Problem 93 - Precalculus with Calculus Previews

Question

What Is the Speed? In order for an airplane to fly due north at \(300 \mathrm{mi} / \mathrm{h}\), it must set a course \(10^{\circ}\) west of north \(\left(\mathrm{N} 10^{\circ} \mathrm{W}\right)\) because of a strong wind blowing due east. What is the speed of the wind?

Step-by-Step Solution

Answer

The speed of the wind is approximately 52.08 mi/h.

Step 1: Define the Problem

We need to find the speed of the wind that affects an airplane's course. The airplane intends to fly due north at a speed of 300 mi/h but must adjust its flight path to 10° west of north. This adjustment is necessary because of an eastern wind.

Step 2: Establish Reference Directions

Consider the north direction as the positive y-axis, and the east direction as the positive x-axis. The plane needs to compensate for the eastward wind by flying 10° west of north, forming a 10° angle with the y-axis.

Step 3: Resolve Airplane's Velocity Components

The airplane's velocity forms two components due to its course: a northward component and a westward component. The northward component (y-axis) is given by 300 cos(10°), and the westward component (x-axis) is given by 300 sin(10°).

Step 4: Set up the Wind Velocity Equation

To cancel the wind effect, the westward component of the airplane's velocity should exactly balance the eastward wind velocity. Therefore, if the wind's speed is W mi/h, we have W = 300 sin(10°).

Step 5: Calculate the Wind Speed

Using the sine function for 10°, calculate the wind speed. \[ W = 300 \cdot \sin(10°) \approx 300 \cdot 0.1736 = 52.08 \text{ mi/h} \]

Step 6: Conclusion

The wind's speed is calculated based on the necessary western component the airplane must have to counteract it, ensuring a true north path.

Key Concepts

Airplane Velocity Components

When dealing with airplane navigation, understanding the velocity components is essential. An airplane moves through the air, which means its actual path or course in the sky is affected by multiple factors like wind.
To navigate correctly, the airplane's velocity is broken into two primary components:

• The Northward (or Y-axis) component: This is the intended path of the airplane. For example, if an airplane wants to move directly towards the north, any deviation must be corrected.
• The Sidewind (or X-axis) component: This occurs when there are wind conditions affecting the airplane’s intended path. In the original exercise, we consider the westward component produced by a strong wind coming from the east.

The airplane aims to fly due north at 300 mi/h, but due to wind, it sets its course 10° west of north. This adjustment introduces new velocity components which are crucial for determining the effect of wind on the airplane's course.

Trigonometry in Physics

Trigonometry comes handy in physics, especially when resolving forces or determining movements in vector-rich environments, such as in aviation. When an airplane sets its course, it incorporates angles to adjust for wind or other forces.
By using trigonometry, pilots can calculate the exact angle needed to maintain a straight flight path. In scenarios like the original exercise, trigonometry allows pilots to break down a single vector (flight path) into horizontal and vertical components using sine and cosine laws:

• Cosine Function: This is primarily used to find the adjacent leg of the angle. For instance, the northward component is calculated using cosine: \( 300 \times \cos(10°) \).
• Sine Function: This calculates the opposite component of the angle. For the exercise, the westward (needed to counteract the eastward wind) component was calculated using \( 300 \times \sin(10°) \).

With this methodical approach, physics uses trigonometry to strategically navigate through an environment filled with forces acting at angles.

Vector Resolution in Navigation

Vector resolution is an important technique used in navigation to determine the effect of various forces on an airplane's path. It involves breaking down a vector into its components, making it easier to analyze.
In the navigation context of the original exercise, the airplane's velocity vector is resolved into two perpendicular components:

• The magnitude of the vector along the north-south direction (indicating the true path desired by the plane).
• The magnitude counteracting wind, which in this case is a western component needed to cancel the eastward wind.

Vector resolution allows for clear visualization of how various forces interact, such as wind, in altering an intended trajectory. Navigators and pilots can then adjust their strategies based on these clear mathematical precise component values, ensuring they can maintain the desired heading, such as the due north path in the exercise.