   Chapter 10.1, Problem 45E

Chapter
Section
Textbook Problem

Suppose that the position of one particle at time t is given by x 1 = 3 sin t y 1 = 2 cos t 0 ≤ t ≤ 2 π and the position of a second particle is given by x 2 = − 3 + cos t y 2 = 1 + sin t 0 ≤ t ≤ 2 π (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision points? In other words, arc the particles ever at the same place at the same time? If so, find the collision points, (c) Describe what happens if the path of the second particle is given by x 2 = 3 + cos   t y 2 = 1 + sin t 0 ≤ t ≤ 2 π

(a)

To determine

To plot: The graph of particles and determine the number of points of intersection.

Explanation

Given data:

The parametric equation for the horizontal position of first particle at time t is as follows.

x1=3sint (1)

The parametric equation for the vertical position of first particle at time t is as follows.

y1=2cost (2)

The range of t is 0 to 2π .

The parametric equation for the horizontal position of second particle at time t is as follows.

x2=3+cost (3)

The parametric equation for the vertical position of second particle at time t is as follows.

y2=1+sint (4)

The range of t is 0 to 2π .

Calculation:

The value of t is increased from 0 to 2π with a step value of π2 and substituted in the parametric equations (1) and (2) to obtain the value of x and y respectively.

Substitute 0 for t in equation (1),

x1=3sint=3sin(0)=0

Substitute 0 for t in equation (2),

y1=2cost=2cos(0)=2(1)=2

The values of x and y for each step value of t is tabulated in the below table.

 t −π2 0 π2 π 2π x −3 0 3 0 0 y 0 2 0 −2 2

Substit

(b)

To determine

To find: The collision points

(c)

To determine

To plot: The graph when the second particle is given by x2=3+cost and y2=1+sint .

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