/*! 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 71 A dog in an open field runs \(12... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 71

A dog in an open field runs \(12.0 \mathrm{~m}\) east and then \(28.0 \mathrm{~m}\) in a direction \(50.0^{\circ}\) west of north. In what direction and how far must the dog then run to end up \(10.0 \mathrm{~m}\) south of her original starting point?

Short Answer

Expert verified
The dog must run \(-10.0m - y_{total}\) meters south and the direction is given by \(\tan^{-1}(-10.0m / x_{total})\) degrees from the x-axis.

Step by step solution

01

Determine the vector components of the initial directions

Start by breaking down the dog's initial movement into vectors. First, the dog runs 12.0m east, which in vector components is represented as \(x = 12.0m\), \(y = 0m\) since it is purely in the x-direction. Secondly, the dog runs 28.0m at \(50.0^{\circ}\) west of north. Using trigonometry, this gives \(x = -28.0m \cdot \cos(50^{\circ})\) and \(y = 28.0m \cdot \sin(50^{\circ})\).
02

Add the vector components together

We can now add the individual vector components resulting from the two movement directions to obtain the total displacement vector. This gives \(x_{total} = 12.0m + -28.0m \cdot \cos(50^{\circ})\) and \(y_{total} = 0m + 28.0m \cdot \sin(50^{\circ})\).
03

Calculate the remaining distance and direction

The dog needs to end up 10.0m south of the starting point. Therefore, the y-component of the final position must be -10.0m, leading to an additional y-movement of \(-10.0m - y_{total}\). As this movement is purely in the y-direction, the x-component stays the same. This remaining y-movement determines how far the dog needs to run. To find the direction, calculate the angle of the total final displacement vector using the arctangent of the ratio of total y-component to total x-component, \(\tan^{-1}(-10.0m / x_{total})\).
04

Summarize the final answer

The final answer indicates the remaining distance the dog has to run and the direction in which it must run to end up at a point 10.0m south of the starting location. The solution is calculated using vector sums and the arctangent function.

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Key Concepts

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

Displacement Vector
Understanding displacement vectors is crucial for solving problems involving motion in physics. A displacement vector represents the change in position of an object from its starting point to its final location, irrespective of the path taken. It is a vector quantity, meaning it has both magnitude and direction.

Let's consider our furry friend from the exercise. The pup's initial jaunt of 12.0m east is a straight shot in the positive x-direction; hence, the y-component of this part of the trip is zero. Then, the dog makes a second move at a quirky angle鈥�50 degrees west of due north. By weaving in some trigonometry, essential for decoding the vector components in any given direction, we break this leg of the journey into x (horizontal) and y (vertical) components, even when the trek isn't aligned with the cardinal directions.

To zero in on the specifics of vector components, imagine splitting the second movement into a right-angled triangle. The x-component lies along the horizontal axis and the y-component along the vertical. Using trigonometric functions sine and cosine, you can calculate the exact lengths of these sides reflecting east-west and north-south movements, respectively.
Trigonometry in Physics
Trigonometry is the mathematical superstar of physics when it comes to angles and triangles, crucial for untangling problems with vectors. In the dog's escapade, that 28.0m move at 50 degrees west of north sends us hunting for a right triangle hidden in the vector diagram.

To extract the components, we use the cosine function for the x-component (adjacent side over hypotenuse) which tells us how much of that 28.0m is headed westward. Conversely, the sine function comes into play for figuring out the y-component (opposite side over hypotenuse) specifying the extent of the northward movement. Here's a little revelation: due to the 'west of north' angle, the x-component emerges negative, a tiny yet pivotal detail indicating the dog's westward swing.

Equipped with trigonometry, physics takes a smoother turn. It enables us to deal with angles, translate them into directional movements, and blend them into the grand tapestry of vector components. Therefore, a firm grasp on sine, cosine, and their buddy tangent, not overlooking their inverse functions for angle retrieval, sets the stage for mastering vector-related problems.
Vector Addition
Finding the overall effect of the dog's exploratory zigzags involves mastery over vector addition. In simple terms, vector addition allows us to combine multiple vectors into a single resultant vector, which here represents our adorable canine's complete displacement.

Let's dive into the math. We meticulously add the x-components and y-components separately. The x-components mingle to form a new x-component of the total displacement, while the y-components unite in a similar fashion. The final displacement vector is the synthesis of these totals, telling us where the pooch ultimately ends up, relative to the starting point.

Vector addition isn't just about piling numbers; it's about understanding how directional movements intertwine. In real life, that means recognizing how multiple forces interact on an object or, in our current furry scenario, it鈥檚 plotting the pup's precise trek across the field. To sum it up, learning how to add vectors coherently unlocks the capability to solve a wide array of physics puzzles, including the path a playful dog might take in an open field.

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Most popular questions from this chapter

Vector \(\vec{A}\) has magnitude \(5.00 \mathrm{~m}\) and lies in the \(x y\) -plane in a direction \(53.0^{\circ}\) from the \(+x\) -axis axis measured toward the \(+y\) -axis. Vector \(\vec{B}\) has magnitude \(8.00 \mathrm{~m}\) and a direction you can adjust. (a) You want the vector product \(\vec{A} \times \vec{B}\) to have a positive \(z\) -component of the largest possible magnitude. What direction should you select for vector \(\overrightarrow{\boldsymbol{B}} ?(\mathrm{~b})\) What is the direction of \(\overrightarrow{\boldsymbol{B}}\) for which \(\overrightarrow{\boldsymbol{A}} \times \overrightarrow{\boldsymbol{B}}\) has the most negative \(z\) -component? (c) What are the two directions of \(\overrightarrow{\boldsymbol{B}}\) for which \(\vec{A} \times \vec{B}\) is zero?

Vectors \(\vec{A}\) and \(\vec{B}\) are in the \(x y\) -plane. Vector \(\vec{A}\) is in the \(+x\) - direction, and the direction of vector \(\overrightarrow{\boldsymbol{B}}\) is at an angle \(\theta\) from the \(+x\) -axis measured toward the \(+y\) -axis. (a) If \(\theta\) is in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\), for what two values of \(\theta\) does the scalar product \(\vec{A} \cdot \vec{B}\) have its maximum magnitude? For each of these values of \(\theta,\) what is the magnitude of the vector product \(\vec{A} \times \vec{B} ?(b)\) If \(\theta\) is in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\) for what value of \(\theta\) does the vector product \(\vec{A} \times \vec{B}\) have its maximum value? For this value of \(\theta,\) what is the magnitude of the scalar product \(\vec{A} \cdot \vec{B} ?(\mathrm{c})\) What is the angle \(\theta\) in the range \(0^{\circ} \leq \theta \leq 180^{\circ}\) for which \(\vec{A} \cdot \vec{B}\) is twice \(|\vec{A} \times \vec{B}| ?\)

Getting Back. An explorer in Antarctica leaves his shelter during a whiteout. He takes 40 steps northeast, next 80 steps at \(60^{\circ}\) north of west, and then 50 steps due south. Assume all of his steps are equal in length. (a) Sketch, roughly to scale, the three vectors and their resultant. (b) Save the explorer from becoming hopelessly lost by giving him the displacement, calculated by using the method of components, that will return him to his shelter.

The Hydrogen Maser. A maser is a laser-type device that produces electromagnetic waves with frequencies in the microwave and radio-wave bands of the electromagnetic spectrum. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency of these waves is 1,420,405,751.786 hertz. (A hertz is another name for one cycle per second.) A clock controlled by a hydrogen maser is off by only \(1 \mathrm{~s}\) in 100,000 years. For the following questions, use only three significant figures. (The large number of significant figures given for the frequency simply illustrates the remarkable accuracy to which it has been measured.) (a) What is the time for one cycle of the radio wave? (b) How many cycles occur in 1 h? (c) How many cycles would have occurred during the age of the earth, which is estimated to be \(4.6 \times 10^{9}\) years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

The volume of a solid cylinder is given by \(V=\pi r^{2} h,\) where \(r\) is the radius and \(h\) is the height. You measure the radius and height of a thin cylindrical wire and obtain the results \(r=0.036 \mathrm{~cm}\) and \(h=12.1 \mathrm{~cm}\). What do your measurements give for the volume of the wire in \(\mathrm{mm}^{3} ?\) Use the correct number of significant figures in your answer.
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