time $t \gt 0$ where the first collision will occur.
The motion of the line in space-time might be considered as a parametric surface (space-time for a $1d$ object, so it is $2d$.)
The math of the lines are:
Line1 X Output: $u_1 \sin(\omega_1 \ t) + \ v_{1x} \ t+ x_{01} $
Line1 Y Output: $u_1 \cos(\omega_1 \ t) + \ v_{1y} \ t+ y_{01} $
and
Line2 X Output: $ u_2 \ \sin(\omega_2 t) + v_{2x} t + {x_{02}}$
Line2 Y Output: $ u_2 \ \cos(\omega_2 t) + v_{2 y} t + {y_{02}}$
You can consider $\omega_1,\omega_2,v_1,v_2,,x_0, y_0$, as "unchanging" or "constant" variables, in the context of this mathematics. t is time that increments over time, u is the span(breadth) of the line, limited to a<u<b. [a,b] are constants that can be arbitrarily defined in the same way of $\omega_1,\omega_2, v_1 \dots $ etc.
What I've tried so far:
I've reformulated some of the variable names to solve for $t_1, t_2$ like this:
$u_1 \ \sin(\omega_1 t_1) + v_{1x} \ t_1 + x_{01} = u_2 \ \sin(\omega_2 t_2) + v_{2x} t_2 + x_{02}$
$ u_1 \ \cos(\omega_1 t_1) + v_{1y} \ t_1 + y_{01} = u_2 \ \cos(\omega_2 t_2) + v_{2y} \ t_2 + y_{02} $
Where do I go from here?
My question is this mathematically impossible, or can this be done? If so, how? My other question is, have I gotten all of my math terminology correct? If I don't have the right math terminology then its harder to find help on search engines.