Problem 88 - Fundamentals of Physics

Question

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?

Step-by-Step Solution

Answer

The jet stream speed is approximately 116.28 km/h.

Step 1: Understand the Problem

We are given a round-trip flight where an airplane flies 4000 km in each direction. In the outgoing direction, it flies with the jet stream, and against it on the return. The airspeed of the plane is 1000 km/h. The problem asks us to find the speed of the jet stream if the total time difference is 70.0 minutes.

Step 2: Establish Equations for Flight Times

Let the jet stream speed be \( v \). The ground speeds are: \( (1000 + v) \) km/h for the outgoing flight, and \( (1000 - v) \) km/h for the return flight. The times taken are: \( t_1 = \frac{4000}{1000 + v} \) hours for outgoing, and \( t_2 = \frac{4000}{1000 - v} \) hours for return.

Step 3: Convert Time Difference

The difference in time \( t_1 - t_2 \) is given as 70.0 minutes. Convert this to hours by dividing by 60: \( \frac{70}{60} = \frac{7}{6} \).

Step 4: Set Up the Equation

The time difference equation is \( \frac{4000}{1000 + v} - \frac{4000}{1000 - v} = \frac{7}{6} \). This equation relates the time difference to the jet stream speed \( v \).

Step 5: Solve for Jet Stream Speed

Rearrange the equation: \( \frac{4000((1000 - v) - (1000 + v))}{(1000 + v)(1000 - v)} = \frac{7}{6} \), simplify to get \( 4000 \times \frac{-2v}{(1000^2 - v^2)} = \frac{7}{6} \). Simplify further: \( \frac{-8000v}{1000000 - v^2} = \frac{7}{6} \).

Step 6: Cross-Multiply and Solve

Cross-multiply to get \( -48000v = 7(1000000 - v^2) \). Expand to: \( -48000v = 7000000 - 7v^2 \) and rearrange to get \( 7v^2 - 48000v - 7000000 = 0 \).

Step 7: Quadratic Equation Solution

Solve \( 7v^2 - 48000v - 7000000 = 0 \) using the quadratic formula, \( v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 7, b = -48000, c = -7000000 \). Calculate the discriminant: \( b^2 - 4ac = 48000^2 + 4 \times 7 \times 7000000 \).

Step 8: Calculate Jet Stream Speed

Solve using the quadratic formula to find \( v \). After substituting and calculating, we find that the positive solution for \( v \) is approximately 116.28 km/h.

Key Concepts

Airspeed

Airspeed is an important concept when it comes to understanding flight dynamics. Airspeed refers to how fast an airplane is moving relative to the air around it. When pilots refer to maintaining a certain speed, they are typically referring to airspeed. For instance, in our exercise, the pilot maintains an airspeed of 1000 km/h. This is the speed without considering the jet stream's effect. Understanding airspeed is crucial as it directly affects lift, stability, and the overall performance of the aircraft in the air.
Airspeed remains constant regardless of the wind direction. It gives pilots a benchmark to ensure they are flying at optimal speeds that maintain safety and efficiency. However, airspeed is not sufficient to determine how quickly an airplane covers ground. That's where ground speed comes in to give the full picture.

Ground Speed

Ground speed is the actual speed at which the aircraft is moving over the Earth's surface. It can differ significantly from the plane’s airspeed due to environmental factors like the jet stream.

• When flying with the jet stream, the ground speed becomes higher than airspeed.
• Conversely, flying against the jet stream results in lower ground speed compared to airspeed.

In the exercise, the ground speed is expressed as either \( (1000 + v) \) km/h or \( (1000 - v) \) km/h, where \( v \) is the jet stream speed. Understanding how airspeed and the jet stream together determine ground speed is crucial. This helps in calculating flight times accurately, and is vital for scheduling, fuel efficiency, and flight safety.

Time Difference

The difference in flight times when flying in the direction of the jet stream versus against it illustrates how wind affects travel. In our exercise, the time difference is given as 70 minutes, which needs to be converted into hours for calculations (\( \frac{70}{60} = \frac{7}{6} \) hours).
This time difference occurs because the plane's ground speed changes depending on whether it benefits from or fights against the wind. When flying with the jet stream, less time is needed to cover the same distance due to increased ground speed; the opposite is true when flying against the wind.
It is essential to factor in the time difference as it impacts airline schedules and fuel usage, requiring adjustments depending on the jet stream's strength. Understanding and predicting these time differences helps airlines plan better and ensures smoother operations.