Solve each problem. See Examples 5 and 6.


Bearing and Ground Speed of a Plane An airline route from San Francisco to Honolulu is on a bearing of 233.0°. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from a direction of 114.0°. Find the resulting bearing and ground speed of the plane.

Verified step by step guidance

1. Identify the vectors involved: the plane's velocity vector and the wind's velocity vector. The plane's velocity is 450 mph on a bearing of 233.0°, and the wind's velocity is 39.0 mph from a bearing of 114.0°. Remember that "from" a direction means the wind is coming from 114.0°, so its direction of travel is 114.0° + 180° = 294.0°.
2. Convert both velocity vectors into their component form using trigonometry. For a vector with magnitude \(v\) and bearing \(\theta\), the components are: \(x = v \times \sin\left(\frac{\pi}{180} \times \theta\right)\) \(y = v \times \cos\left(\frac{\pi}{180} \times \theta\right)\) Calculate the components for the plane and the wind separately.
3. Add the corresponding components of the plane and wind vectors to find the resultant ground velocity vector: \(x_{result} = x_{plane} + x_{wind}\) \(y_{result} = y_{plane} + y_{wind}\)
4. Calculate the magnitude of the resultant ground speed using the Pythagorean theorem: \(\text{Ground Speed} = \sqrt{x_{result}^2 + y_{result}^2}\)
5. Determine the resulting bearing of the plane by finding the angle of the resultant vector relative to north. Use the inverse tangent function: \(\theta = \arctan\left(\frac{x_{result}}{y_{result}}\right)\) Adjust the angle to the correct quadrant and convert it back to a bearing between 0° and 360°.

Key Concepts

Bearing and Direction in Navigation

Bearing is the angle measured clockwise from the north direction to the line of travel, expressed in degrees from 0° to 360°. Understanding bearings is essential for navigation problems, as it helps represent directions precisely, such as the plane's route at 233° and wind direction at 114°.

Vector Addition of Velocities

The plane's velocity relative to the ground is the vector sum of its airspeed vector and the wind velocity vector. Adding these vectors requires breaking them into components, summing the components, and then recombining to find the resultant velocity's magnitude and direction.

Trigonometric Resolution of Vectors

Resolving vectors into components uses sine and cosine functions based on their bearings. By converting speeds and directions into x (east-west) and y (north-south) components, trigonometry allows calculation of the resultant vector's ground speed and bearing accurately.