An airplane is flying in the direction 148° with an airspeed of u = 920 kilometers per hour. Because of the wind, its groundspeed and direction are v = 820 kilometers per hour and 140°, respectively (see figure). Find the direction and speed of the wind.
To find the direction and speed of the wind, we can use vector addition.
Let's represent the airplane's velocity vector as A and the wind's velocity vector as W.
The groundspeed vector, which is the vector sum of the airplane's airspeed vector and the wind's velocity vector, is given by:
V = A + W
We are given the magnitudes of vectors A and V:
|A| = 920 km/h
|V| = 820 km/h
We can break down vector A and vector V into their x and y components, using trigonometry.
For vector A:
- The x-component (Ax) is given by |A| * cos(148°)
- The y-component (Ay) is given by |A| * sin(148°)
For vector V:
- The x-component (Vx) is given by |V| * cos(140°)
- The y-component (Vy) is given by |V| * sin(140°)
Now we can set up two equations based on the x and y components:
Ax + Wx = Vx -- (1)
Ay + Wy = Vy -- (2)
Substituting the formulas for the x and y components:
|A| * cos(148°) + |W| * cos(θ) = |V| * cos(140°) -- (1)
|A| * sin(148°) + |W| * sin(θ) = |V| * sin(140°) -- (2)
We need to solve these equations to find the direction and speed of the wind.
Let's calculate the components of vectors A and V using the given values:
Ax = 920 km/h * cos(148°) ≈ -468.22 km/h
Ay = 920 km/h * sin(148°) ≈ 830.30 km/h
Vx = 820 km/h * cos(140°) ≈ -490.80 km/h
Vy = 820 km/h * sin(140°) ≈ 725.64 km/h
Substituting these values into equations (1) and (2) gives us:
-468.22 + |W| * cos(θ) = -490.80 -- (3)
830.30 + |W| * sin(θ) = 725.64 -- (4)
Now we can isolate |W| * cos(θ) and |W| * sin(θ) in equations (3) and (4):
|W| * cos(θ) = -490.80 + 468.22 -- (5)
|W| * sin(θ) = 725.64 - 830.30 -- (6)
Solving equations (5) and (6) will give us |W| * cos(θ) and |W| * sin(θ).
|W| * cos(θ) ≈ -22.58
|W| * sin(θ) ≈ -104.66
Now, we can find the magnitude of vector W using the Pythagorean theorem:
|W| = sqrt((-22.58)^2 + (-104.66)^2) ≈ 106.88 km/h
Finally, we can find the direction of the wind using the inverse tangent function:
θ = atan((-104.66) / (-22.58)) ≈ 79.8°
Therefore, the direction of the wind is approximately 79.8° and the speed of the wind is approximately 106.88 km/h.
To solve this problem, we need to break down the airplane's velocity into its horizontal and vertical components.
Let's represent the airplane's wind velocity as (v_wx, v_wy), where v_wx is the horizontal component and v_wy is the vertical component.
Given that the airplane's airspeed is u = 920 km/h and its groundspeed is v = 820 km/h, we have the following equations:
Horizontal Component: v_wx = u cos(148°)
Vertical Component: v_wy = u sin(148°)
Now, let's analyze the components of the wind velocity, taking into account its direction of 140°:
Wind Horizontal Component: v_wx = v cos(140°)
Wind Vertical Component: v_wy = v sin(140°)
Since the wind is affecting the airplane's velocity, the sum of the wind components and the airplane's component should be equal:
v_wx + u cos(148°) = v cos(140°)
v_wy + u sin(148°) = v sin(140°)
Now, we can substitute the given values and solve the system of equations to find the wind velocity.
v cos(140°) - u cos(148°) = v_wx
v sin(140°) - u sin(148°) = v_wy
Substituting values:
820 cos(140°) - 920 cos(148°) = v_wx
820 sin(140°) - 920 sin(148°) = v_wy
Using a calculator, we get:
v_wx ≈ -119.268 km/h (negative sign indicates wind is blowing in opposite direction of airplane)
v_wy ≈ -257.936 km/h (negative sign indicates wind is blowing in opposite direction of airplane)
Therefore, the wind's direction is opposite to the airplane's direction, i.e., 148° - 180° = -32°. And the wind's speed can be calculated using the Pythagorean theorem:
Speed of Wind = √(v_wx² + v_wy²)
Speed of Wind ≈ √((-119.268 km/h)² + (-257.936 km/h)²)
Speed of Wind ≈ 285.268 km/h
So, the wind's direction is approximately -32°, and its speed is approximately 285.268 km/h.
(50,152)
if the wind's vector is w, then
u+w = v
<487,-780> + <x,y> = <527,-628>
So, now you can figure x and y, then get the speed and direction.