91Ó°ÊÓ

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.

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 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.