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Chapter 3: Problem 31

Peggy drives from Cornwall to Atkins Glen in 45 min. Cornwall is $73.6 \mathrm{km}\( from Illium in a direction \)25^{\circ}$ west of south. Atkins Glen is \(27.2 \mathrm{km}\) from Illium in a direction \(15^{\circ}\) south of west. Using Illium as your origin, (a) draw the initial and final position vectors, (b) find the displacement during the trip, and (c) find Peggy's average velocity for the trip.

Short Answer

Expert verified
Answer: The three main steps to solve the problem are: (a) draw the initial and final position vectors, (b) find the displacement during the trip, and (c) find Peggy's average velocity for the trip.

Step by step solution

01

(Step 1: Draw initial and final position vectors)

Convert the given distances and angles to rectangular coordinates. For the initial position (Cornwall), the coordinates can be found using the relationships: $$\begin{cases} x_1 = 73.6 \times \cos{(25^\circ)}\\ y_1 = -(73.6 \times \sin(25^\circ)) \end{cases}$$ And for the final position (Atkins Glen), the coordinates are: $$\begin{cases} x_2 = -(27.2 \times \sin{(15^\circ)})\\ y_2 = -(27.2 \times \cos(15^\circ)) \end{cases}$$ Calculate these coordinates.
02

(Step 2: Find displacement)

Subtract the initial position coordinates from the final position coordinates, to get the displacement vector components: $$\begin{cases} \Delta x = x_2 - x_1\\ \Delta y = y_2 - y_1 \end{cases}$$ Calculate these components.
03

(Step 3: Magnitude and direction of displacement)

Calculate the magnitude of the displacement vector, using the Pythagorean theorem: $$D = \sqrt{(\Delta x)^2 + (\Delta y)^2}$$ Next, find the angle of the displacement vector using the inverse tangent function: $$\theta = \arctan{\frac{\Delta y}{\Delta x}}$$ Make sure to consider the appropriate quadrant based on the signs of \(\Delta x\) and \(\Delta y\). Calculate the magnitude \(D\) and direction \(\theta\) of the displacement.
04

(Step 4: Find average velocity)

Calculate the average velocity by dividing the magnitude of the displacement vector (\(D\)) by the given time (45 minutes, converted to hours): $$v_{avg} = \frac{D}{t}$$ Calculate the average velocity of Peggy's trip. Now we have drawn the initial and final position vectors, found the displacement during the trip, and calculated Peggy's average velocity for the trip.

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