Problem 40 - Trigonometry

Question

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. True Course and Speed A plane is flying with an airspeed of 244 miles per hour with heading \(272.7^{\circ}\). The wind currents are running at a constant \(45.7\) miles per hour in the direction \(262.6^{\circ}\). Find the ground speed and true course of the plane.

Step-by-Step Solution

Answer

The ground speed is approximately 200.7 mph with a true course of about 270°.

Step 1: Understanding the Problem

We need to find the ground speed and the true course of the plane, considering its airspeed, heading, and the wind speed and direction. To do this, we'll use vector addition, where the plane's speed and direction and the wind's speed and direction are considered as vectors.

Step 2: Identify Vector Components

The plane's heading and wind direction give us angles, which we translate into vector components on a Cartesian plane. We'll resolve the plane's speed ( 244 mph) and the wind's speed ( 45.7 mph) into north and east components using the given angles.

Step 3: Resolve Plane's Speed into Components

The heading of the plane is 272.7^{\circ}, measured clockwise from North. So the components can be calculated as:\[ V_{plane,x} = 244 \cdot \sin(272.7^{\circ}) \] \[ V_{plane,y} = 244 \cdot \cos(272.7^{\circ}) \].

Step 4: Resolve Wind's Speed into Components

Similarly, resolve the wind speed into components. The wind direction is given as 262.6^{\circ}:\[ V_{wind,x} = 45.7 \cdot \sin(262.6^{\circ}) \] \[ V_{wind,y} = 45.7 \cdot \cos(262.6^{\circ}) \].

Step 5: Calculate Ground Speed Components

Combine the components of the plane's velocity and the wind to find the ground speed components:\[ V_{ground,x} = V_{plane,x} + V_{wind,x} \] \[ V_{ground,y} = V_{plane,y} + V_{wind,y} \].

Step 6: Determine Ground Speed Magnitude

Calculate the magnitude of the ground speed vector:\[ \text{Ground Speed} = \sqrt{(V_{ground,x})^2 + (V_{ground,y})^2} \].

Step 7: Determine True Course

Determine the true course by calculating the angle of the ground speed vector using the arctangent function:\[ \text{True Course} = \text{atan2}(V_{ground,x}, V_{ground,y}) \]. Ensure the final angle is converted into degrees from the unit circle setup (0 degrees corresponds to North).

Key Concepts

Vector Addition

In aviation, understanding how to combine different speed vectors is crucial to finding a plane's ground speed and true course. Vectors represent quantities that have both magnitude and direction, such as speed. When dealing with problems involving vectors, such as the airspeed of a plane and the wind speed, the concept of vector addition becomes essential.
To imagine this, think of each speed and direction (such as a plane's heading and wind direction) as an arrow. The length of the arrow represents the magnitude (speed), and the direction of the arrow corresponds to the direction in which the object moves. To determine the resultant vector — in this case, the ground speed vector — you simply add these arrows together. This involves summing their components along the north-south and east-west axes. By doing so, you find out how fast and in what direction the plane actually moves over the earth’s surface.

Bearing and Heading

Aircraft navigation often uses terms like 'bearing' and 'heading.' Understanding these terms is key to solving problems related to the plane's course.

• Bearing: The angle between a reference direction (usually north) and the line pointing to a target, measured clockwise. It's like a compass direction from one point to another.
• Heading: The direction in which an aircraft's nose is pointed or intended to travel. It differs from the bearing because it’s influenced by factors like wind.

The heading may not always be where the aircraft ends up due to wind drift, which is why true course, the actual path over ground, can differ significantly. In exercise scenarios like those given, including both heading and wind factors is necessary to derive the true course.

Resolving Vectors

Resolving vectors is a process where you break down a vector into its component parts. These components typically lie along a set of perpendicular axes (e.g., north-south and east-west). It allows for simpler calculation and combination of vectors.
To resolve vectors:

• For an angle \(\theta\) measured clockwise from north, the north component of a vector is found using cosine, and the east component uses sine.
• Mathematically, if a vector has a magnitude \(V\) and direction \(\theta\), its components are \[V_x = V \cdot \sin(\theta)\] \[V_y = V \cdot \cos(\theta)\]

By calculating these components for both the plane's velocity and the wind, you gain the detailed information needed to perform vector addition correctly. This step is fundamental in determining the resultant velocity vector, which leads to calculating ground speed and true course.

Ground Speed Calculation

Once vectors have been resolved into their components, calculating ground speed is the next stage. Ground speed is the actual speed of the aircraft over the ground, influenced by both the aircraft's airspeed and the wind.
To calculate ground speed:

• Add up the north-south components from both the plane and wind to get the total north-south component of the ground speed.
• Similarly, add the east-west components from both vectors to get the total east-west component.
• Finally, apply the Pythagorean theorem to these components to find the magnitude of the ground speed vector: \[ \text{Ground Speed} = \sqrt{(V_{ground,x})^2 + (V_{ground,y})^2}\]

Along with the ground speed, you calculate the true course, which is the actual direction of travel over the ground. Using the components, the arctangent function gives you the angle relative to the north, thereby completing the navigation task.