Problem 104 - A Graphical Approach to Precalculus with Limits

Question

A plane flies 650 mph on a heading of \(175.3^{\circ} .\) A 25 -mph wind from a direction of \(266.6^{\circ}\) blows against the plane. Find the final bearing of the plane.

Step-by-Step Solution

Answer

The final bearing of the plane is approximately \(171.4^\circ\).

Step 1: Understanding the Problem

We have a plane moving at 650 mph on a heading of \(175.3^\circ\). There is also a wind blowing at 25 mph from \(266.6^\circ\). To find the final bearing, we need to use vector addition to account for both the speed and the effect of the wind.

Step 2: Representing the Vectors in Components

First, represent the speed of the plane as a vector. We'll denote the plane's velocity vector as \(\vec{V_p} = 650 \text{ mph at } 175.3^\circ\). The components are given by: \[V_{px} = 650 \cos(175.3^\circ), \quad V_{py} = 650 \sin(175.3^\circ)\].Similarly, represent the wind as vector \(\vec{V_w} \text{ from } 266.6^\circ\), which is equivalent to vector \(25 \text{ mph at } 86.6^\circ\) (by considering opposite direction):\[V_{wx} = 25 \cos(86.6^\circ), \quad V_{wy} = 25 \sin(86.6^\circ)\].

Step 3: Calculating the Components

Calculate the components using trigonometric functions:\[V_{px} = 650 \cos(175.3^\circ) = -643.65\ (approx), \quad V_{py} = 650 \sin(175.3^\circ) = 72.82\ (approx)\].For the wind:\[V_{wx} = 25 \cos(86.6^\circ) = 0.44\ (approx), \quad V_{wy} = 25 \sin(86.6^\circ) = 24.99\ (approx)\].

Step 4: Adding the Vectors

To find the resultant vector \(\vec{V_r}\), add the components:\[V_{rx} = V_{px} + V_{wx} = -643.65 + 0.44 = -643.21\],\[V_{ry} = V_{py} + V_{wy} = 72.82 + 24.99 = 97.81\].

Step 5: Determine the Magnitude and Direction of the Resultant

The magnitude of the resultant vector is \[|\vec{V_r}| = \sqrt{V_{rx}^2 + V_{ry}^2} = \sqrt{(-643.21)^2 + (97.81)^2}\] = \(650.56\text{ mph}\) (approx).The direction (bearing) is obtained using \[\theta = \tan^{-1}\left(\frac{V_{ry}}{V_{rx}}\right)\],which gives a negative angle because \(V_{rx}\) is negative.

Step 6: Adjust the Bearing for Standard Compass Directions

Convert the negative angle from step 5 to a proper bearing angle by adding \(180^\circ\) to get the final bearing. This ensures that the bearing is in the correct quadrant on the compass.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools when dealing with vector addition, especially in problems involving direction and magnitude. These functions include sine, cosine, and tangent, and they help in breaking down vectors into their components. For instance, any vector that is represented at an angle can be split into horizontal (\( x \)-axis) and vertical (\( y \)-axis) components using cosine and sine, respectively.

• Cosine (\( \cos \)) is used to find the adjacent side of a right triangle when the hypotenuse and an angle are known.
• Sine (\( \sin \)) is helpful for determining the opposite side given the same information.
• Tangent (\( \tan \)) links the opposite and adjacent sides, often used to find angles.

When considering a vector's direction as an angle \( \theta \) from the x-axis, the components can be calculated as:

• Horizontal component: \( V_x = V \cos(\theta) \)
• Vertical component: \( V_y = V \sin(\theta) \)

These calculations allow us to analyze each vector's effect along the axes and combine them to determine a resultant vector.

Resultant Vector

A resultant vector is the single vector that has the same effect as the combination of two or more vectors. In vector problems, such as calculating the final path of a plane under the influence of wind, finding the resultant vector is crucial.
The process involves:

• Breaking each vector into its x and y components using trigonometric functions as mentioned earlier.
• Adding respective components: sum the x-components together to find the resultant vector's x-component, and do the same for the y-components.

Mathematically, if you have two vectors, \( \vec{V_1} \) and \( \vec{V_2} \), with components \( V_{1x}, V_{1y}, V_{2x}, \) and \( V_{2y} \), the resultant vector's components are:

• \( V_{rx} = V_{1x} + V_{2x} \)
• \( V_{ry} = V_{1y} + V_{2y} \)

Finally, the magnitude of the resultant vector is found using the Pythagorean theorem:\[ |\vec{V_r}| = \sqrt{V_{rx}^2 + V_{ry}^2} \]This magnitude represents the total speed or length of the resultant vector. Finding the direction involves calculating the angle with the x-axis using the arctan function of the y-component over the x-component:\[ \theta = \tan^{-1}\left(\frac{V_{ry}}{V_{rx}}\right) \].This angle might need adjustment based on the quadrant to provide the correct bearing.

Bearing Angle

The bearing angle is a crucial aspect of navigation, reflecting the direction of travel based on compass measurements. It is measured in degrees clockwise from due north towards the east. Understanding the bearing angle involves converting the calculated angle from mathematical solutions into a format that is useful for navigation.
When calculating a bearing angle, you should consider:

• The quadrant of the calculated angle: If your x-component is negative, it indicates the angle lies in the second or third quadrant, requiring adjustment to convert it to a standard compass bearing.
• By including 180 degrees for angles in the second and third quadrants, you correctly align the angle with north.

For example, if a negative angle arises from a vector calculation, you add 180° to ensure it falls within the typical compass range. This system allows pilots and navigators to understand their actual travel direction in terms of standard compass bearings. By applying these principles, you can effectively interpret the final bearing as a direction that can be easily communicated and understood in practical navigation scenarios.