Answered: A cruise ship leaving port travels 53.0…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

Goal Add vectors algebraically and find the resultant vector.

Problem A hiker begins a trip by first walking 25.0 km 45.0° south of east from her base camp. On the second day she walks 40.0 km in a direction 60.0°north of east, at which point she discovers a forest ranger's tower. (a) Determine the components of the hiker's displacements in the first and second days. (b) Determine the components of the hiker's total displacement for the trip. (c) Find the magnitude and direction of the displacement from base camp.

Strategy This problem is just an application of vector addition using components. We denote the displacement vectors on the first and second days by and , respectively. Using the camp as the origin of the coordinates, we get the vectors shown in Figure a. After finding x- and y-components for each vector, we add them "componentwise." Finally, we determine the magnitude and direction of the resultant vector , using the Pythagorean theorem and the inverse tangent function.

SOLUTION

(a) Find the components of .

Use the demonstrated equations to find the components of .

A_x = A cos (-45.0°) = (25.0 km)(0.707)= 17.7 km

A_y = A sin -(45.0°) = (-25.0 km)(0.707)= -17.7 km

Find the components of .

B_x = B cos 60.0° = (40.0 km)(0.500)= 20.0 km

B_y = B sin 60.0° = (40.0 km)(0.866)= 34.6 km

(b) Find the components of the resultant vector, = + .

To find R_x, add the x-components of and .

R_x = A_x + B_x = 17.7 km + 20.0 km = 37.7 km

To find R_y, add the y-components of and .

R_y = A_y + B_y = -17.7 km + 34.6 km = 16.9 km

Use the Pythagorean theorem to get the magnitude.

R = √R_x² + R_y² = √ (37.7 km² + (16.9 km)² = 41.3 km

Calculate the direction of using the inverse tangent function.

θ = tan^-1	(	16.9 km	)	= 24.1°
37.7 km

LEARN MORE

Remarks Figure b shows a sketch of the components of and their directions in space. The magnitude and direction of the resultant can also be determined from such a sketch.

Question A second hiker follows the same path the first day, but then walks 15.0 km east on the second day before turning and reaching the ranger's tower. How does the second hiker's resultant displacement vector compare with that of the first hiker? List all aspects that apply. (Select all that apply.)

The two displacements have the same direction.The second hiker's displacement has the same magnitude as the first.The second hiker's displacement has a smaller magnitude than the first.The second hiker's displacement is greater in magnitude.The two displacements have different directions.

PRACTICE IT

Use the worked example above to help you solve this problem. A hiker begins a trip by first walking 24.5km southeast from her base camp. On the second day she walks 44.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger's tower.

(a) Determine the components of the hiker's displacements in the first and second days.

A_x =	km
A_y =	km
B_x =	km
B_y =	km

(b) Determine the components of the hiker's total displacement for the trip.

R_x =	km
R_y =	km

(c) Find the magnitude and direction of the displacement from base camp.

Magnitude	km
Direction	° north of east

EXERCISEHINTS: GETTING STARTED | I'M STUCK!

A cruise ship leaving port travels 53.0 km 45.0° north of west and then 71.0 km at a heading 30.0° north of east.

(a) Find the ship's displacement vector.

R_x =	km
R_y =	km

(b) Find the displacement vector's magnitude and direction.

Magnitude	km
Direction	° north of east

### Vector Addition Diagrams

#### Diagram (a): Vector Addition in Cartesian Coordinates

- **Axes**:
- The x-axis represents distance in kilometers (km) east-west.
- The y-axis represents distance in kilometers (km) north-south.

- **Points**:
- `Camp` at (0, 0)
- `Tower` at approximately (40, 30)

- **Vectors**:
- `\(\vec{A}\)` extends from `Camp` to a point at approximately (20, -10) with an angle of 45 degrees below the x-axis within the fourth quadrant.
- `\(\vec{B}\)` extends from the end of vector `\(\vec{A}\)` to a point at approximately (50, -10) with an angle of 60 degrees below the x-axis.
- `\(\vec{R}\)` extends directly from `Camp` to `Tower` (the resultant vector).

- **Directions**: A compass indicating the cardinal directions (N, S, E, W).

#### Diagram (b): Vector Components

- **Axes**:
- The x-axis represents distance in kilometers (km) east-west.
- The y-axis represents distance in kilometers (km) north-south.

- **Vectors**:
- `\(\vec{R}\)` extends from the origin (0, 0) to approximately (40, 17) - same resultant vector as in diagram (a).
- `\(\vec{R_x}\)` is the x-component of `\(\vec{R}\)` extending horizontally from (0, 0) to approximately (40, 0) representing a distance of 37.7 km.
- `\(\vec{R_y}\)` is the y-component of `\(\vec{R}\)` extending vertically from (40, 0) to approximately (40, 17) representing a distance of 16.9 km.

These diagrams demonstrate vector addition and decomposition in a two-dimensional plane, making use of angles and distances to depict the resultant vector and its components in Cartesian coordinates.

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