An airplane flies due north at 235 km/h relative to the air. There is a wind blowing at 65 km/h to the northeast relative to the ground. What are the plane's speed and direction relative to the ground?
To find the plane's speed and direction relative to the ground, we can resolve the airplane's velocity vector into its northward and eastward components.
First, let's consider the airplane's velocity relative to the ground.
Given:
Airplane's velocity relative to air (V_air) = 235 km/h due north
Wind velocity relative to ground (V_wind) = 65 km/h to the northeast
To find the airplane's velocity relative to the ground, we need to add the velocities of the airplane and the wind.
1. Resolve the wind velocity into its northward and eastward components:
We can treat the northeast direction as the combination of north and east directions.
The wind's northward component (V_wind_north) = V_wind * sin(45 degrees)
V_wind_north = 65 km/h * sin(45 degrees) ≈ 45.9 km/h
The wind's eastward component (V_wind_east) = V_wind * cos(45 degrees)
V_wind_east = 65 km/h * cos(45 degrees) ≈ 45.9 km/h
2. Add the northward components of the airplane's velocity and the wind's velocity:
The airplane's northward velocity (V_air_north) = V_air = 235 km/h
The total northward velocity (V_total_north) = V_air_north + V_wind_north
V_total_north = 235 km/h + 45.9 km/h ≈ 280.9 km/h
3. Add the eastward components of the airplane's velocity and the wind's velocity:
The airplane's eastward velocity (V_air_east) = 0 km/h (since the airplane is flying due north)
The total eastward velocity (V_total_east) = V_air_east + V_wind_east
V_total_east = 0 km/h + 45.9 km/h = 45.9 km/h
Now that we have the northward (V_total_north) and eastward (V_total_east) components of the airplane's velocity relative to the ground, we can determine the total speed and direction using the Pythagorean theorem.
4. Determine the total speed:
The total speed (V_total) = √(V_total_north^2 + V_total_east^2)
V_total = √(280.9 km/h^2 + 45.9 km/h^2) ≈ 285 km/h
5. Determine the direction:
The direction can be determined using trigonometry:
The angle (θ) = atan(V_total_east / V_total_north)
θ = atan(45.9 km/h / 280.9 km/h) ≈ 9.3 degrees
Therefore, the plane's speed relative to the ground is approximately 285 km/h, and its direction relative to the ground is approximately 9.3 degrees east of north.
To find the plane's speed and direction relative to the ground, we can use vector addition.
1. First, let's break down the wind velocity into its northward and eastward components.
- The northeast wind blows at an angle of 45 degrees with respect to the north. Therefore, the northward component of the wind velocity would be: 65 km/h * cos(45°) = 45.9 km/h.
- Similarly, the eastward component of the wind velocity would be: 65 km/h * sin(45°) = 45.9 km/h.
2. Next, let's add the northward component of the wind velocity and the plane's velocity relative to the air.
- The northward component of the plane's velocity relative to the air is 235 km/h.
So, the sum of the northward components is: 235 km/h + 45.9 km/h = 280.9 km/h (rounded to one decimal place).
3. Since there is no wind blowing directly from east to west, the eastward component of the plane's velocity relative to the ground would remain the same.
Therefore, the plane's speed and direction relative to the ground are approximately 280.9 km/h in the north direction and 45.9 km/h in the east direction.