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 34a

Chapter 3, Problem 34a

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.

Verified step by step guidance

Begin by identifying the vectors involved in the problem. The plane's airspeed vector, $ \mathbf{v}_{P/A} $, is 35 m/s due south, and the wind vector, $ \mathbf{v}_{A/E} $, is 10 m/s toward the southwest.

Understand that the velocity of the plane relative to the earth, $ \mathbf{v}_{P/E} $, is the vector sum of the plane's velocity relative to the air, $ \mathbf{v}_{P/A} $, and the air's velocity relative to the earth, $ \mathbf{v}_{A/E} $. This can be expressed as: $ \mathbf{v}_{P/E} = \mathbf{v}_{P/A} + \mathbf{v}_{A/E} $.

Draw the vector $ \mathbf{v}_{P/A} $ pointing directly south. This vector is vertical on the diagram, with a magnitude of 35 m/s.

Next, draw the vector $ \mathbf{v}_{A/E} $ pointing southwest. This vector forms a 45-degree angle with the south direction, as southwest is halfway between south and west. The magnitude of this vector is 10 m/s.

Finally, use the tip-to-tail method to add the vectors. Place the tail of $ \mathbf{v}_{A/E} $ at the tip of $ \mathbf{v}_{P/A} $. The resultant vector $ \mathbf{v}_{P/E} $ is drawn from the tail of $ \mathbf{v}_{P/A} $ to the tip of $ \mathbf{v}_{A/E} $. This vector represents the velocity of the plane relative to the earth.

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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 a method used to combine two or more vectors to determine a resultant vector. In this context, it involves adding the plane's velocity vector and the wind's velocity vector to find the plane's velocity relative to the earth. This is typically done using graphical methods like the head-to-tail method or by breaking vectors into components.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular frame of reference. Here, it refers to the velocity of the plane relative to the earth, which is affected by both the plane's airspeed and the wind's speed and direction. Understanding relative velocity is crucial for determining how the plane's movement is influenced by external factors like wind.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system, typically the x and y axes. Breaking vectors into components simplifies calculations, especially when dealing with vectors at angles. In this problem, the wind's southwest direction requires resolving it into south and west components to accurately perform vector addition.

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Related Practice

Textbook Question

Two piers, A and B, are located on a river; B is 1500 m downstream from A (Fig. E3.32). Two friends must make round trips from pier A to pier B and return. One rows a boat at a constant speed of 4.00 km/h relative to the water; the other walks on the shore at a constant speed of 4.00 km/h. The velocity of the river is 2.80 km/h in the direction from A to B. How much time does it take each person to make the round trip?

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Textbook Question

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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Textbook Question

A 'moving sidewalk' in an airport terminal moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does it take her to reach the opposite end if she walks In the opposite direction?

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Textbook Question

A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

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Textbook Question

A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. How much time is required to cross the river?

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Textbook Question

A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. What is your velocity (magnitude and direction) relative to the earth?

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