Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 29

A flock of geese is attempting to migrate due south, but the wind is blowing from the west at \(5.1 \mathrm{m} / \mathrm{s}\). If the birds can fly at \(7.5 \mathrm{m} / \mathrm{s}\) relative to the air, what direction should they head?

Short Answer

Expert verified
The geese should aim to fly a few degrees west of due south to compensate for the westward wind. The exact angle can be calculated using the equation from Step 3.

Step by step solution

01

Identify given vectors

The geese attempt to fly due south, and the velocity of the geese relative to the air is \(7.5 \mathrm{m} / \mathrm{s}\), which we'll call \( \vec{v_{g}}\). The wind velocity is blowing from the west towards the east at \(5.1 \mathrm{m}/\mathrm{s}\), and we will denote it as \( \vec{v_{w}}\). The overall velocity of the geese \( \vec{v}\) can be written as the vector sum \( \vec{v_{g}} + \vec{v_{w}}\).
02

Create a triangle using the vectors

One can depict the situation as a triangle. The hypotenuse represents the geese's velocity, the side opposite to the angle we need, represents the wind's velocity, and the adjacent side represents the desired direction of geese. This triangle indicates that the angle should be calculated with the help of trigonometry (specifically, the tangent function).
03

Calculate the direction

We know from trigonometry that tan of the angle theta can be obtained by dividing the side opposite to theta by the adjacent side. Therefore, we can use the wind velocity vector as the opposite side and the unknown geese velocity in the southward direction as the adjacent side. As a result, the equation to find the desired direction is \[ tan(\theta) = \frac{v_{w}}{v_{g}} \]\ Applying the inverse tangent function (arctan or tan^-1) on both sides we have \ \[ \theta = arctan\left(\frac{v_{w}}{v_{g}}\right) \] Substituting the given values into the equation gives us the direction that the geese should aim for: \[ \theta = arctan\left(\frac{5.1}{7.5}\right) \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How is it possible for an object to be moving in one direction but accelerating in another?

A basketball player is 15 ft horizontally from the center of the basket, which is \(10 \mathrm{ft}\) off the ground. At what angle should the player aim the ball from a height of \(8.2 \mathrm{ft}\) with a speed of \(26 \mathrm{ft} / \mathrm{s} ?\)

Attempting to stop on a slippery road, a car moving at \(80 \mathrm{km} / \mathrm{h}\) skids at \(30^{\circ}\) to its initial motion, stopping in 3.9 s. Determine the average acceleration in \(\mathrm{m} / \mathrm{s}^{2},\) in coordinates with the \(x\) -axis in the direction of the original motion and the \(y\) -axis toward the side to which the car skids.

Let \(A=15^{T}-40 \mathrm{J}\) and \(B=31 \mathrm{J}+18 \mathrm{k} .\) Find \(C\) such that \(\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}\)

What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4 -cm-long hour hand of a clock in the interval from noon to 6 PM? Use unit vector notation, with the \(x\) -axis pointing toward 3 and the \(y\) -axis toward noon.
See all solutions

Recommended explanations on Physics Textbooks

Space Physics

Read Explanation

Scientific Method Physics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Fields in Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free