A pilot is flying at 200 miles per hour (airspeed), and wants his flight path (course or actual direction) to be on a bearing of 237 degrees. A wind is blowing for the north at 40 miles per hour. Find the bearing he should fly, and find the plane's ground speed. Include a labeled drawing.
Vp + 40mi/h[180] = 200mi/h[237o].
Vp + 40*sin180 + i40*Cos180 = 200*sin237 + i200*Cos237,
Vp + 0 -40i = -167.7 - 108.9i,
Vp = -167.7 - 68.9i =181.3mi/h[67.7o] S. of W. = 181.3mi/h[202.3o] CW.
To find the bearing the pilot should fly and the plane's ground speed, we can use vector addition.
Step 1: Draw a labeled diagram.
Draw a diagram representing the situation. Place an arrow pointing north to represent the wind speed and label it as "40 mph," and draw another arrow pointing northeast to represent the plane's airspeed of 200 mph. Label it as "200 mph."
```
(↑) Wind (40 mph)
|
|
| 237°
|
→ (↑↗) Plane (200 mph)
```
Step 2: Resolve vectors into components.
Since the wind is blowing directly north (upwards), there is no horizontal component for the wind. The plane's airspeed vector needs to be resolved into horizontal and vertical components.
The vertical component of the plane's airspeed is 200 * sin(45°) = 200 * √2 / 2 ≈ 141.4 mph.
The horizontal component of the plane's airspeed is 200 * cos(45°) = 200 * √2 / 2 ≈ 141.4 mph.
Step 3: Add the horizontal components for both vectors.
Since there is no horizontal component for the wind, we can add the horizontal component of the plane's airspeed directly to find the resultant horizontal component.
Total horizontal component = 141.4 mph + 0 mph (from the wind) = 141.4 mph.
Step 4: Add the vertical components for both vectors.
Since the wind is blowing directly upwards (north), we can add the vertical component of the plane's airspeed directly to find the resultant vertical component.
Total vertical component = 141.4 mph + 40 mph (from the wind) = 181.4 mph.
Step 5: Find the resultant magnitude and angle.
To find the magnitude of the resultant vector (the plane's ground speed), we use the Pythagorean theorem.
Resultant magnitude (ground speed) = √(141.4^2 + 181.4^2) ≈ 229.6 mph.
To find the angle of the resultant vector (the bearing the pilot should fly), we use the inverse tangent (arctan) function.
Resultant angle = arctan(181.4 mph / 141.4 mph) ≈ 51.3°.
Step 6: Check the quadrant and adjust the angle if needed.
Since the resultant vector is pointing northeast, and 237° is in the southwest quadrant, we need to subtract the angle we found from 180° to get the correct bearing the pilot should fly.
Correct bearing = 180° - 51.3° ≈ 128.7°.
Therefore, the pilot should fly on a bearing of approximately 128.7 degrees, and the plane's ground speed is approximately 229.6 mph.