A plane is flying 25 degrees north of west at 190 km/h and encounters a wind from 15 degrees north of east at 45 km/h. What is the planes new velocity with respect to the ground in standard position?
v=190 km/h, α= 25°,
u=45 km/h, β =15°.
V(x) =v•cos α - u•sin β =…
V(y)= v•sinα+u•sinβ=…
V=sqrt{V(x)²+V(y)²}=...
Sine Law:
V/sin(α+ β) =u/sinγ
Solve for γ.
V directed (α+γ)° north of west
lol can you solve this for me? We have done this multiple times (we think) and we are getting a different answer than that's in the back of the book
And your answer doesn't match the back of the book either.
The back of the book says 227 km/h at 162.4 degrees
V(x) = - v•cos α +u•cosβ = - 190cos25 + 45cos15=172.2+43.5= -128.7 km/h
V(y)= v•sinα+u•sinβ= 190sin25 + 45sin15= 80.3 + 11.6 = 91.9 km/h
V=sqrt{V(x)²+V(y)²}= 158.1 km/h
To solve this problem, we need to break down the velocities into their horizontal (east-west) and vertical (north-south) components.
Let's start with the plane's velocity. We are given that it is flying 25 degrees north of west at 190 km/h. This means that the plane has a northward component and a westward component.
Using trigonometry, we can find the components as follows:
Northward component = velocity * sin(angle)
Westward component = velocity * cos(angle)
Applying this to the given values, we have:
Northward component = 190 km/h * sin(25 degrees)
Westward component = 190 km/h * cos(25 degrees)
Next, let's consider the wind's velocity. We are given that it is coming from 15 degrees north of east at 45 km/h. This means that the wind has an eastward component and a southward component.
Using the same trigonometric approach:
Eastward component = velocity * cos(angle)
Southward component = velocity * sin(angle)
Applying this to the given values, we have:
Eastward component = 45 km/h * cos(15 degrees)
Southward component = 45 km/h * sin(15 degrees)
Now, we need to add up the respective components of the plane's velocity and the wind's velocity to find the resultant velocity (the plane's new velocity with respect to the ground).
Resultant eastward component = plane's eastward component + wind's eastward component
Resultant northward component = plane's northward component + wind's southward component
Calculating these components:
Resultant eastward component = (190 km/h * cos(25 degrees)) + (45 km/h * cos(15 degrees))
Resultant northward component = (190 km/h * sin(25 degrees)) + (45 km/h * sin(15 degrees))
Now, we can calculate the magnitude (speed) of the resultant velocity using the Pythagorean theorem:
Resultant velocity = √((Resultant eastward component)^2 + (Resultant northward component)^2)
Plugging in the values:
Resultant velocity = √(((190 km/h * cos(25 degrees)) + (45 km/h * cos(15 degrees)))^2 + ((190 km/h * sin(25 degrees)) + (45 km/h * sin(15 degrees)))^2)
After calculating the above expression, you will get the resultant velocity of the plane with respect to the ground in standard position.