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Chapter 1: Problem 41

A disoriented physics professor drives 3.25 \(\mathrm{km}\) north, then 4.75 \(\mathrm{km}\) west, and then 1.50 \(\mathrm{km}\) south. Find the magnitude and direction of the resultant displacement, using the method of components. In a vector addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained using the method of components.

Short Answer

Expert verified
The resultant displacement is 5.05 km at 20.04掳 north of west.

Step by step solution

01

Assigning Vectors to the Professor's Path

Let's assign vectors to each leg of the professor's trip. Starting from the origin: \( \vec{A} = 3.25\, \text{km} \) north, \( \vec{B} = 4.75\, \text{km} \) west, and \( \vec{C} = 1.50\, \text{km} \) south.
02

Decompose Vectors into Components

Convert each vector into its x and y components, assuming north and east are positive directions. The components are: - \( \vec{A} \) has components: \( A_x = 0 \), \( A_y = +3.25 \) km.- \( \vec{B} \) has components: \( B_x = -4.75 \), \( B_y = 0 \) km.- \( \vec{C} \) has components: \( C_x = 0 \), \( C_y = -1.50 \) km.
03

Calculate Resultant Components

Add together the corresponding components to find the resultant vector's components: - \( R_x = A_x + B_x + C_x = 0 - 4.75 + 0 = -4.75 \)- \( R_y = A_y + B_y + C_y = 3.25 + 0 - 1.50 = 1.75 \) km.
04

Calculate the Magnitude of the Resultant Displacement

Use the Pythagorean theorem to find the magnitude of the resultant displacement vector: \[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-4.75)^2 + (1.75)^2} \approx 5.05 \, \text{km} \]
05

Calculate the Direction of the Resultant Displacement

Calculate the direction \( \theta \) with respect to the west direction using the arctangent function: \[ \theta = \tan^{-1}\left( \frac{R_y}{|R_x|} \right) = \tan^{-1}\left( \frac{1.75}{4.75} \right) \approx 20.04^\circ \] This means the displacement is approximately 20.04掳 north of west.
06

Vector Addition Diagram

Draw a vector diagram showing the professor's path and resultant vector. Start with the first vector \( \vec{A} \) pointing north, then connect \( \vec{B} \) to \( \vec{A} \) pointing west, and finally \( \vec{C} \) pointing south. The resultant vector \( \vec{R} \) should appear from the origin to the final point; it should qualitatively match our calculated 5.05 km at 20.04掳 north of west displacement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Displacement
Displacement is a vector quantity that describes how far and in what direction an object is from its starting point. Unlike distance, which merely accounts for the pathway traveled, displacement considers the shortest route between two points in a specified direction.
In our exercise, the professor's displacement is not equal to the sum of distances traveled north, west, and south but rather represents the straight line path from the start to the endpoint.
This concept is crucial as it combines both magnitude and direction aspects of motion, providing a clearer understanding of total movement on a plane.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle in geometry, widely used to calculate the magnitude of the resultant vectors. It's represented by the equation: \[ c = \sqrt{a^2 + b^2} \], where \(c\) is the hypotenuse of a right-angled triangle, and \(a\) and \(b\) are its other sides.
In the context of vector addition, particularly if the motion components form a right triangle, the Pythagorean theorem helps find the magnitude of the displacement vector.
For the professor's journey, it allowed us to find the resultant displacement from the separate horizontal and vertical components, calculated as approximately 5.05 km.
Vector components
Breaking down vectors into their components is essential for simplifying vector addition. Each vector can be represented in terms of its horizontal (x-axis) and vertical (y-axis) parts.
This concept, known as vector decomposition, helps manage vectors pointing in various directions by only adding corresponding components, thus making it easier to find a resultant vector.
In the exercise, the vector path taken by the professor was broken into x and y components: north, west, and south. By associating these paths into phase-aligned parts, we obtained an overall overview before calculating the full displacement.
Resultant vector
The concept of a resultant vector refers to a single vector representing the net effect of combining multiple vectors. In essence, it tells us the overall change in position when several vectors act on a point.
For our professor's journey, we began by converting each directional path into vector components. Then, by calculating the sum of these components, we derived the resultant vector, representing both the magnitude and the angle of displacement from the start point.
We determined the resultant displacement vector to be approximately 5.05 km, directed 20.04掳 north of west, providing a clear visualization of the professor's overall movement.

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