Problem 86 - Algebra and Trigonometry

Question

A plane is flying at a speed of 540 miles per hour on a bearing of \(\mathrm{S} 36^{\circ} \mathrm{E}\). Its ground speed is 500 miles per hour and its true bearing is \(\mathrm{S} 44^{\circ} \mathrm{E}\). Find the speed, to the nearest mile per hour, and the direction angle, to the nearest tenth of a degree, of the wind.

Step-by-Step Solution

Answer

First find the components (southward and eastward) of both the airspeed and ground speed vectors. Then find the components of the wind vector by subtracting the airspeed components from the ground speed components. The speed of the wind is found using the Pythagorean theorem applied to its components, and its direction angle is calculated using the arctangent function of the ratio of the components. Finally, convert the direction angle to degrees.

Step 1: Find Components of Plane's Motion in the Air

Break down the plane's airspeed vector into its southward and eastward components using sine and cosine respectively. The southward component is \(540 \sin(36^\circ)\) miles per hour. The eastward component is \(540 \cos(36^\circ)\) miles per hour.

Step 2: Find Components of Plane's Motion over the Ground

Similarly, break down the plane's ground speed vector into its southward and eastward components. The southward component is \(500 \sin(44^\circ)\) miles per hour. The eastward component is \(500 \cos(44^\circ)\) miles per hour.

Step 3: Find Components of Wind's Velocity

The wind velocity components are found by subtracting the plane's air components from the ground components. The southward component of wind's velocity is \(500 \sin(44^\circ) - 540 \sin(36^\circ)\) miles per hour. The eastward component is \(500 \cos(44^\circ) - 540 \cos(36^\circ)\) miles per hour.

Step 4: Compute the Speed of the Wind

The speed of wind is the magnitude of the wind vector, which can be calculated using the Pythagorean theorem: \(\sqrt{((500 \sin(44^\circ) - 540 \sin(36^\circ))^2 + (500 \cos(44^\circ) - 540 \cos(36^\circ))^2}\) miles per hour.

Step 5: Find the Direction of the Wind

The direction of wind is the angle whose tangent is the ratio of the southward component to the eastward component. Use the arctangent function to get this angle. If both components of the wind vector are positive, use the formula: \(\text{direction angle} = \tan^{-1} (\frac{SouthComponent}{EastComponent})\). Convert the result in radians to degrees and round to the nearest tenth of a degree.

Key Concepts

Decomposition of Vectors

Understanding the decomposition of vectors is essential in breaking down a vector into its component parts. In trigonometry, this is often done by using the sine and cosine functions to separate a vector into its vertical and horizontal components. For example, an airplane flying through the air has an airspeed (the speed and direction of the plane relative to the air) which can be decomposed into how fast it's moving toward the south and how fast it's moving toward the east.

In our exercise, the plane's airspeed vector is decomposed into southward and eastward components using the angles given with respect to bearer S36°E. The sine function helps us find the southward (vertical) component, while the cosine function determines the eastward (horizontal) component. This principle is widely applicable, whether it's for analyzing forces in physics or calculating velocities in navigation.

Ground Speed and Airspeed

Ground speed and airspeed are two terms that anyone involved in aviation must understand. Airspeed is the velocity of an aircraft through the air, whereas ground speed is the speed of the aircraft relative to the ground. It's crucial to note that wind can affect these two speeds differently. If the wind is in the same direction as the flight, it can increase the ground speed. Conversely, a wind blowing against the flight direction can reduce the ground speed.

For our exercise, we consider the airspeed given as 540 miles per hour and the ground speed as 500 miles per hour. To deduce the wind's influence, we have to understand how these speeds are impacted by external factors like wind, by employing vector analysis.

Bearing and Direction Angles

The bearing of an aircraft is the direction in which it is moving, usually expressed in degrees from a reference direction, such as north or south. In navigation, understanding bearing is critical for charting a course. A direction angle, on the other hand, describes the angle a vector makes with the positive direction of an axis.

In the solution to our textbook problem, we deal with bearings from a directional perspective (S36°E and S44°E) and utilize angles to decompose the velocity of the plane and wind into components. These bearings help us calculate both the vertical and horizontal components of each vector. Correct determination of bearing and direction angles is essential for accurate navigation and determining the impact of prevailing winds.

Wind Velocity Calculation

Calculating wind velocity is a fundamental skill in navigation, especially in aviation. It involves finding both the speed and the direction of the wind. You can do this by using the components of the plane's motion in the air and its motion over the ground, as we have decomposed them in previous steps. The wind velocity components are determined with a vector difference of these two motions.

To get the wind's velocity, we subtract the airspeed components from the ground speed components. This reveals how the wind alters the plane's trajectory. With both the southward and eastward components of the wind velocity, we can compute the speed and direction of the wind using the Pythagorean theorem and the arctangent function, ultimately giving us a full picture of the wind's influence on the plane's flight path.