Problem 61 - College Physics

Question

A jetliner has a cruising air speed of \(600 \mathrm{mi} / \mathrm{h}\) relative to the air. How long does it take this plane to fly round trip from San Francisco to Chicago, an east-west flight of \(2000 \mathrm{mi}\) each way, (a) if there is no wind blowing and (b) if the wind is blowing at \(150 \mathrm{mi} / \mathrm{h}\) from the west to the east?

Step-by-Step Solution

Answer

(a) 6.67 hours, (b) 7.11 hours.

Step 1: Calculating Time with No Wind

First, determine the time taken for a one-way trip with no wind. Use the formula for time, which is \( \text{time} = \frac{\text{distance}}{\text{speed}} \). Here, the distance is 2000 miles and the speed is 600 miles per hour. \[ \text{Time for one way} = \frac{2000 \text{ mi}}{600 \text{ mi/h}} = \frac{10}{3} \text{ hours} \approx 3.33 \text{ hours} \]Since it's a round trip, multiply the time for one way by 2:\[ \text{Round trip time} = 2 \times \frac{10}{3} \text{ hours} = \frac{20}{3} \text{ hours} \approx 6.67 \text{ hours} \]

Step 2: Calculating Time with Wind to the East

Now, calculate the time for each leg of the trip when there is a wind blowing at 150 mi/h from west to east.On the way from San Francisco to Chicago:- The wind assists the plane, so the effective speed is \( 600 + 150 = 750 \text{ mi/h} \).- Time for this leg:\[ \text{Time eastbound} = \frac{2000 \text{ mi}}{750 \text{ mi/h}} = \frac{8}{3} \text{ hours} \approx 2.67 \text{ hours} \]

Step 3: Calculating Time with Wind to the West

For the return trip from Chicago to San Francisco:- The wind opposes the plane, so the effective speed is \( 600 - 150 = 450 \text{ mi/h} \).- Time for this leg:\[ \text{Time westbound} = \frac{2000 \text{ mi}}{450 \text{ mi/h}} = \frac{20}{9} \text{ hours} \approx 2.22 \times 4 \text{ hours} \approx 4.44 \text{ hours} \]

Step 4: Calculating Total Time with Wind

Add the time for both legs of the journey to get the total round trip time with wind:\[ \text{Total round trip time} = \frac{8}{3} + \frac{20}{9} \text{ hours} = \frac{24}{9} + \frac{20}{9} \text{ hours} = \frac{44}{9} \text{ hours} \approx 4.89 \times 2 \text{ hours} \approx 7.11 \text{ hours} \]

Key Concepts

Airplane Speed Calculations

To comprehend how fast an airplane is moving, you need to understand that speed is calculated by dividing the distance the airplane travels by the time it takes to cover that distance. This is expressed through the formula:

• \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

For example, if a jetliner travels 2000 miles at a speed of 600 miles per hour, you can use this formula to determine how much time the journey will take. Specifically, since the jetliner’s speed is 600 miles per hour, you need to divide the 2000 miles by 600 miles per hour to find the time for one way. This calculation shows us the time needed for a journey when the speed of the airplane doesn't change due to external factors like wind. Understanding how speed, time, and distance relate provides a solid foundation for further concepts like relative velocity and effects of wind on travel.

Relative Velocity

Relative velocity is how the speed of one object appears to change when measured from another moving object. In the case of airplanes, you often measure the speed of the airplane relative to the air around it. This is straightforward when there is no wind, as the airplane moves relative to a stationary air mass. However, the situation becomes complex with wind influence.
For example, if the wind blows in the same direction the airplane is traveling, it affects the airplane’s speed positively. The effective speed the airplane experiences, known as relative velocity, becomes the sum of its original speed and the wind speed. Conversely, if there is a headwind (wind blowing against the travel direction), the effective speed decreases, as the relative velocity is the airplane's speed minus the wind speed. This principle is crucial for calculating total travel times in exercises where wind plays a role in the motion.

Wind Effects on Motion

Wind can significantly influence the motion of airplanes, adding complexity to speed and time calculations. Its effects depend primarily on the direction and speed of the wind. When wind comes from behind (tailwind), it aids the airplane, allowing it to move faster over the ground thus reducing travel time. Conversely, a headwind slows it down, increasing the time needed to reach a destination.

• Tailwind: Increases effective speed (airplane speed + wind speed).
• Headwind: Decreases effective speed (airplane speed - wind speed).

Considering the wind is crucial for pilots both for fuel efficiency and scheduling. In practice, control towers and pilots adjust flight paths and speeds according to current wind conditions. This concept ensures safer and more effective flights.

Round Trip Travel Time

Calculating round trip travel time allows for determining the total journey time for an airplane to travel to a destination and back. The formula for a round trip is essentially twice the time taken for a one-way trip, assuming consistent conditions. However, wind complicates this calculation.
When wind is involved, the speed for each leg of the trip differs, hence altering the time. You must calculate the time for each direction separately:

• Time from origin to destination with tailwind
• Time for return trip with headwind

Adding these two durations gives the total round trip time. These calculations assist passengers and airline operators in planning and providing accurate travel schedules, even when faced with challenging weather conditions like strong winds.