The nose of an ultralight plane is pointed due south, and its airspeed indicator shows 35 m/s35\text{ m/s}. The plane is in a 10 m/s10\text{ m/s} wind blowing toward the southwest relative to the earth. Let xx be east and yy be north, and find the components of v→P/E\overrightarrow{v}_{P/E} .

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 34b

Chapter 3, Problem 34b

The nose of an ultralight plane is pointed due south, and its airspeed indicator shows $35 m slash s$ . The plane is in a $10 m slash s$ wind blowing toward the southwest relative to the earth. Let $x$ be east and $y$ be north, and find the components of ${\vec{v}}_{P / E}$ .

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First, identify the velocity of the plane relative to the earth, $ \mathbf{v}_{P/E} $, which is the vector sum of the plane's velocity relative to the air, $ \mathbf{v}_{P/A} $, and the wind's velocity relative to the earth, $ \mathbf{v}_{A/E} $.

The plane's velocity relative to the air, $ \mathbf{v}_{P/A} $, is 35 m/s due south. In terms of components, this is $ \mathbf{v}_{P/A} = (0, -35) $ m/s, since it has no eastward component and a southward component.

The wind's velocity relative to the earth, $ \mathbf{v}_{A/E} $, is 10 m/s toward the southwest. Southwest is 45 degrees from both south and west, so the components are $ \mathbf{v}_{A/E} = (-10 \cos 45^\circ, -10 \sin 45^\circ) $ m/s.

Calculate the components of $ \mathbf{v}_{A/E} $ using trigonometric identities: $ \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} $. Thus, $ \mathbf{v}_{A/E} = (-10 \times \frac{\sqrt{2}}{2}, -10 \times \frac{\sqrt{2}}{2}) $ m/s.

Add the components of $ \mathbf{v}_{P/A} $ and $ \mathbf{v}_{A/E} $ to find $ \mathbf{v}_{P/E} $: $ \mathbf{v}_{P/E} = (0 + (-10 \times \frac{\sqrt{2}}{2}), -35 + (-10 \times \frac{\sqrt{2}}{2})) $ m/s.

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Key Concepts

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

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In this context, the plane's velocity vector and the wind's velocity vector must be added to find the plane's actual velocity relative to the Earth. This involves breaking each vector into its components and summing them accordingly.

Component Vectors

Component vectors are the projections of a vector along the axes of a coordinate system, typically x and y. To solve the problem, the plane's southward velocity and the wind's southwest velocity must be expressed in terms of their eastward and northward components, using trigonometric functions like sine and cosine.

Trigonometry in Physics

Trigonometry is used in physics to resolve vectors into components. For a vector at an angle, the cosine function helps find the adjacent side (x-component), and the sine function helps find the opposite side (y-component). In this problem, trigonometry is essential to determine the components of the wind's southwest direction.

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Textbook Question

The nose of an ultralight plane is pointed due south, and its airspeed indicator shows $35$\text{ m/s}$$ . The plane is in a $10$\text{ m/s}$$ wind blowing toward the southwest relative to the earth. In a vector-addition diagram, show the relationship of $$\overrightarrow{v}$_{P/E}$ (the velocity of the plane relative to the earth) to the two given vectors.

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