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.

Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 7a

Chapter 3, Problem 7a

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.

Verified step by step guidance

Identify the equations of motion for the bird: x(t) = αt and y(t) = 3.0 m − βt², where α = 2.4 m/s and β = 1.2 m/s².

Substitute the given values of α and β into the equations: x(t) = 2.4t and y(t) = 3.0 − 1.2t².

Determine the coordinates of the bird at specific time intervals between t = 0 and t = 2.0 s. For example, calculate x(0), y(0), x(1), y(1), x(2), and y(2).

Plot these coordinates on the xy-plane to visualize the path. Start by marking the points (x(0), y(0)), (x(1), y(1)), and (x(2), y(2)).

Connect the plotted points with a smooth curve to represent the bird's path, noting that the x-coordinate increases linearly while the y-coordinate follows a parabolic trajectory.

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often time (t). In this problem, x(t) = αt and y(t) = 3.0 m − βt² describe the bird's position in the xy-plane over time. Understanding these equations helps in plotting the trajectory by calculating x and y for various t values.

Kinematics in Two Dimensions

Kinematics in two dimensions involves analyzing motion along two axes, typically x and y. The bird's motion is described by its horizontal and vertical components, x(t) and y(t). This concept is crucial for understanding how the bird's position changes over time and how to sketch its path accurately.

Graphical Representation of Motion

Graphical representation involves plotting the path of an object based on its position equations. By calculating the bird's x and y coordinates at different time intervals, one can sketch its trajectory on the xy-plane. This visual representation helps in understanding the overall motion pattern and direction of the bird.

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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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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 magnitude and direction of the bird's velocity and acceleration at 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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