A pilot is flying at a heading of 96° at 225 miles per hour wind is blowing from the south west at a heading of 37° find the ground speed and courses of the plane
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To find the ground speed and course of the plane, we can break down the problem into its components and use trigonometry.
1. Determine the horizontal and vertical components of the wind speed:
- Horizontal component = wind speed * cos(wind heading)
- Vertical component = wind speed * sin(wind heading)
In this case, the wind speed is 225 mph, and the wind heading is 37°.
- Horizontal component = 225 * cos(37°)
= 225 * 0.7986
≈ 179.68 mph
- Vertical component = 225 * sin(37°)
= 225 * 0.6018
≈ 135.89 mph
2. Find the ground speed components:
- Ground speed in the east-west (x) direction = plane speed in x direction - wind speed in x direction
- Ground speed in the north-south (y) direction = plane speed in y direction - wind speed in y direction
In this case, the plane is flying at a heading of 96°:
- Plane speed in the x direction = plane speed * cos(plane heading)
= 225 * cos(96°)
= 225 * (-0.1736)
≈ -39.06 mph (negative because the plane is flying west)
- Plane speed in the y direction = plane speed * sin(plane heading)
= 225 * sin(96°)
= 225 * 0.9848
≈ 221.08 mph
- Ground speed in the east-west (x) direction = -39.06 mph - 179.68 mph
= -218.74 mph
- Ground speed in the north-south (y) direction = 221.08 mph - 135.89 mph
= 85.19 mph
3. Calculate the resultant ground speed:
- Ground speed = square root((ground speed in x direction)^2 + (ground speed in y direction)^2)
= square root((-218.74 mph)^2 + (85.19 mph)^2)
= square root(47879.0276 + 7253.3761)
= square root(55132.4037)
≈ 234.81 mph
4. Determine the course of the plane:
- Course = arctan(ground speed in x direction / ground speed in y direction)
In this case, the ground speed in the east-west (x) direction is negative (-218.74 mph), and the north-south (y) direction is positive (85.19 mph):
- Course = arctan(-218.74 mph / 85.19 mph)
= arctan(-2.569)
≈ -68.62°
Note: Arctan function gives the angle in the fourth quadrant, so the course would be -68.62° relative to the north. To determine the course relative to a compass heading (0° being north, 90° being east), we can subtract the course angle from 360°:
- Course = 360° - 68.62°
≈ 291.38°
Therefore, the ground speed is approximately 234.81 mph, and the course of the plane is approximately 291.38° (relative to a compass heading).