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Chapter 3: Problem 84

City A lies directly west of city \(B\) . When there is no wind, an airliner makes the 5550 -kmround-trip fight between them in 6.60 \(\mathrm{h}\) of flying time while traveling at the same speed in both directions. When a strong, steady \(225-\mathrm{km} / \mathrm{h}\) wind is blowing from west to east and the airliner has the same airspeed as before, how long will the trip take?

Short Answer

Expert verified
The trip will take 7.12 hours.

Step by step solution

01

Determine Airliner's Airspeed

When there is no wind, the total round-trip distance is 5550 km, completed in 6.60 hours. The airliner's airspeed can be calculated using \(\text{speed} = \frac{\text{distance}}{\text{time}}\). Hence, the airspeed \( v = \frac{5550 \text{ km}}{6.60 \text{ h}} = 840 \text{ km/h}\).
02

Calculate Ground Speed with Wind

When wind is blowing from west to east at 225 km/h, the wind affects the ground speed differently in each direction. - **From A to B**: Airspeed relative to ground = 840 km/h + 225 km/h = 1065 km/h. - **From B to A**: Airspeed relative to ground = 840 km/h - 225 km/h = 615 km/h.
03

Calculate Time for Each Leg of the Trip

The distance for one way is half of the round trip: \(\frac{5550 \text{ km}}{2} = 2775 \text{ km}\).- **Time from A to B**: \(\frac{2775 \text{ km}}{1065 \text{ km/h}} \approx 2.61 \text{ h}\).- **Time from B to A**: \(\frac{2775 \text{ km}}{615 \text{ km/h}} \approx 4.51 \text{ h}\).
04

Calculate Total Round-Trip Time

Add the times for each leg of the journey: \(2.61 \text{ h} + 4.51 \text{ h} = 7.12 \text{ h}\).Thus, the total time for the trip with the wind is 7.12 hours.

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Key Concepts

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

Airspeed Calculation
To understand airspeed, it's important to know that this is the speed an aircraft travels relative to the air around it. When we're talking about an airplane flying without any wind, airspeed is simply the distance covered divided by the time taken. In the exercise, the airliner travels a total round-trip distance of 5550 km without wind in 6.60 hours. Using the formula for speed, \(\text{speed} = \frac{\text{distance}}{\text{time}}\), we find that the airspeed is \(\frac{5550 \ \text{km}}{6.60 \ \text{h}} = 840 \ \text{km/h}\). This is crucial because this airspeed of 840 km/h remains unchanged, regardless of whether there is wind or not. Airspeed only measures how fast the plane is moving through the air, not over the ground.
Ground Speed
Ground speed refers to how fast an aircraft moves relative to the ground. This speed can differ from airspeed when the wind is present. The ground speed accounts for both the speed of the airplane and the wind speed working with or against it. In the given problem, when the wind blows from west to east at 225 km/h, it alters the ground speed as follows:
  • From City A to City B: Here, the wind aids the airplane, increasing its effective speed over the ground. The ground speed is the sum of the airspeed and wind speed: \(840 \ \text{km/h} + 225 \ \text{km/h} = 1065 \ \text{km/h}\).

  • From City B to City A: In this direction, the wind opposes the airplane, reducing its speed over the ground. The ground speed is the difference between the airspeed and wind speed: \(840 \ \text{km/h} - 225 \ \text{km/h} = 615 \ \text{km/h}\).
The keen importance of calculating ground speed is that it provides the actual travel speed of the airplane along its path.
Wind Speed Impact on Flight Time
Wind can significantly impact flight time, as it affects the ground speed during a journey. Positive or tailwinds help an aircraft move faster over the ground, potentially reducing travel time, while headwinds work against it, increasing travel time. From the problem's calculations:
  • Flying from City A to City B: With a helpful tailwind, the plane covers 2775 km at 1065 km/h. This results in a travel time of \(\frac{2775 \ \text{km}}{1065 \ \text{km/h}} \approx 2.61 \ \text{hours}\).

  • Flying from City B to City A: Against the headwind, the same distance of 2775 km is covered at a reduced ground speed of 615 km/h, increasing the travel time to \(\frac{2775 \ \text{km}}{615 \ \text{km/h}} \approx 4.51 \ \text{hours}\).
When we add these two times together, the total round-trip time comes out to be approximately 7.12 hours, illustrating the impact of wind on overall travel time. This is longer compared to the 6.60 hours when there was no wind, demonstrating how substantial wind's role can be in flight duration.

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