/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 A small aircraft is headed due s... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 69

A small aircraft is headed due south with a speed of \(57.8 \mathrm{~m} / \mathrm{s}\) with respect to still air. Then, for \(9.00 \times 10^{2}\) s a wind blows the plane so that it moves in a direction \(45.0^{\circ}\) west of south, even though the plane continues to point due south. The plane travels \(81.0 \mathrm{~km}\) with respect to the ground in this time. Determine the velocity (magnitude and direction) of the wind with respect to the ground. Determine the directional angle relative to due south.

Short Answer

Expert verified
The wind velocity is 71.5 m/s at an angle of 63.4掳 west of south.

Step by step solution

01

Determine the Total Displacement

First, convert the plane's travel distance from kilometers to meters, which gives us the displacement: \( 81.0 \text{ km} = 81000 \text{ meters} \).
02

Find the Average Velocity with Respect to the Ground

Calculate the average velocity of the plane with respect to the ground using the formula \( v = \frac{d}{t} \), where \( d \) is displacement and \( t \) is time. Thus, \( v = \frac{81000 \text{ m}}{900 \text{ s}} = 90 \text{ m/s} \).
03

Break Down Plane's Velocity into Components

The plane is moving due south at \( 57.8 \text{ m/s} \). Decompose this velocity into south and west components because the wind's effect results in a southwest direction. Southward component remains \( 57.8 \text{ m/s} \).
04

Set up the Westward Component Equation

Using trigonometric principles for a vector moving at \( 45.0^{\circ} \) west of south, the westward movement can be expressed as \( v_{\text{west}} = v_{\text{ground}} \cdot \sin(45^{\circ}) = 90 \times \sin(45^{\circ}) = 63.64 \text{ m/s} \).
05

Calculation of Wind Speed

The wind velocity must account for moving the plane westward and adjusting the southward apex to \( 90 \text{ m/s} \) in resultant direction. Use vector addition to solve for westward wind component: \( v_{\text{wind-west}} = 63.64 \text{ m/s} \).
06

Calculate Resultant Magnitude of Wind Velocity

Calculate the magnitude of the wind's velocity, using Pythagorean theorem for components as: \( v_{\text{wind}} = \sqrt{v_{\text{south-change}}^2 + v_{\text{west}}^2} = \sqrt{(32.2)^2 + (63.64)^2} = 71.5 \text{ m/s} \).
07

Determine the Direction of Wind

Find the angle relative to south using \( \tan(\theta) = \frac{v_{\text{west}}}{v_{\text{south-change}}} \), solve for \( \theta \): \( \theta = \tan^{-1}(\frac{63.64}{32.2}) = 63.4^{\circ} \).

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Key Concepts

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

Vector Components
When analyzing motion, vectors describe quantities that have both direction and magnitude. In this exercise, the airplane's velocity and the wind's effect are treated as vectors.
  • The plane's velocity relative to still air is purely southward.
  • When the wind blows, it adds a westward component to the plane's velocity.
To understand the resulting motion, we break down these vectors into components:
  • A southward component remained from the plane's original path.
  • A westward component came from the wind's influence.
Using this approach allows the establishment of a complete picture of how the two velocities result in a single new direction and speed.
Trigonometry in Physics
Trigonometry helps us resolve vectors into perpendicular components, usually along the x (horizontal) and y (vertical) axes. In our problem, trigonometric functions are used to determine how much of the total plane velocity is directed west and how much remains south.The plane is observed moving at an angle of 45掳 west of south.
  • The sine function (\(\sin(45^{\circ})\)) was used to find the westward component.This tells us how much of the plane's resultant velocity is moving westward due to the wind.
  • Similarly, functions such as cosine could be used to find other components if needed.
By applying these trigonometric principles, complex vector problems can be simplified into manageable calculations.
Vector Addition
Vector addition allows us to combine different velocity components to find a single resultant vector. Here, the plane's southward speed had to be combined with the wind's westward push. The resultant vector represents the plane's actual path on the ground.
  • Conceptually, this is done by placing the tail of one vector at the head of the previous one.
  • Mathematically, we apply the Pythagorean theorem to compute the magnitude of the resultant vector:
\[v_{\text{resultant}} = \sqrt{v_{\text{south}}^2 + v_{\text{west}}^2}\]This process helps reveal the ultimate velocity of any object subject to multiple forces.
Wind Effect on Motion
The wind impacts the motion of objects by altering their intended path and speed. Here, the plane travels with a pure south velocity, but the wind changes this dynamically by adding a new vector component.
  • The westward component introduced by the wind affects the plane's trajectory.
  • Resulting in a skewed path compared to the original direction.
Despite the plane pointing southward, it moves diagonally southwest with the ground. This teaches us about real-world motion, emphasizing perception vs. reality. Understanding the wind's effect highlights the importance of accounting for external forces when predicting movement outcomes.

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Most popular questions from this chapter

In a mall, a shopper rides up an escalator between floors. At the top of the escalator, the shopper turns right and walks \(9.00 \mathrm{~m}\) to a store. The magnitude of the shopper's displacement from the bottom of the escalator is \(16.0 \mathrm{~m}\). The vertical distance between the floors is \(6.00 \mathrm{~m}\). At what angle is the escalator inclined above the horizontal?

In a football game a kicker attempts a field goal. The ball remains in contact with the kicker's foot for \(0.050 \mathrm{~s}\), during which time it experiences an acceleration of \(340 \mathrm{~m} / \mathrm{s}^{2}\). The ball is launched at an angle of \(51^{\circ}\) above the ground. Determine the horizontal and vertical components of the launch velocity.

A car drives straight off the edge of a cliff that is \(54 \mathrm{~m}\) high. The police at the scene of the accident observe that the point of impact is \(130 \mathrm{~m}\) from the base of the cliff. How fast was the car traveling when it went over the cliff?

A major-league pitcher can throw a baseball in excess of \(41.0 \mathrm{~m} / \mathrm{s}\). If a ball is thrown horizontally at this speed, how much will it drop by the time it reaches a catcher who is \(17.0 \mathrm{~m}\) away from the point of release?

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