If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions r1(t) = t2, 11t − 18, t2 r2(t) = 17t − 72, t2, 13t − 36 for t ≥ 0. Find the values of t at which the particles collide. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

Let be [tex]r_{1}(t) = \langle t^{2}, 11\cdot t - 18, t^{2}\rangle[/tex] and [tex]r_{2}(t) = \langle 17\cdot t-72, t^{2}, 13\cdot t - 36\rangle[/tex] the parametric equations of the trajectories of two particles, where [tex]t\geq 0[/tex]. The following condition must be met when the particles collide:

[tex]r_{1}(t) = r_{2}(t)[/tex]

Which implies this system of equations:

[tex]t^{2} = 17\cdot t -72[/tex] and [tex]11\cdot t -18 = t^{2}[/tex] and [tex]t^{2} = 13\cdot t -36[/tex]

And equivalent to the following second-order polynomials:

[tex]t^{2}-17\cdot t +72 = 0[/tex]

[tex]t^{2}-11\cdot t +18 = 0[/tex]

[tex]t^{2}-13\cdot t +36 = 0[/tex]

Another condition is that at least a common root must exists in order to confirm a possible collision. All roots are found by factorizing each polynomial:

[tex](t-8)\cdot (t-9) = 0[/tex]

[tex](t-2)\cdot (t-9) = 0[/tex]

[tex](t-4) \cdot (t-9) = 0[/tex]

There is one common root ([tex]t = 9[/tex]), which means that both particles collide at [tex]t = 9[/tex].