Chapter 3: Problem 38
An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the south. (a) If the airspeed of the plane (its speed in still air) is 320.0 km/h (about 200 mi/h), in which direction should the pilot head? (b) What is the speed of the plane over the ground? Draw a vector diagram.
Short Answer
Step by step solution
Understand the Problem
Identify the Known Values
Use Vector Addition for Direction
Set Up the Vector Equation
Calculate Direction with Trigonometry
Calculate Ground Speed
Vector Diagram
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Airplane Navigation
In our airplane navigation problem, the pilot needs to fly due west. However, a southward wind blows off the original course. To navigate correctly, the pilot must adjust the airplane's heading, which means the direction the nose of the plane points in flight, to counteract the wind.
By adjusting the heading, the pilot ensures the resultant movement of the airplane across the ground aligns with the intended route. This is achieved through careful calculations involving vectors and trigonometry.
Vector Components
To solve for the airplane's new heading, we break down these vectors into components along the north-south and east-west axes. This process involves adjusting the initial course by applying vector components, handling the northward and westward directions separately.
Vectors are represented as arrows where the length denotes magnitude and the arrowhead indicates direction. By understanding and manipulating vector components, pilots can derive the needed direction and speed to stay on their desired path.
Trigonometry in Physics
In our scenario, the airplane's movement involves finding the correct angle to travel due west despite a southward wind. By using trigonometric functions, we determine how much the airplane needs to deviate by calculating the angle between its desired and actual direction.
Specifically, the sine function helps find the angle by setting up equations like \[ V_{north} = 320 \sin(\theta) = 80 \text{ km/h} \]. Solving this reveals the precise heading needed to adjust for the wind's effect, ensuring the plane travels straight westward.