Chapter 13.1, Problem 49E

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8th Edition

James Stewart

ISBN: 9781305266643

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 functions

**r**_{1}(*t*) = ⟨*t*^{2},7*t* − 12, t^{2}⟩

**r**_{2}(*t*) = ⟨4*t* − 3, t^{2}, 5*t* − 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

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.

Substitute

Equate each component.

Solve equation (1) as follows

Therefore, the value of *t* is either 1 or 3

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