Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 4b

Chapter 3, Problem 4b

The position of a squirrel running in a park is given by $\vec{r}\) = $\left$[ (0.280~$\mathrm{m/s}$)t + (0.0360~$\mathrm{m/s^2}$)t^2 $\right$] $\hat{i}$ + (0.0190~$\mathrm{m/s^3}$)t^3 \(\hat{j}$ . At $t equals 5.00 s$ , how far is the squirrel from its initial position?

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Identify the position function given: $ \mathbf{r}(t) = [(0.280 \text{ m/s})t + (0.0360 \text{ m/s}^2)t^2]\hat{\mathbf{i}} + (0.0190 \text{ m/s}^3)t^3\hat{\mathbf{j}} $. This function describes the position of the squirrel in terms of time $ t $.

To find the position of the squirrel at $ t = 5.00 \text{ s} $, substitute $ t = 5.00 \text{ s} $ into the position function $ \mathbf{r}(t) $. Calculate each component separately: $ \hat{\mathbf{i}} $ and $ \hat{\mathbf{j}} $.

For the $ \hat{\mathbf{i}} $ component, substitute $ t = 5.00 \text{ s} $ into $ (0.280 \text{ m/s})t + (0.0360 \text{ m/s}^2)t^2 $ and compute the value.

For the $ \hat{\mathbf{j}} $ component, substitute $ t = 5.00 \text{ s} $ into $ (0.0190 \text{ m/s}^3)t^3 $ and compute the value.

Combine the computed $ \hat{\mathbf{i}} $ and $ \hat{\mathbf{j}} $ components to find the position vector $ \mathbf{r}(5.00 \text{ s}) $. The magnitude of this vector will give the distance from the initial position at $ t = 0 \text{ s} $. Use the formula $ \sqrt{(\text{component } \hat{\mathbf{i}})^2 + (\text{component } \hat{\mathbf{j}})^2} $ to find the distance.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Vector

The position vector describes the location of an object in space relative to a reference point, often the origin. In this problem, the position vector r is given as a function of time, with components in the î and ĵ directions, representing the squirrel's position in a two-dimensional plane.

Displacement

Displacement is the vector quantity that represents the change in position of an object. It is calculated by finding the difference between the final and initial position vectors. In this problem, the displacement at t = 5.00 s is determined by evaluating the position vector at this time and subtracting the initial position.

Vector Addition

Vector addition involves combining vectors by adding their corresponding components. To find the squirrel's displacement, the components of the position vector at t = 5.00 s are added algebraically. This process involves summing the individual î and ĵ components to determine the total displacement vector.

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Textbook Question

The position of a squirrel running in a park is given by $\vec{r}\) = $\left$[ (0.280~$\mathrm{m/s}$)t + (0.0360~$\mathrm{m/s^2}$)t^2 $\right$] $\hat{i}$ + (0.0190~$\mathrm{m/s^3}$)t^3 \(\hat{j}$ . (a) What are $v_{x} (t)$ and $v_{y} (t)$ , the $x$ -and $y$ -components of the velocity of the squirrel, as functions of time?

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A web page designer creates an animation in which a dot on a computer screen has position $\vec{r} = [4.0 cm + (2.5 {cm/s}^{2}) t^{2}] i + (5.0 cm/s) t j$ . Find the magnitude and direction of the dot's average velocity between $t equals 0$ and $t equals 2.0 s$ .

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Textbook Question

The coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m − βt², where α = 2.4 m/s and β = 1.2 m/s². (a) Sketch the path of the bird between t = 0 and t = 2.0 s.

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Textbook Question

The coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m − βt², where α = 2.4 m/s and β = 1.2 m/s². Calculate the velocity and acceleration vectors of the bird as functions of time.

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Textbook Question

A dog running in an open field has components of velocity v_x = 2.6 m/s and v_y = −1.8 m/s at t1 = 10.0 s. For the time interval from t₁ = 10.0 s to t₂ = 20.0 s, the average acceleration of the dog has magnitude 0.45 m/s² and direction 31.0° measured from the +x–axis toward the +y–axis. At t₂ = 20.0 s, what are the x- and y-components of the dog's velocity?

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