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Chapter 4: Problem 55

Phnom Penh is \(470 \mathrm{~km}\) east and \(250 \mathrm{~km}\) south of Bangkok. Hanoi is \(60 \mathrm{~km}\) east and \(1030 \mathrm{~km}\) north of Phnom Penh. (a) Choose a coordinate system, and translate these data into \(\Delta x\) and \(\Delta y\) values with the proper plus and minus signs. (b) Find the components of the \(\Delta \mathbf{r}\) vector pointing from Bangkok to Hanoi.(answer check available at lightandmatter.com)

Short Answer

Expert verified
\(\Delta \mathbf{r} = (530 \, \text{km}, 780 \, \text{km})\).

Step by step solution

01

Establish a Coordinate System

Let's choose a coordinate system where Bangkok ( the origin) is at (0,0). The eastward direction is positive x, and the northward direction is positive y.
02

Calculate Bangkok to Phnom Penh

From Bangkok to Phnom Penh, the east direction is given as 470 km, which we take as positive x. The south direction is given as 250 km, which should be taken as negative y. So, the coordinates are d for Phnom Penh relative to Bangkok: \(\Delta x_{BP} = +470 \, \text{km}, \, \Delta y_{BP} = -250 \, \text{km}\).
03

Calculate Phnom Penh to Hanoi

From Phnom Penh to Hanoi, we have an eastward distance of 60 km, which remains positive. The north direction is 1030 km, so it is positive y. Thus, for Hanoi relative to Phnom Penh, \(\Delta x_{PH} = +60 \, \text{km}, \, \Delta y_{PH} = +1030 \, \text{km}\).
04

Combine Displacements from Bangkok to Hanoi

Combine the displacements from Bangkok to Phnom Penh and Phnom Penh to Hanoi:For the x-component: \[\Delta x_{BH} = \Delta x_{BP} + \Delta x_{PH} = 470 \, \text{km} + 60 \, \text{km} = 530 \, \text{km}\]For the y-component:\[\Delta y_{BH} = \Delta y_{BP} + \Delta y_{PH} = -250 \, \text{km} + 1030 \, \text{km} = 780 \, \text{km}\]
05

Construct the Resultant Displacement Vector

The resulting vector \(\Delta \mathbf{r}\) from Bangkok to Hanoi is composed of the components calculated:\[\Delta \mathbf{r} = (530 \, \text{km}, 780 \, \text{km})\].

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Key Concepts

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

Coordinate System
To solve problems like these, we first need to establish a coordinate system. Imagine a graph with two axes: the x-axis, running left to right, and the y-axis, running up and down. In our coordinate system, these lines will help you visualize directions and distances.
  • The x-axis typically represents east-west directions.
  • The y-axis usually represents north-south directions.
Here, Bangkok is at the origin, meaning it is at the point (0,0). Scaling positive values towards the east on the x-axis and positive values towards the north on the y-axis gives a simple structure that builds the foundation for understanding displacement in two-dimensional space.
Vector Components
Vectors are mathematical tools used to represent directions and distances. They have both a magnitude (how far) and a direction (which way). In this exercise, we are translating the given directions (east, south, north) into vector components, denoted by \( \Delta x \) and \( \Delta y \), for the changes in the x and y coordinates, respectively.
  • For east direction, \( \Delta x \) is positive because it is along the positive x-axis.
  • For south direction, \( \Delta y \) is negative because it is along the negative y-axis.
  • For north direction, \( \Delta y \) is positive since it moves along the positive y-axis.
By understanding vector components, you can calculate how far something has traveled in either direction over a point of origin.
Resultant Vector
A resultant vector serves as a single vector that represents the sum of two or more vectors. In this exercise, it shows us the total displacement from Bangkok to Hanoi by combining the movements from two legs of the journey.
After breaking down the vectors into components, our task is to combine these parts into a single vector:
  • Combine x-components: \( \Delta x_{BH} = 470 \, \text{km} + 60 \, \text{km} = 530 \, \text{km} \).
  • Combine y-components: \( \Delta y_{BH} = -250 \, \text{km} + 1030 \, \text{km} = 780 \, \text{km} \).
The final resultant displacement vector is noted as \( \Delta \mathbf{r} = (530 \, \text{km}, 780 \, \text{km}) \). This sums up the entire movement from Bangkok to Hanoi, providing the total shift in position.
East and North Directions
Understanding the east and north directions in a coordinate system is crucial because they dictate the sign of your vector components. Both these directions add a positive component to your vectors: east adds to the x-coordinate, and north adds to the y-coordinate.
For clarity:
  • Moving east means you add to \( \Delta x \) and proceed along the coordinate system that runs left to right.
  • Moving north means you add to \( \Delta y \) in the up direction, which raises the value on the y-axis.
Grasp of these directions helps translate real-world directions into a format that can be easily computed and visualized on a coordinate plane.

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Most popular questions from this chapter

A baseball pitcher throws a pitch clocked at \(v_{x}=73.3 \mathrm{mi} / \mathrm{h}\). He throws horizontally. By what amount, \(d\), does the ball drop by the time it reaches home plate, \(L=60.0 \mathrm{ft}\) away? (a) First find a symbolic answer in terms of \(L, v_{x}\), and \(g\). (answer check available at lightandmatter.com) (b) Plug in and find a numerical answer. Express your answer in units of ft. (Note: \(1 \mathrm{ft}=12\) in, \(1 \mathrm{mi}=5280 \mathrm{ft}\), and \(1 \mathrm{in}=2.54 \mathrm{~cm}\) ) (answer check available at lightandmatter.com)

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