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Problem 49 Hard Difficulty

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
$ r_1 (t) = \langle t^2, 7t - 12, t^2 \rangle $ $ r_2 (t) = \langle 4t - 3, t^2, 5t - 6 \rangle $
for $ t \ge 0 $. Do the particles collide?

Answer

Yes

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Video Transcript

Okay. So we have two particles traveling along these two paths and the question is will they collide or not? And so what we gotta do is we're going to find out if there are any time where they're at the same place. Okay. So I just started with the X coordinates. So if I set them equal to each other then I'm finding a time or times where there are both at the same X value. So we get t squared minus 40 plus three equals zero t minus three, T minus one equals zero. So T equals three and T equals one at those two times both these objects are at the same X value. So we got to see if they're at the same white and Z value too. So are one of one is one negative 51. Okay. So that's that's the point where the first object is at time one And then our two of one is at 1 1 -1. So it's not at the same place. So they don't yeah collide At T. equals one. All right, so let's check out three, three were at nine 9 9. The first object is and then the second object is at 9959. Oh. So at equals three. Both objects are at The .999. So yes they do collide. Okay. Okay. So do we have to do the same thing with the Y values and the Z values? Well no because if they were going to collide they have to be at the same X value. So we would have had to find them in the first place by by finding what time they are in the same X values. And the only time they're in the same X place are these two times. And so this is the only time they could collide.