A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?

Verified step by step guidance

1. <Step 1: Represent the plane's velocity vector.> The plane's airspeed is 520 mph, and it wants to fly on a bearing of 310°. Convert this bearing to a standard position angle by subtracting it from 360° to get 50°. The velocity vector of the plane can be represented as: ${\overrightarrow{v}}_p=520⟨\cos (50°),\sin (50°)⟩$.
2. <Step 2: Represent the wind's velocity vector.> The wind is blowing from a bearing of 212°, which means it is blowing towards a bearing of 32° (since 212° + 180° = 392°, and 392° - 360° = 32°). Convert this to a standard position angle by subtracting it from 360° to get 328°. The wind's velocity vector is: ${\overrightarrow{v}}_w=37⟨\cos (328°),\sin (328°)⟩$.
3. <Step 3: Determine the resultant ground velocity vector.> The ground velocity vector ${\overrightarrow{v}}_g$ is the sum of the plane's velocity vector and the wind's velocity vector: ${\overrightarrow{v}}_g={\overrightarrow{v}}_p+{\overrightarrow{v}}_w$.
4. <Step 4: Calculate the magnitude of the ground velocity vector.> Use the Pythagorean theorem to find the magnitude of ${\overrightarrow{v}}_g$: $|{\overrightarrow{v}}_g|=\sqrt{(v_{gx}{)}^2+(v_{gy}{)}^2}$, where $v_{gx}$ and $v_{gy}$ are the components of ${\overrightarrow{v}}_g$.
5. <Step 5: Determine the direction of the ground velocity vector.> Use the inverse tangent function to find the direction angle $\theta$ of the ground velocity vector: $\theta ={\tan }^{-1}\left(\frac{v_{gy}}{v_{gx}}\right)$. Convert this angle to a bearing if necessary.>

Key Concepts

Bearing

Bearing is a navigation term that describes the direction of one point from another, measured in degrees from North. In this context, bearings are used to indicate the direction the plane and wind are coming from. A bearing of 310° means the direction is 310 degrees clockwise from true North, while a bearing of 212° indicates the wind's direction. Understanding bearings is crucial for determining the resultant direction of the plane's flight.

Vector Addition

Vector addition is a mathematical process used to combine two or more vectors to determine a resultant vector. In this scenario, the plane's airspeed and the wind speed are both vectors that need to be added to find the actual ground speed and direction of the plane. This involves breaking down each vector into its components (usually using sine and cosine functions) and then summing these components to find the overall effect on the plane's trajectory.

Ground Speed

Ground speed is the speed of an aircraft relative to the ground, which can differ from its airspeed due to wind effects. It is calculated by considering both the airspeed of the plane and the wind speed and direction. In this problem, the pilot needs to adjust her flight path to account for the wind, which will affect her ground speed and the direction she must fly to maintain her intended course.