Suppose a plane flies at a constant groundspeed of 500 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction

To find the airspeed and direction required for the plane to maintain its groundspeed of 500 miles per hour due east, despite encountering a 50-mile-per-hour wind from the northwest, follow these steps:

### Step 1: Understand the Components of the Wind
- The wind is coming from the northwest, which is at a 45° angle from both the north and west directions.
- The wind has two components:
  - **Westward component** (opposing the eastward motion of the plane)
  - **Southward component**

The magnitude of each component of the wind is:
\[
\text{Wind component} = 50 \times \cos(45^\circ) = 50 \times \frac{1}{\sqrt{2}} \approx 35.36 \text{ mph}
\]
Thus, both the westward and southward components are approximately 35.36 mph.

### Step 2: Determine the Required Airspeed Components
To maintain a due east direction:
- **Eastward Component of Airspeed**: The plane must counteract the westward wind component and maintain a 500 mph groundspeed:
\[
\text{Required Eastward Component} = 500 + 35.36 = 535.36 \text{ mph}
\]
- **Northward Component of Airspeed**: The plane needs to cancel out the southward component of the wind by flying slightly northward:
\[
\text{Required Northward Component} = 35.36 \text{ mph}
\]

### Step 3: Calculate the Total Airspeed
Now, use the Pythagorean theorem to calculate the total airspeed:
\[
\text{Airspeed} = \sqrt{(535.36)^2 + (35.36)^2}
\]
\[
\text{Airspeed} = \sqrt{286614.13 + 1250.33} \approx \sqrt{287864.46} \approx 536.52 \text{ mph}
\]

### Step 4: Determine the Compass Direction
The compass direction (angle north of east) can be calculated using the arctan function:
\[
\theta = \tan^{-1}\left(\frac{35.36}{535.36}\right) \approx \tan^{-1}(0.066) \approx 3.8^\circ \text{ north of east}
\]

### Final Answer
- **Airspeed**: The plane needs to fly at approximately **536.52 mph**.
- **Compass Direction**: The plane should be oriented about **3.8° north of east** to maintain its due east groundspeed of 500 mph.

Would you like more details, or do you have any questions?

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**Relative Questions:**

1. How would the required airspeed change if the wind were stronger, say 70 mph from the northwest?
2. What would the airspeed be if the plane were flying in a different direction, such as southeast, with the same wind?
3. How does changing the wind direction to due west affect the required airspeed and direction?
4. What happens to the plane's groundspeed if the pilot doesn't adjust the airspeed and direction?
5. How would the calculations change if the wind speed were given in knots instead of miles per hour?

**Tip:** Always break down wind velocity into its components when calculating the effects on airspeed and direction.

Learn how to calculate the required airspeed and direction for a plane flying at a groundspeed of 500 mph due east encountering a 50 mph crosswind from the northwest. Understand vector addition, trigonometric functions, and the Pythagorean theorem applied in aviation. Suitable for advanced high school students and above.

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