Question

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

The pilot should head 14.48° north of west, and the ground speed is 310.13 km/h.

Step-by-step solution

Understand the Problem

The airplane needs to fly due west, but there's a wind blowing toward the south at 80 km/h. The airspeed of the plane is 320 km/h. We need to find the direction the plane should head to effectively travel due west and the resulting ground speed.

Identify the Known Values

From the problem, the known values are: - The wind speed (southwards): 80 km/h - The airspeed of the plane: 320 km/h.

Use Vector Addition for Direction

Since the plane must compensate for the southward wind, the resultant vector of the wind and the plane's heading should point due west. The airplane's heading is the combination of its own speed and the wind's speed as vectors.

Set Up the Vector Equation

Let the westward component of the plane's heading be represented as a vector to the west (plane's airspeed), with magnitude 320 km/h. Let the southward wind be represented as a vector, pointing south, with magnitude 80 km/h. We need to find the direction \( \theta \) the plane should head such that the resultant vector is purely westward.

Calculate Direction with Trigonometry

To counteract the southward wind, the plane must have a northward component equal to the wind's speed. Hence, the north-south component of the plane's movement should equal 80 km/h. This is accomplished by setting up the equation:\[ V_{north} = 320 \sin(\theta) = 80 \text{ km/h} \]Solve for \( \theta \):\[ \sin(\theta) = \frac{80}{320} = \frac{1}{4} \]\[ \theta = \arcsin\left(\frac{1}{4}\right) \approx 14.48^\circ \]Therefore, the pilot should head \( 14.48^\circ \) north of west.

Calculate Ground Speed

The ground speed is the westward component of the plane's velocity. The westward velocity can be found using:\[ V_{west} = 320 \cos(\theta) \]Substitute \( \theta = 14.48^\circ \) into:\[ V_{west} = 320 \cos(14.48^\circ) \approx 310.13 \text{ km/h} \]

Vector Diagram

Draw a vector diagram with the following components:- A vector of 80 km/h pointing south.- A vector of 320 km/h at an angle \( \theta \) pointing slightly north of west.- The resultant vector should point precisely west, representing the true path of the airplane across the ground.

Key concepts

Airplane Navigation

When navigating an airplane, pilots must account for various external factors like wind. These factors can steer the airplane off course if not properly managed. The goal of airplane navigation is often to arrive at a specific destination in the most efficient path despite these influences.

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

Vectors are quantities that have both a magnitude (size) and a direction, such as velocity or force. In the context of our problem, the airplane's velocity and the wind's velocity can be considered vectors.

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

Trigonometry is vital in physics when dealing with angles, especially in navigation. It involves using functions like sine, cosine, and tangent to relate angles to side lengths in right triangles.

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.