$\text{comet1} = x_1(t), y_1(t)$
$\text{comet2} = x_2(t), y_2(t)$
- set $x_1(t) = x_2(t)$ and solve for $t$. Since $t$ had a square, I had 2 possible values for $t$ ($t_1$ and $t_2$).
- substitute $t_1$ into $x_1(t_1)$ and $x_2(t_1)$ to see if values are a match. then substitute $t_2$ into $x_1(t_2)$ and $x_2(t_2)$ to see if thats a match.
- if it is a match for either time, substitute $t$ value into the $y_1$ and $y_2$. If those match, the comets collide?
What to do if $t$ has a imaginary number? like 7.3 + .12345436756223i?
What do derivatives of comet $x_1'$, $y_1'$ show me? Does this simply show the rate of change of the path the comets have?
That approach is fine. We can also see when (if ever) their $y$-coordinates coincide first, then check their $x$-coordinates to see if they collided.
If your solutions $t$ to $x_1(t)=x_2(t)$ are imaginary, then the comets never have the same $x$-coordinate at the same time (since we parameterize time with real numbers), so the comets certainly don't collide.
$x_1'(t)$ (if defined) is the "horizontal" velocity of the first comet at time $t$. $y_1'(t)$ is likewise the "vertical" velocity of the first comet at time $t$.