/*! 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 88 Long flights at midlatitudes in ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 88

Long flights at midlatitudes in the Northern Hemisphere encounter the jet stream, an eastward airflow that can affect a plane's speed relative to Earth's surface. If a pilot maintains a certain speed relative to the air (the plane's airspeed), the speed relative to the surface (the plane's ground speed) is more when the flight is in the direction of the jet stream and less when the flight is opposite the jet stream. Suppose a round-trip flight is scheduled between two cities separated by \(4000 \mathrm{~km}\), with the outgoing flight in the direction of the jet stream and the return flight opposite it. The airline computer advises an airspeed of \(1000 \mathrm{~km} / \mathrm{h},\) for which the difference in flight times for the outgoing and return flights is \(70.0 \mathrm{~min} .\) What jet-stream speed is the computer using?

Short Answer

Expert verified
The jet-stream speed is 200 km/h.

Step by step solution

01

Understanding the given information

The total round-trip distance is the sum of the distances for both the outgoing and return trips, i.e., \( 4000 \, \text{km} \) each way, so \( 8000 \, \text{km} \) overall. The airspeed is given as \( 1000 \, \text{km/h} \), and the difference in flight times is \( 70.0 \, \text{minutes} \), which translates to \( 70.0 \, \text{minutes} = \frac{70}{60} \, \text{hours} = \frac{7}{6} \, \text{hours} \). We are asked to identify the jet-stream speed.
02

Define variables and relation

Let's denote the jet-stream speed as \( v_j \) (in km/h). The speed of the plane relative to Earth's surface or ground speed is \( 1000 + v_j \) km/h in the direction of the jet stream and \( 1000 - v_j \) km/h against the jet stream.
03

Set up equations for flight times

The time taken for the outgoing flight in the direction of the jet stream is \( t_1 = \frac{4000}{1000 + v_j} \) hours. The time for the return flight against the jet stream is \( t_2 = \frac{4000}{1000 - v_j} \) hours. The difference in these times is given: \( t_2 - t_1 = \frac{7}{6} \).
04

Solve the equation for jet-stream speed

Substitute the expressions for \( t_1 \) and \( t_2 \):\[ \frac{4000}{1000 - v_j} - \frac{4000}{1000 + v_j} = \frac{7}{6} \].Simplify by multiplying through by the common denominator \( (1000 - v_j)(1000 + v_j) \):\[ 4000(1000 + v_j) - 4000(1000 - v_j) = \frac{7}{6} (1000^2 - v_j^2) \].This produces:\[ 8000v_j = \frac{7}{6} (1000000 - v_j^2) \].Multiply through by 6:\[ 48000v_j = 7000000 - 7v_j^2 \].Rearrange to get:\[ 7v_j^2 + 48000v_j - 7000000 = 0 \].
05

Solve quadratic equation

Use the quadratic formula \( v_j = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):Here, \( a = 7 \), \( b = 48000 \), \( c = -7000000 \).Calculate the discriminant:\[ b^2 - 4ac = 48000^2 - 4(7)(-7000000) \].Calculate \( v_j \) using positive root for physically meaningful (non-negative) speed. Solve to obtain the jet stream speed.

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Key Concepts

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

Airspeed
Airspeed refers to how fast an airplane is moving relative to the air around it. Think of it as the speed of a plane within the air, regardless of the movement of the air itself. In aviation, keeping a constant airspeed is crucial for maintaining the aircraft's performance and efficiency. For example, in the stated problem, the airspeed is a consistent 1000 km/h, meaning that this speed doesn't change whether the plane is flying with or against the jet stream.
Airspeed is essential for pilots to manage so that they can predict fuel consumption and flight times accurately. It ensures that they keep the aircraft stable and provides a basis for navigation and scheduling. Having a clear understanding of airspeed helps align with safety standards and contributes significantly to the successful management of a flight.
Ground Speed
Ground speed is the speed of an airplane relative to the Earth's surface. It considers both the airspeed and the wind speed affecting the aircraft. If there's no wind, both ground speed and airspeed match. However, when a plane encounters wind, especially high-speed currents like the jet stream, airspeed and ground speed differ.
In the example problem, the plane's ground speed increases when flying in the direction of the jet stream because the wind boosts its progress. Conversely, the ground speed diminishes when the plane flies against the jet stream as it faces a headwind. By knowing the ground speed, pilots can estimate the actual travel time and ensure they stay on schedule. It helps them in planning their routes and keeping passengers informed about time arrival and departure.
Northern Hemisphere
The Northern Hemisphere is the top half of the Earth above the equator. It's where countries like the United States, most of Europe, and much of Asia are located. In this hemisphere, the jet streams, strong winds found high in the atmosphere, primarily blow from west to east. This affects air travel significantly.
For flights scheduled in the Northern Hemisphere, pilots often encounter these jet streams, which can provide a tailwind or a headwind depending on the direction of travel. For example, a flight heading east might have a boosted ground speed due to a tailwind from the jet stream, whereas flying west might become more challenging due to opposing winds. Understanding wind patterns such as those in the Northern Hemisphere is crucial for efficiently planning flight paths and ensuring safe travel.
Round-trip Flight Distance
When discussing round-trip flight distance, it means the entire journey from the departure point to the destination and back again. In this problem example, the total round-trip distance is 8000 km, with each leg of the journey being 4000 km.
Calculating accurate round-trip distances is essential for several reasons. It allows airlines to plan for necessary fuel, estimate total travel time, and set ticket prices. Additionally, understanding the impact of different segments, like when one leg benefits from a tailwind or faces a headwind, can lead to more efficient flight planning.
Knowing the exact conditions experienced during each direction of a round trip can determine crew schedules and passenger itineraries. This ensures that the airline runs smoothly and passengers are made aware of their travel times, helping manage everyone鈥檚 expectations and experiences.

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Most popular questions from this chapter

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