A plane flies due north (90° from east) with a velocity of 100 km/h for 3 hours. During this time, a steady wind blows southeast at 30 km/h at an angle of 315° from due east. After 3 hours, where will the plane’s position be relative to its starting point? Show your work.
V = 100km/h[90o] + 30km/h[315o]
X = 100*cos90 + 30*cos315 = 21.21 km/h
Y = 100*sin90 + 30*sin315 = 78.79 km/h
tan A = Y/X = 78.79/21.21 = 3.71461
A = 74.93o
V=Y/sinA = 78.79/sin74.93=81.6 km/h
[78.79]
D=V*t = 81.6[78.79] * 3 = 245 km[78.8o]
CORRECTION:
V = 81.6km/h[74.93o]
D = 245 km[74.93o]
yes
To determine the plane's position after 3 hours, we need to combine the velocities of the plane and the wind.
First, let's break down the velocities into their x and y components.
The plane's velocity is due north, which means it has a velocity only in the y-direction (upward) and no velocity in the x-direction (sideways). Therefore, the plane's velocity components are:
Vx (plane) = 0 km/h (no x component)
Vy (plane) = 100 km/h (y component)
The wind's velocity is blowing southeast, which means it has both x and y components. To find the components, we need to use the given angle of 315° from due east.
To find the x and y components of the wind's velocity, we need to break down the velocity vector using trigonometry:
Vx (wind) = V (wind) * cos(θ)
Vy (wind) = V (wind) * sin(θ)
Given that V (wind) = 30 km/h and θ = 315°, we can calculate the components:
Vx (wind) = 30 km/h * cos(315°)
Vy (wind) = 30 km/h * sin(315°)
Now let's calculate these components:
Vx (wind) = 30 km/h * cos(315°) ≈ 30 km/h * (-0.7071) ≈ -21.2132 km/h
Vy (wind) = 30 km/h * sin(315°) ≈ 30 km/h * (-0.7071) ≈ -21.2132 km/h
Next, we can add the components of the plane's velocity and the wind's velocity to get the final velocity components:
Vx (total) = Vx (plane) + Vx (wind)
Vy (total) = Vy (plane) + Vy (wind)
Vx (total) = 0 km/h + (-21.2132 km/h) ≈ -21.2132 km/h
Vy (total) = 100 km/h + (-21.2132 km/h) ≈ 78.7868 km/h
Now we have the total x and y components of the velocity after considering the plane's velocity and the wind's velocity.
Finally, we can find the displacement of the plane after 3 hours by multiplying the components by the time:
Displacement in the x-direction = Vx (total) * time
Displacement in the y-direction = Vy (total) * time
Displacement in the x-direction = -21.2132 km/h * 3 h ≈ -63.6396 km
Displacement in the y-direction = 78.7868 km/h * 3 h ≈ 236.3604 km
The plane's final position relative to its starting point is approximately 63.64 km to the west and 236.36 km to the north.