An airplane is flying at an airspeed of 345 km/h on a heading of 040 degrees. The wind is blowing at 18 km/h from a bearing of 087 degrees. Determine the ground velocity of the airplane.
I am also confused about this question, but I do know that the velocity of the plane is [345cos50,345sin50], the velocity of the wind is needed, and you need to add the two to get the resultant. I am just really bad with converting angles and can't figure out what numbers to use for the wind.
To determine the ground velocity of the airplane, we need to find the horizontal components of both the airspeed and the wind velocity.
1. Convert the airspeed of 345 km/h to meters per second:
345 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 95.83 m/s
2. Convert the wind velocity of 18 km/h to meters per second:
18 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 5 m/s
3. Calculate the horizontal component of the airspeed using trigonometry:
Horizontal component of airspeed = Airspeed * cos(heading)
Horizontal component of airspeed = 95.83 m/s * cos(40 degrees) ≈ 73.48 m/s
4. Calculate the horizontal component of the wind velocity using trigonometry:
Horizontal component of wind velocity = Wind velocity * cos(bearing)
Horizontal component of wind velocity = 5 m/s * cos(87 degrees) ≈ 0.81 m/s
5. Add the horizontal components of the airspeed and the wind velocity to get the ground velocity:
Ground velocity = Horizontal component of airspeed + Horizontal component of wind velocity
Ground velocity = 73.48 m/s + 0.81 m/s ≈ 74.29 m/s
Therefore, the ground velocity of the airplane is approximately 74.29 m/s.
To determine the ground velocity of the airplane, we need to find the resultant vector of the airplane's airspeed and the wind velocity.
Step 1: Convert the airspeed and wind velocity from km/h to m/s.
- Airspeed: 345 km/h = 345000 m/3600 s = 95.83 m/s (rounded to two decimal places)
- Wind velocity: 18 km/h = 18000 m/3600 s = 5 m/s
Step 2: Split the airspeed and wind velocity into their respective components.
- Airspeed component in the north/south direction: Airspeed * sin(heading)
north/south component = 95.83 m/s * sin(40 degrees)
= 95.83 m/s * 0.6428
≈ 61.64 m/s (rounded to two decimal places)
- Airspeed component in the east/west direction: Airspeed * cos(heading)
east/west component = 95.83 m/s * cos(40 degrees)
= 95.83 m/s * 0.7660
≈ 73.52 m/s (rounded to two decimal places)
- Wind component in the north/south direction: Wind velocity * sin(wind bearing)
north/south component = 5 m/s * sin(87 degrees)
= 5 m/s * 0.9998
≈ 4.99 m/s (rounded to two decimal places)
- Wind component in the east/west direction: Wind velocity * cos(wind bearing)
east/west component = 5 m/s * cos(87 degrees)
= 5 m/s * 0.0175
≈ 0.09 m/s (rounded to two decimal places)
Step 3: Calculate the total north/south and east/west components.
- Total north/south component = Airspeed north/south component + Wind north/south component
= 61.64 m/s + 4.99 m/s
≈ 66.63 m/s (rounded to two decimal places)
- Total east/west component = Airspeed east/west component + Wind east/west component
= 73.52 m/s + 0.09 m/s
≈ 73.61 m/s (rounded to two decimal places)
Step 4: Calculate the magnitude and direction of the ground velocity vector using the Pythagorean theorem and inverse tangent (arctan) function.
- Ground velocity magnitude = √(Total north/south component^2 + Total east/west component^2)
= √(66.63 m/s)^2 + (73.61 m/s)^2
= √(4429.73 + 5410.10)
≈ √9840.83
≈ 99.20 m/s (rounded to two decimal places)
- Ground velocity direction = arctan(Total north/south component / Total east/west component)
= arctan(66.63/73.61)
≈ arctan(0.904)
≈ 42.68 degrees (rounded to two decimal places)
Therefore, the ground velocity of the airplane is approximately 99.20 m/s at a heading of approximately 42.68 degrees.