   Chapter 13.1, Problem 49E

Chapter
Section
Textbook Problem

# 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 functionsr1(t) = ⟨t2,7t − 12, t2⟩r2(t) = ⟨4t − 3, t2, 5t − 6⟩for t ≥ 0. Do the particles collide?

To determine

To find: Whether the vector functions of particles collide.

Explanation

Given:

The vector functions of trajectories of two particles are r1(t)=t2,7t12,t2 and r2(t)=4t3,t2,5t6 .

Equate the components of vector equations of two trajectories and find the value of parameter t. If the determined value of t satisfies all the equated expressions, then the trajectories collide at that point.

Equate the two trajectories of two particles.

r1(t)=r2(t)

Substitute t2,7t12,t2 for r1(t) and 4t3,t2,5t6 for r2(t) ,

t2,7t12,t2=4t3,t2,5t6

Equate each component.

t2=4t3

t24t+3=0 (1)

7t12=t2

t27t+12=0 (2)

t2=5t6

t25t+6=0 (3)

Solve equation (1) as follows

t23tt+3=0t(t3)1(t3)=0(t1)(t3)=0t=1(or)3

Therefore, the value of t is either 1 or 3

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