The nose of a plane is pointing west with an airspeed of 350 km/h. The plane’s resultant ground velocity is 315 km/h [S75°W]. Determine the speed and direction of the wind, correct to 2 d.p. Include a labeled diagram with your solution.
I got 81.53 km/h [S] but I think I may be wrong
For wind velocity I will use the direction the wing is GOING
( In other words a North wind goes South)
Wind velocity East = Ve
Wind velocity South = Vs
Total velocity west = 350 - Ve = 315 sin 75
Total velocity south = Vs = 315 cos 75
According to my sketch, if the speed of the wind is x km/h
by the cosine law:
x^2 = 350^2 + 315^2 - 2(350)(315)cos15°
x = appr 93.479 km/h
then by the sine law, if the angle opposite the side 315 is A
sinA/315 = sin15/93.479
I get A = 60.71°
so the wind has a speed of 93.48 km/h in the direction of S 29.29° E
check my arithmetic, I did not write any of this down first.
To determine the speed and direction of the wind, we can use vector addition. Let's denote the speed of the wind as "W" and its direction as "θ".
Given that the plane's airspeed is 350 km/h towards the west, and its resultant ground velocity is 315 km/h [S75°W], we can break down these velocities into their respective components.
1. Determine the east-west component (ground velocity):
The resultant ground velocity is pointing S75°W, which can be broken down into east and west components. Using trigonometry, we can find the east-west component as follows:
East-West Component = Ground Velocity * cos(θ)
∴ East-West Component = 315 km/h * cos(75°)
= 315 km/h * (-cos(75°)) (since the component is towards the west)
≈ -79.49 km/h (rounding to two decimal places)
2. Determine the north-south component (ground velocity):
Using trigonometry again, we can find the north-south component as follows:
North-South Component = Ground Velocity * sin(θ)
∴ North-South Component = 315 km/h * sin(75°)
≈ 303.58 km/h (rounding to two decimal places)
Now, we can consider the wind's effect on the plane's velocity. Since the nose of the plane is pointing directly west, the wind affects only the east-west component of the plane's velocity.
3. Determine the east-west component (plane velocity):
The resulting east-west component of the plane's velocity can be written as:
East-West Component (plane velocity) = East-West Component (ground velocity) + East-West Component (wind velocity)
∴ -79.49 km/h = 350 km/h + East-West Component (wind velocity)
Solving for the East-West Component (wind velocity), we get:
East-West Component (wind velocity) = -79.49 km/h - 350 km/h
= -429.49 km/h
4. Determine the speed of the wind:
The speed of the wind can be found by taking the magnitude of the East-West Component (wind velocity):
Speed of the wind = |East-West Component (wind velocity)|
= |-429.49 km/h|
= 429.49 km/h
Therefore, the speed of the wind is approximately 429.49 km/h.
5. Determine the direction of the wind:
To find the direction of the wind, we need to calculate the angle it forms with the north direction (clockwise). We can use the tangent function:
tan(θ) = North-South Component (ground velocity) / East-West Component (ground velocity)
∴ tan(θ) = 303.58 km/h / (-79.49 km/h) (Note: -79.49 km/h since the component is towards the west)
Solving for θ, we get:
θ ≈ -74.94°
Since the angle is measured clockwise from the north, the direction of the wind is approximately 74.94° clockwise from north, which can be written as:
Direction of the wind ≈ N74.94°E
In conclusion, the speed of the wind is approximately 429.49 km/h and its direction is approximately N74.94°E.
To determine the speed and direction of the wind, we can use vector addition.
Let's consider the plane's airspeed as one vector (350 km/h west) and the wind's velocity as another vector that we need to find. The resultant ground velocity is the sum of these two vectors.
We have the resultant ground velocity as 315 km/h [S75°W]. We can break this resultant vector into its north-south and east-west components using trigonometry.
The north-south component is given by 315 km/h * sin(75°), which is approximately 306.6 km/h south.
The east-west component is given by 315 km/h * cos(75°), which is approximately 82.07 km/h west.
Now, let's draw a labeled diagram to visualize the solution:
Wind Velocity
------------------------> (from west to east)
|
|
|
v
Wind ---- Plane x -----------------------> (from west to east)
Speed Ground Velocity
^ (from south to north)
|
|
|
|
|
v
We want to find the length of the wind velocity vector (speed of the wind) and its directional angle with respect to north (angle the wind is blowing from).
Using vector addition, we can find the resultant vector of the wind speed and the plane's airspeed. The resultant vector will be equal to the ground velocity vector.
To find the speed of the wind, we can use the Pythagorean theorem:
Wind speed = sqrt((315 km/h)^2 - (350 km/h)^2)
≈ 71.18 km/h
Now, to find the direction of the wind, we can use trigonometry. The directional angle of the wind (angle it is blowing from) can be found using the inverse tangent function:
Angle = atan(306.6 km/h / 82.07 km/h)
≈ 74.84°
Therefore, the speed of the wind is approximately 71.18 km/h, and the wind is coming from a direction approximately 74.84° west of north.