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Chapter 6: Problem 19

A plane in crosswinds. You are trying to steer an airplane towards the north. The airspeed of your plane is \(300 \mathrm{~km} / \mathrm{h}\). However, there is a strong wind from the west, with a wind speed of \(60 \mathrm{~km} / \mathrm{h}\). (a) In what direction should you direct the plane so that it travels towards the north? Illustrate your argument with a diagram. (b) What is the speed of the plane relative to the ground?

Short Answer

Expert verified
Direct the plane at 101.54 degrees to the west of north. The speed relative to the ground is 294 km/h.

Step by step solution

01

Understand the problem

The plane needs to travel north, but there is a crosswind from the west pushing it to the east. We need to determine the direction the plane should be steered and its speed relative to the ground.
02

Represent velocities as vectors

Let the plane's airspeed vector be \(\vec{v_p} = 300 \, \text{km/h}\) directed north. The wind velocity vector is \(\vec{v_w} = 60 \, \text{km/h}\) directed east. We need to find the resultant vector \(\vec{v_r}\) that points north.
03

Set up vector components

Let the angle \(\theta\) represent the deviation to the west that the plane must steer to counteract the wind. The goal is for the resultant vector to have no east-west component.
04

Resolve the plane's velocity vector

The plane's velocity vector in terms of its components will be \(\vec{v_p} = 300\cos(\theta)\hat{i} + 300\sin(\theta)\hat{j}\). The wind vector is \(60\hat{i}\).
05

Set up the equation for zero east-west component

For the resultant vector to have zero east-west component, the sum of the east-west components must equal zero: \(300\cos(\theta) + 60 = 0\).
06

Solve for \(\theta\)

Solving \(300\cos(\theta) + 60 = 0\) gives \(\cos(\theta) = -\frac{60}{300} = -0.2\). This yields \(\theta = 101.54^\circ\).
07

Determine the northward speed

To find the speed relative to the ground, use the northward component of the airspeed: \(300\sin(\theta)\). Using \(\theta = 101.54^\circ\), \(\sin(101.54^\circ) = 0.98\), thus, the speed is approximately \(300 \times 0.98 = 294 \, \text{km/h}\).
08

Draw a vector diagram

Draw the vectors representing the plane's direction, the wind, and the resultant vector to the north. The plane's vector should be slanted to the west by 101.54 degrees relative to the north.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
In mechanics, a vector is a quantity that has both magnitude and direction. The velocity of a plane and the speed of the wind are both vectors.

To make sense of complicated movement, we often break vectors into components. Component vectors are simpler vectors that add up to the original vector.

For example, a plane's velocity in the north direction can be split into two components: one going east and the other going west.

In this exercise, the wind's impact also needs to be considered as a vector that influences the plane’s direction and speed.
Crosswinds
Crosswinds are winds that blow perpendicular to the direction of travel. They can significantly affect the trajectory of any moving object, such as an airplane.

For a plane aiming to travel north while experiencing a strong wind from the west, crosswinds push the plane eastward.

Pilots must adjust their course to counteract this effect. They can do this by steering slightly into the wind, a process known as crabbing.

Understanding crosswinds involves mastering the mechanics of how these forces interact and affect the resultant motion of the plane.
Resultant Velocity
The resultant velocity is the actual direction and speed of an object when all influencing factors are accounted for.

In this example, we need to find the resultant velocity of the plane relative to the ground, which includes both the plane's airspeed and the effect of the westward wind.

By resolving the vectors involved, we can determine the total impact on the plane's movement. Calculating the resultant velocity helps the pilot understand how to adjust for crosswinds.

The key is to add the velocity components vectorially to ascertain the true path of the airplane.
Trigonometry in Physics
Trigonometry is a branch of mathematics dealing with the relationships between angles and sides of triangles. It is extensively used in physics to resolve vector components.

In the given problem, we use trigonometric functions to break down the plane's velocity into northward and westward components.

The equation involving cosines and sines allows us to calculate the precise angle at which the pilot should steer to counteract the wind.

Specifically, we used the cosine function to solve for the steering angle \(\theta\) and the sine function to find the northward speed of the plane.

Applying these trigonometric principles ensures that the resultant vector points accurately toward the desired north direction, irrespective of the crosswind.

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