Question

f 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, 13t − 36, t2 r2(t) = 7t − 12, t2, 5t − 4

Answer 1

The two objects will collide with the same position vector for all three components at exactly t = 4 s

Step-by-step explanation:

For two particles starting out at the same time to collide, their position Vector's at the time of collision must be exactly the same.

So, at the collision point, position vector of object 1 is equated to that of object 2.

r₁ = (t², 13t-36, t²)

r₂ = (7t-12, t², 5t-4)

At he point of collision

t² = 7t - 12

t² - 7t + 12 = 0

t² - 4t - 3t + 12 = 0

t(t - 4) - 3(t - 4) = 0

t = 3s or t = 4s

13t - 36 = t²

t² - 13t + 36 = 0

t² - 4t - 9t + 36 = 0

t(t - 4) - 9(t - 4) = 0

t = 9s or 4s

t² = 5t - 4

t² - 5t + 4 = 0

t² - 4t - t + 4 = 0

t(t - 4) - 1(t - 4) = 0

t = 1s or t = 4s

The three components intersect at other times, but at t = 4s, they all intersect at the same time! Meaning that, at this point the two objects are at the same place with the same position vector at that time.