Question

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, 6t − 5, t2 r2(t) = 9t − 20, t2, 11t − 30 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.)

Answer 1

Answer:

  t = {5}

Step-by-step explanation:

The trajectories intersect at exactly one time, t=5. The point of collision is ...

  r1(5) = r2(5) = (25, 25, 25)

__

The particles collide when the difference between the coordinates of one of them and the coordinates of the other one is zero. That is, ...

  r1(t) - r2(t) = 0

  (t^2, 6t -5, t^2) - (9t -20, t^2, 11t -30) = 0

This resolves to three (3) quadratic equations:

1. t^2 -9t +20 = 0 . . . . (t -4)(t -5) = 0
2. -5 +6t -t^2 = 0 . . . . . -(t -1)(t -5) = 0
3. t^2 -11t +30 = 0 . . . . (t -5)(t -6) = 0

These have a common factor of t-5, so will all be zero when t=5.

The particles collide only at t = 5.