Parametric plots: Determine if 2 comets collide at a given time. Am I solving it correctly?

Question

There are $2$ comets

comet 1 $(x(t), y(t))$, comet 2 $(x_1(t), y_1(t))$

I need to determine if these two comets collide. From reading my steps below, is this the proper way to solve this?

$1.$ set $x$ function = to $x_1$ function.

$2.$ solve for $t$.

$3.$ use $t$ and plug it into $y(t)$ and $y_1(t)$.

$4.$ if both $y$ functions equal each other, the comets collide?

Thank you

Answer 1

Your thought is good. If you get there, they do collide. The potential problem comes in step $2$. You might not be able to solve for an explicit $t$, even if you know it exists. There might also be more than $1 \ t$, but you can check them all if the number is finite. There could be infinite matches-turn off gravity and think of the comets both on straight line trajectories. They all go through all $x$ values, so you have all time to check.