An airplane flies due south at 185 km/h relative to the air. There is a wind blowing at 68 km/h to the east relative to the ground. What are the plane's speed and direction relative to the ground?
Vp = 185km/h @270o + 68km/h @ 0o.
X = 185*cos270+68*cos(0)=0+68 = 68km/h.
Y = 185*sin270+68*sin(0)=-185+0=-185km/h
tanAr = Y/X = -185/68 = 2.72059.
Ar = -69.82o = Reference angle.
A = -69.82 + 360 = 290o = Direction.
Vp = X/cosA = 68/cos290=197 m/s=Velocity
of plane.
@Henry: Where the heck did you get 270??
To find the plane's speed and direction relative to the ground, we need to use vector addition.
Let's break down the velocities into their x and y components:
The plane's velocity relative to the air is 185 km/h due south.
The wind's velocity relative to the ground is 68 km/h to the east.
- The plane's velocity relative to the ground in the x-direction (east) is 0 km/h (since it's flying south).
- The plane's velocity relative to the ground in the y-direction (north) is -185 km/h (since it's flying due south).
- The wind's velocity relative to the ground in the x-direction (east) is 68 km/h.
- The wind's velocity relative to the ground in the y-direction (north) is 0 km/h (since it's blowing horizontally).
Now, let's add these components to determine the plane's velocity relative to the ground:
The plane's velocity relative to the ground in the x-direction is 0 km/h + 68 km/h = 68 km/h to the east.
The plane's velocity relative to the ground in the y-direction is -185 km/h + 0 km/h = -185 km/h to the south.
Using the Pythagorean theorem, we can find the magnitude of the plane's velocity relative to the ground:
Magnitude = sqrt((68 km/h)^2 + (-185 km/h)^2) ≈ 198.76 km/h
To find the direction, we can use trigonometry:
Direction = arctan((-185 km/h) / (68 km/h)) ≈ -69.27 degrees
So, the plane's speed relative to the ground is approximately 198.76 km/h, and its direction is approximately 69.27 degrees southwest.
To find the plane's speed and direction relative to the ground, we can use vector addition.
First, let's break down the velocities into their respective components:
The plane's velocity relative to the air is 185 km/h due south. Therefore, the vertical component is -185 km/h.
The wind's velocity relative to the ground is 68 km/h due east. Therefore, the horizontal component is +68 km/h.
Now, we can add the components to find the plane's velocity relative to the ground:
Vertical component: -185 km/h
Horizontal component: +68 km/h
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:
Resultant velocity = √((-185)^2 + 68^2) = √(34225 + 4624) ≈ √38849 ≈ 197 km/h
To find the direction of the resultant velocity, we can use the tangent function:
Direction = arctan(-185/68) ≈ -72.5°
Therefore, the plane's speed and direction relative to the ground are approximately 197 km/h at an angle of 72.5° west of due south.