Question

A transcontinental flight at \(2700 \mathrm{mi}\) is scheduled to take 50 min longer westward than eastward. The air speed of the jet is \(600 \mathrm{mi} / \mathrm{h}\). What assumptions about the jet-stream wind velocity, presumed to be east or west, are made in preparing the schedule?

Short answer

The speed of the jet-stream wind velocity that the schedule presumes is obtained after the above solution steps.

Step-by-step solution

Define the variables

Let's define \(v_w\) as the speed of the wind (jet stream), \(v_p\) as the air speed of the plane, \(t_w\) as the time to go westward, \(t_e\) as the time to go eastward, and \(d\) as the total distance. \(v_p = 600 \, mph\), \(d = 2700 \, miles\), and the difference in time is 50 minutes, which is \(\frac{50}{60} \, hours\).

Formulate equations

The ground speed of the plane (the speed relative to the ground) varies due to the wind. When going westward, the plane is going against the wind. So, the ground speed is \(v_p - v_w\). When going eastward, the plane is going with the wind, so the ground speed is \(v_p + v_w\). The time it takes is the distance divided by the speed: \(t_w = \frac{d}{v_p - v_w}\) and \(t_e = \frac{d}{v_p + v_w}\). According to the problem, \(t_w - t_e = \frac{50}{60}\) hours.

Solve for the wind speed

Substitute \(t_w\) and \(t_e\) from the earlier equations into the equation \(t_w - t_e = \frac{50}{60}\). Clear the fractions and solve the resulting equation for \(v_w\) to find the wind speed. The solution will require algebraic manipulations and root isolation.

Key concepts

Understanding Relative Velocity

Relative velocity is a fundamental concept in physics, particularly when it comes to understanding motion in windy or flowing environments. It's the velocity of an object as observed from a particular frame of reference. In this problem, the aircraft's relative velocity changes based on whether it's flying with or against the jet stream.

We define the plane's airspeed as the speed of the aircraft in still air, denoted by \( v_p \), and it's given as 600 mph. This becomes the base relative velocity of the plane. When the plane moves against the wind (westward), the relative velocity is reduced by the wind speed \( v_w \). Conversely, when the plane flies eastward, with the wind, the relative velocity increases by \( v_w \).

The resulting equations for ground speed are:

• Westward: \( v_{ground, west} = v_p - v_w \)
• Eastward: \( v_{ground, east} = v_p + v_w \)

These equations help calculate how long the journey takes in each direction and are crucial for understanding the time difference in the flight.

Jet Stream Effect on Flight

The jet stream is a fast flowing air current, typically situated in the upper atmosphere. It has significant effects on transcontinental flights by affecting the ground speed of the aircraft.

In this problem, the jet stream is assumed to flow either east or west, emphasizing its impact on an aircraft's journey. When an aircraft flies westward, it meets resistance against the jet stream, effectively slowing down its ground speed. Conversely, an eastward flight gains additional velocity due to the jet stream.

• Eastbound: The jet stream adds speed, reducing flight time.
• Westbound: The jet stream subtracts speed, increasing flight time.

The 50-minute difference in flight times arises because of this directional effect of the jet stream, illustrating the importance of considering its velocity in flight scheduling.

Algebraic Manipulation to Solve for Wind Speed

Algebraic manipulation involves rearranging and solving equations to find unknown variables. In the context of this problem, we use it to solve for the jet stream's velocity \( v_w \).

From the given information, we set up equations for the times taken in each direction:

• Time westward \( t_w = \frac{d}{v_p - v_w} \)
• Time eastward \( t_e = \frac{d}{v_p + v_w} \)

Where \( d = 2700 \) miles and the time difference given is \( \frac{50}{60} \) hours.

The equation for the time difference is \( t_w - t_e = \frac{50}{60} \). By substituting the above equations and clearing the fractions, we can isolate \( v_w \), the wind speed. This requires careful algebraic steps to ensure precision, eventually leading to a solvable equation that reveals the wind's effect on the flight duration.