Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 6a

Chapter 3, Problem 6a

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?

Verified step by step guidance

Start by understanding the given information: the initial velocity components at t1 = 10.0 s are υx = 2.6 m/s and υy = −1.8 m/s. The average acceleration over the time interval from t1 to t2 is 0.45 m/s² at an angle of 31.0° from the +x-axis toward the +y-axis.

Break down the average acceleration into its x and y components using trigonometry. The x-component of acceleration (ax) can be found using ax = a * cos(θ), and the y-component (ay) using ay = a * sin(θ), where a = 0.45 m/s² and θ = 31.0°.

Calculate the change in velocity for each component using the formula Δυ = a * Δt, where Δt = t2 - t1 = 20.0 s - 10.0 s = 10.0 s. Apply this formula separately for the x and y components using the respective acceleration components calculated in the previous step.

Determine the final velocity components at t2 = 20.0 s by adding the change in velocity to the initial velocity components. For the x-component, use υx_final = υx_initial + Δυx, and for the y-component, use υy_final = υy_initial + Δυy.

Summarize the process: You have calculated the x and y components of the dog's velocity at t2 by considering the initial velocity, the average acceleration, and the time interval. This approach allows you to find the final velocity components without directly calculating the numerical values.

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Key Concepts

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

Velocity Components

Velocity components refer to the projection of velocity along the coordinate axes, typically x and y. In this problem, the dog's velocity is given in terms of its x-component (υx = 2.6 m/s) and y-component (υy = −1.8 m/s). Understanding these components is crucial for calculating changes in velocity due to acceleration over time.

Average Acceleration

Average acceleration is the change in velocity over a given time interval, expressed as a vector with both magnitude and direction. Here, the average acceleration is 0.45 m/s² at an angle of 31.0° from the +x-axis toward the +y-axis. This concept helps determine how the velocity components change from t1 to t2.

Vector Addition

Vector addition involves combining vectors to find resultant vectors, such as velocity or acceleration. In this context, it is used to calculate the new velocity components at t2 by adding the initial velocity components to the change in velocity due to acceleration, considering both magnitude and direction.

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Related Practice

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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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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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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 magnitude and direction of the bird's velocity and acceleration at t = 2.0 s.

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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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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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