A plane head north with an airspeed of 420 km/h. However, relative to the ground, it travels in a direction 7.0 west of north. if the wind's direction is towards the southwest (45.0 south of west), what is the wind speed?
vector diagram would be the 420 heading north + unknown head NW = unknown heading 7 degrees W of N
7 degrees in bottom corner 45 + 90= 135 above that and 180-7-135 = 38 in the top left corner
using sine law Vw/sin(7)= 420/sin(38) where Vw = velocity of wind = 83km/hr
Well, it seems like the plane is having a "windy" situation up there! Let's break it down.
First, we need to find the component of the wind speed towards the north and towards the west.
If the plane is heading 7.0 degrees west of north, then the component of the airspeed towards the north would be 420 km/h multiplied by the cosine of 7.0 degrees. Similarly, the component towards the west would be 420 km/h multiplied by the sine of 7.0 degrees.
Now, putting on our detective hats, let's figure out how this relates to the wind's direction!
If the wind's direction is towards the southwest (45.0 degrees south of west), we can find the component of the wind speed towards the west by multiplying the wind speed by the cosine of 45.0 degrees.
Since the plane's airspeed and the wind's speed combine to give us the ground speed, we can equate the component towards the west from the plane's airspeed with the component towards the west from the wind.
By doing some math magic and equating the two westward components, we can solve for the wind speed.
So, grab your calculator and give it a whirl! Crunch those numbers, and you'll land on the wind speed. Happy calculating!
To find the wind speed, we need to break down the plane's motion into its north and west components.
Given:
Airspeed = 420 km/h
Plane's direction relative to the ground = 7.0° west of north
Wind's direction = 45.0° south of west
Step 1: Find the northward component of the plane's motion.
The northward component can be found using trigonometry. We need to find the component of the airspeed in the north direction, which is the cosine of the angle.
Northward component = Airspeed × cos(angle)
Northward component = 420 km/h × cos(7.0°)
Step 2: Find the westward component of the plane's motion.
The westward component can also be found using trigonometry. We need to find the component of the airspeed in the west direction, which is the sine of the angle.
Westward component = Airspeed × sin(angle)
Westward component = 420 km/h × sin(7.0°)
Step 3: Find the wind's north and south component.
The wind's north and south component can be found using trigonometry, with the given wind direction.
Wind's north component = Wind speed × cos(angle)
Wind's south component = Wind speed × sin(angle)
Step 4: Find the wind's speed.
Since we know that the plane's northward component is equal to the wind's north component and the plane's westward component is equal to the sum of the wind's west component and the wind's south component, we can set up two equations.
Northward component = Wind's north component
Westward (plane's) component = Wind's west component + Wind's south component
Substituting the component equations, we have:
420 km/h × cos(7.0°) = Wind speed × cos(45.0°)
420 km/h × sin(7.0°) = Wind speed × sin(45.0°)
Solving these equations will give us the wind speed.
(Note: I will perform the calculations, but the final answer might be rounded for simplicity.)
Calculating the north component:
Northward component = 420 km/h × cos(7.0°)
Northward component = 420 km/h × 0.992
Northward component ≈ 416.64 km/h
Calculating the south component:
South component = Wind speed × sin(45.0°)
416.64 km/h = Wind speed × sin(45.0°)
Dividing both sides by sin(45.0°):
Wind speed ≈ 416.64 km/h ÷ sin(45.0°)
Wind speed ≈ 589.19 km/h
Therefore, the wind speed is approximately 589.19 km/h.
To find the wind speed, we need to determine the velocity of the plane relative to the ground using the given information.
Let's break down the given data:
- The plane's airspeed is 420 km/h, heading north.
- However, relative to the ground, the plane travels in a direction 7.0° west of north.
- The wind's direction is towards the southwest, 45.0° south of west.
To solve this problem, we can use vector addition. We can break down the velocities into their north-south (vertical) and east-west (horizontal) components.
First, let's find the north-south component of the plane's velocity relative to the ground:
The airspeed of the plane in the north direction is equal to its north-south component.
So, the north-south component of the plane's velocity is 420 km/h.
Next, let's find the east-west component of the plane's velocity relative to the ground:
To determine the east-west component, we need to calculate the horizontal component of the airspeed (which is 420 km/h) due to the wind's impact.
The wind blows towards the southwest, which is 45.0° south of west.
So, we need to find the cosine component of the wind's angle, which gives us the horizontal component.
Cos(45°) = adjacent/hypotenuse = east-west component/horizontal component.
By rearranging the equation, we get: east-west component = cos(45°) * 420 km/h.
Solving this equation gives us the east-west component of the plane's velocity, which is approximately 297.5 km/h south of west.
Now that we have both the north-south and east-west components of the plane's velocity, we can add them using vector addition:
To find the resultant velocity (plane's velocity relative to the ground), we use the Pythagorean theorem:
Resultant velocity = sqrt((north-south component)^2 + (east-west component)^2)
Plugging in the values, we get:
Resultant velocity = sqrt((420 km/h)^2 + (297.5 km/h)^2).
The magnitude of the resultant velocity represents the speed of the plane relative to the ground, which is influenced by both the airspeed and the wind. We need to find the wind speed, which is the difference between the resultant velocity and the airspeed of the plane.
Wind speed = Resultant velocity - Airspeed of the plane
Now we can calculate the wind speed by plugging in the values we found:
Wind speed = (sqrt((420 km/h)^2 + (297.5 km/h)^2)) - 420 km/h
Calculating this expression will give us the wind speed.
All angles are CCW from +x-axis.
Vp + Vw = 420km/h[97o].
420[90o] + Vw[225o] = 420[97o]
420i + Vw*Cos225+Vw*sin225 = 420*Cos97 +420*sin97 = 420i - 0.707Vw - i0.707Vw = -51.2 + 417i = -51.2 -3i.
0.707Vw (-1-1i) = -51.2 - 3i.
0.707Vw(1.41[225o]) = 51.3[183.4o].
Vw[225o] = 51.3[183.4o]
Vw = 51.3km/h[-41.6]
Speed of wind = 51.3km/h.