/*! 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 12 A truck is traveling due north a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 12

A truck is traveling due north at \(40 \mathrm{~km} / \mathrm{hr}\) approaching a crossroad. On a perpendicular road a police car is traveling west toward the intersection at \(35 \mathrm{~km} / \mathrm{hr}\). Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car. displacement \(\vec{d}=\)_________

Short Answer

Expert verified
The displacement vector of the truck with respect to the police car is: \(\vec{d} = 40 \, km \, \hat{i} + 35 \, km \, \hat{j}\)

Step by step solution

01

Find the distance each vehicle will travel

In this problem, we know the velocity of both vehicles and the time it takes to reach the intersection. We can use the formula: distance = velocity * time to find the total distance traveled by each vehicle in one hour. For the truck: Distance = Velocity * Time Distance = 40 km/hr * 1 hr = 40 km (North) For the police car: Distance = Velocity * Time Distance = 35 km/hr * 1 hr = 35 km (West)
02

Write the vectors for each vehicle

Now that we have the distance both vehicles will travel within one hour, we can write their displacement vectors. Truck vector: \(\vec{A} = 40 \, km \, \hat{i}\) (since the truck is traveling north) Police car vector: \(\vec{B} = -35 \, km \, \hat{j}\) (since the police car is traveling west)
03

Calculate the displacement vector

To find the displacement vector of the truck with respect to the police car, we can subtract the police car vector from the truck vector. \(\vec{d} = \vec{A} - \vec{B}\) \(\vec{d} = (40 \, km \, \hat{i} + 0 \, km \, \hat{j}) - (0 \, km \, \hat{i} - 35 \, km \, \hat{j})\) \(\vec{d} = 40 \, km \, \hat{i} + 35 \, km \, \hat{j}\) The displacement vector of the truck with respect to the police car is: \(\vec{d} = 40 \, km \, \hat{i} + 35 \, km \, \hat{j}\)

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.

Multivariable Calculus
Multivariable calculus is an extension of single-variable calculus where functions of more than one variable are involved. When dealing with vector displacement in the context of multivariable calculus, we explore the displacement in a space defined by two or more dimensions, typically on an 'xy' plane for two-dimensional space or in 'xyz' space for three-dimensional analyses.

For example, in the given exercise, the movement of the truck and police car can be illustrated on a two-dimensional plane with positions and displacements represented by vectors. These vectors account for movement in multiple directions鈥攏orth and west鈥攚hich correspond to the 'i' and 'j' unit vectors, respectively, in a Cartesian coordinate system. Students can visualize problems like these by plotting vectors on a graph and applying vector operations to determine the resulting displacement.
Vector Addition and Subtraction
Understanding vector addition and subtraction is crucial for solving physics and engineering problems where forces, velocities, and displacements are often represented as vectors. Vectors are mathematical objects with both magnitude and direction. They can be added or subtracted by aligning them head-to-tail and drawing a resulting vector.

In the exercise, the displacement of the truck with respect to the police car is found by subtracting the displacement vector of the police car from that of the truck. This is a straightforward application where we simply take the corresponding components in each direction鈥攏orth and west鈥攁nd perform the subtraction algebraically. The exercise demonstrates how the resulting vector gives us the relative position from one point to another, essentially showing how far and in what direction one must travel to get from the police car's position to the truck's location.
Relative Motion in Physics
The concept of relative motion refers to the motion of an object as observed from a particular frame of reference. In physics, it's often important to compare the motion of two or more objects with respect to each other to understand their interactions and trajectories.

For the given problem, we are interested in the truck's displacement relative to the police car. By calculating the vector displacement from the police car to the truck, we are effectively switching to a frame of reference where the police car is at rest, and the truck is moving. This shifts the problem from absolute motion (how each vehicle is moving in relation to the crossroads) to relative motion (how the truck's position is changing in relation to the police car). It's a foundational concept for understanding how different observers may perceive motion differently, depending on their own state of motion.

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

Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle\) be vectors in \(\mathbb{R}^{3}\). In this exercise we investigate properties of the triple scalar product \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\) a. Show that \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=\left|\begin{array}{ccc}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3}\end{array}\right|\). b. Show that \(\left|\begin{array}{lll}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3}\end{array}\right|=-\left|\begin{array}{ccc}v_{1} & v_{2} & v_{3} \\\ u_{1} & u_{2} & u_{3} \\ w_{1} & w_{2} & w_{3}\end{array}\right| .\) Conclude that interchanging the first two rows in a \(3 \times 3\) matrix changes the sign of the determinant. In general (although we won't show it here), interchanging any two rows in a \(3 \times 3\) matrix changes the sign of the determinant. c. Use the results of parts (a) and (b) to argue that $$ (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=(\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v}=(\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u} $$ d. Now suppose that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) do not lie in a plane when they eminate from a common initial point. a. Given that the parallepiped determined by \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) must have positive volume, what can we say about \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} ?\) b. Now suppose that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) all lie in the same plane. What value must \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\) have? Why? c. Explain how (i.) and (ii.) show that if \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) all eminate from the same initial point, then \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) lie in the same plane if and only if \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}=0 .\) This provides a straightforward computational method for determining when three vectors are co-planar.

Consider the function \(h\) defined by \(h(x, y)=8-\sqrt{4-x^{2}-y^{2}}\). a. What is the domain of \(h ?\) (Hint: describe a set of ordered pairs in the plane by explaining their relationship relative to a key circle.) b. The range of a function is the set of all outputs the function generates. Given that the range of the square root function \(g(t)=\sqrt{t}\) is the set of all nonnegative real numbers, what do you think is the range of \(h ?\) Why? c. Choose 4 different values from the range of \(h\) and plot the corresponding level curves in the plane. What is the shape of a typical level curve? d. Choose 5 different values of \(x\) (including at least one negative value and zero), and sketch the corresponding traces of the function \(h\). e. Choose 5 different values of \(y\) (including at least one negative value and zero), and sketch the corresponding traces of the function \(h\). f. Sketch an overall picture of the surface generated by \(h\) and write at least one sentence to describe how the surface appears visually. Does the surface remind you of a familiar physical structure in nature?

Find the length of the vectors (a) \(3 \tilde{i}-\tilde{j}-3 \tilde{k}\) : length \(=\)_________ (b) \(-1.6 \tilde{i}+0.4 \tilde{j}-1.2 \tilde{k}:\) length \(=\)_________

Find a parametrization of the curve \(x=-5 z^{2}\) in the xz-plane. Use \(t\) as the parameter for all of your answers. \(x(t)=\)________ \(y(t)=\)________ \(z(t)=\)________

Which is traveling faster, a car whose velocity vector is \(26 \vec{i}+31 \vec{j},\) or a car whose velocity vector is \(40 \vec{i}\), assuming that the units are the same for both directions? \((\square\) the first car \(\square\) the second car) is the faster car. At what speed is the faster car traveling? speed \(=\)_________
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.