Range of an Airplane An air rescue plane averages 300 miles...

Range of an Airplane An air rescue plane averages 300 miles per hour in still air. It carries enough fuel for 5 hours of flying time. If, upon takeoff, it encounters a head wind of $30 \mathrm{mi} / \mathrm{h},$ how far can it fly and return safely? (Assume that the wind remains constant.)

Key Concepts

Relative Velocity

This concept involves understanding that when an object moves in a medium that is also moving (like an airplane in wind), the effective speed relative to the ground is the vector sum of its speed in still air and the speed of the wind. For headwinds, the wind speed is subtracted from the object's speed, and for tailwinds, it is added, altering the ground speed for different legs of the journey.

Distance, Speed, and Time Relationships

At the heart of this problem is the fundamental relationship that distance equals speed multiplied by time. When speed changes due to external factors such as wind, the time taken for each leg of a trip must be computed separately, and their sum equated to the total available time (or fuel constraints) to determine the feasible distance.

Impact of Wind on Flight Range

Wind plays a critical role in aerial navigation by affecting the duration of each leg of a round trip differently. A headwind reduces the effective ground speed which increases flight time for the outbound leg, while a tailwind increases ground speed decreasing flight time on return. Balancing these effects is essential to calculating the maximum range when fuel is limited.

Fuel Constraints on Range

Fuel capacity imposes a strict limit on the amount of time a journey can take, regardless of the distances involved. When calculating the maximum distance an airplane can travel for a round trip, it is necessary to account for the fact that both the outbound and return legs together must not exceed the total available flight time, which is typically determined by fuel quantity.

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