/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 Emergency Landing. A plane leave... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 70

Emergency Landing. A plane leaves the airport in Galisteo and flies 170 \(\mathrm{km}\) at \(68^{\circ}\) east of north and then changes direction to fly 230 \(\mathrm{km}\) at \(48^{\circ}\) south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to this plane?

Short Answer

Expert verified
The rescue crew should fly 329.3 km in the direction 19掳 south of east.

Step by step solution

01

Break Down the Problem

We want to find a direct path (magnitude and direction) to the plane from the starting point. The plane's path involves two segments: 170 km at 68掳 east of north, and 230 km at 48掳 south of east. We'll use vector addition to combine these two paths and find the resultant vector from the starting point to the plane's final position.
02

Define the Coordinate System

Let positive y represent north and positive x represent east. We will decompose each segment of the journey into x and y components.
03

Decompose the Vector Segments

For the first segment: 170 km at 68掳 east of north, \(x_1 = 170 \cdot \sin(68掳)\),\(y_1 = 170 \cdot \cos(68掳)\).For the second segment: 230 km at 48掳 south of east, \(x_2 = 230 \cdot \cos(48掳)\),\(y_2 = -230 \cdot \sin(48掳)\) because it's south of east.
04

Calculate the Components

Start by computing each component:\(x_1 = 170 \cdot 0.9272 \approx 157.6\) km,\(y_1 = 170 \cdot 0.3746 \approx 63.7\) km,\(x_2 = 230 \cdot 0.6691 \approx 153.9\) km,\(y_2 = -230 \cdot 0.7431 \approx -171.0\) km.
05

Sum the Components

Add the x-components and y-components respectively:\(x_{total} = x_1 + x_2 = 157.6 + 153.9 = 311.5\) km, \(y_{total} = y_1 + y_2 = 63.7 - 171.0 = -107.3\) km.
06

Calculate the Resultant Magnitude

Use the Pythagorean theorem to find the magnitude of the resultant vector:\[ R = \sqrt{x_{total}^2 + y_{total}^2} = \sqrt{(311.5)^2 + (-107.3)^2} \approx 329.3 \text{ km} \].
07

Find the Direction of the Resultant

Calculate the angle \( \theta \) from the east direction using \( \tan \theta = \frac{|y|}{x} = \frac{107.3}{311.5} \). Thus, \( \theta = \tan^{-1}(0.344) \approx 19.0^{\circ} \). Since the resultant vector is in the southeast quadrant, the direction is 19掳 south of east.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coordinate System
When working with vector quantities in physics, it's important to use a coordinate system to determine direction and magnitude precisely. A coordinate system typically involves an origin, usually marked as the point (0,0), from which you can define positions using axes. For problems involving navigation, a common choice is:
  • The positive x-axis: representing east.
  • The positive y-axis: representing north.
You can think about this as placing a map flat and determining directions based on compass points. This setup helps to accurately resolve vector components into manageable parts, further aiding calculations, especially when dealing with trigonometric functions.
Vector Components
Understanding vector components is crucial when applying vectors in physics, as each vector can be split into two parts that align with the coordinate axes. By breaking down a vector into its components, we can handle them like simple numbers.
  • The x-component represents the projection onto the east-west axis.
  • The y-component represents the projection onto the north-south axis.
For example, in vector addition problems, a vector traveling a certain distance at an angle needs to be decomposed into its respective components. Using trigonometric functions such as sine and cosine, we can find these components, offering a clearer way to continue with calculations.
Pythagorean Theorem
The Pythagorean theorem is a foundational principle in mathematics that is often used in physics when dealing with vectors. It relates the sides of a right triangle, making it perfect for determining the magnitude of a resultant vector in a two-dimensional plane.When you have the components of a vector, you can use the formula:\[ R = \sqrt{x^2 + y^2} \]Where \( R \) is the hypotenuse or the resultant vector, \( x \) is the length along the x-axis, and \( y \) is the length along the y-axis. This gives the straight-line distance from the starting point to the endpoint, crucial in navigation problems, such as finding a direct route to a location.
Trigonometry in Physics
Trigonometry is a powerful tool in physics, especially when resolving vectors and calculating angles. It involves functions like sine, cosine, and tangent, which relate the angles and sides of a triangle.
  • Cosine is used to find the adjacent side of an angle in a right triangle, which corresponds to the x-component.
  • Sine is used to find the opposite side, which corresponds to the y-component.
To determine the direction of a vector, the tangent function is often used. By calculating:\[ \tan \theta = \frac{|y|}{x} \]We find the angle \( \theta \) related to the adjacent side, giving insight into the vector's orientation regarding the coordinate system. When dealing with navigation or physics problems, understanding this relationship is key to finding both direction and accurately representing movement.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Iron has a property such that a \(1.00-\mathrm{m}^{3}\) volume has a mass of \(7.86 \times 10^{3} \mathrm{kg}\) (density equals \(7.86 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} ) .\) You want to manufacture iron into cubes and spheres. Find (a) the length of the side of a cube of iron that has a mass of 200.0 \(\mathrm{g}\) and (b) the radius of a solid sphere of iron that has a mass of 200.0 \(\mathrm{g}\) .

By making simple sketches of the appropriate vector products, show that \((a) \vec{A} \cdot \vec{B}\) can be interpreted as the product of the magnitude of \(\overrightarrow{\boldsymbol{A}}\) times the component of \(\overrightarrow{\boldsymbol{B}}\) along \(\overrightarrow{\boldsymbol{A}}\), or the magnitude of \(\vec{B}\) times the component of \(\vec{A}\) along \(\overrightarrow{\boldsymbol{B}}\) (b) \(|\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}|\) can be interpreted as the product of the magnitude of \(\overrightarrow{\boldsymbol{A}}\) times the component of \(\overrightarrow{\boldsymbol{B}}\) perpendicular to \(\overrightarrow{\boldsymbol{A}},\) or the magnitude of \(\overrightarrow{\boldsymbol{B}}\) times the component \(\overrightarrow{\boldsymbol{A}}\) perpendicular to \(\overrightarrow{\boldsymbol{B}}\).

With a wooden ruler you measure the length of a rectangular piece of sheet metal to be 12 \(\mathrm{mm}\) . You use micrometer calipers to measure the width of the rectangle and obtain the value 5.98 \(\mathrm{mm}\). Give your answers to the following questions to the correct number of significant figures. (a) What is the area of the rectangle? (b) What is the ratio of the rectangle's width to its length? (c) What is the perimeter of the rectangle? (d) What is the difference between the length and width? (e) What is the ratio of the length to the width?

(a) Is the vector \((\hat{\imath}+\hat{j}+\hat{k})\) a unit vector? Justify your answer. (b) Can a unit vector have any components with magnitude greater than unity? Can it have any negative components? In each case justify your answer. (c) If \(\overrightarrow{\boldsymbol{A}}=a(3.0 \hat{\imath}+4.0 \hat{\mathbf{y}}),\) where \(\boldsymbol{a}\) is a constant, determine the value of \(a\) that makes \(\vec{A}\) a unit vector.

Let the angle \(\theta\) be the angle that the vector \(\vec{A}\) makes with the \(+x\) -axis, measured counterelockwise from that axis. Find the angle \(\theta\) for a vector that has the following components: (a) \(A_{x}=2.00 \mathrm{m},\) \(A_{y}=-1.00 \mathrm{m}\) (b) \(A_{x}=2.00 \mathrm{m}, A_{y}=1.00 \mathrm{m}\) (c) \(A_{x}=-2.00 \mathrm{m}\), \(A_{y}=1.00 \mathrm{m}\) (d) \(A_{x}=-200 \mathrm{m}, A_{y}=-1.00 \mathrm{m}\).
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Famous Physicists

Read Explanation

Astrophysics

Read Explanation

Measurements

Read Explanation

Kinematics Physics

Read Explanation

Translational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.