An airplane has air speed of 540 mph and a heading of 50*. The wind is blowing from the north at 27 mph. Find the plane's ground speed and course direction.
540 @ 50° = <413.66,347.11>
wind is <0,-27>
add them up to get <413.66,320.11>
which is 523 @ 37.7°
To find the airplane's ground speed and course direction, we need to break down the airplane's velocity into its components: the airspeed and the wind speed.
1. First, let's determine the wind's effect on the airplane's velocity. Since the wind is blowing from the north at 27 mph, it will have an effect on both the speed and the direction of the airplane.
2. To calculate the effect of the wind on the speed, we subtract the wind speed from the airplane's airspeed:
Ground Speed = Airspeed - Wind Speed
Ground Speed = 540 mph - 27 mph
Ground Speed = 513 mph
3. Next, let's determine the effect of the wind on the course direction. Since the wind is coming from the north, it will push the airplane off course slightly.
4. We can use trigonometry to determine the angle between the airplane's heading and the actual course direction. The angle between the heading and the course direction is equal to the arctangent of the wind speed divided by the airspeed:
Angle = arctan(Wind Speed / Airspeed)
Angle = arctan(27 / 540)
Angle ≈ 2.85°
5. To find the actual course direction, we subtract the angle determined in the previous step from the airplane's heading:
Course Direction = Heading - Angle
Course Direction = 50° - 2.85°
Course Direction ≈ 47.15°
So, the plane's ground speed is approximately 513 mph, and its course direction is approximately 47.15°.