An airplane is traveling 725 m/s at 48.8° degrees north of west when a wind starts to blow. The velocity of the wind is 106.7 m/s and it is blowing 21.8° east of north. What is the speed of the airplane relative to the ground?
Unless otherwise indicated, all angles are measured CCW from +x-axis.
Vr = 725m/s[131.2o] + 106.7m/2[68.2o].
X = 725*Cos131.2 + 106.7*Cos68.2= -438 m/s.
Y = 725*sin131.2 + 106.7*sin68.2 = 645 m/s.
Tan A = Y/X = 645/-438 = -1.47162.
A = 55.8o N. of W.
Vr = Y/sin A = 645/sin55.8 = 780 m/s.
= Resultant velocity.
To find the speed of the airplane relative to the ground, we need to determine the resultant vector by adding the vectors of the airplane's velocity and the velocity of the wind.
Step 1: Break down the vectors into their x and y components.
The airplane's velocity can be split into its x and y components using trigonometry.
Given: Velocity of the airplane = 725 m/s at 48.8° N of W.
Velocity of the airplane in the x-direction:
Vx = v * cos(θ)
Vx = 725 m/s * cos(48.8°) ≈ 467.46 m/s
Velocity of the airplane in the y-direction:
Vy = v * sin(θ)
Vy = 725 m/s * sin(48.8°) ≈ 548.35 m/s
Similarly, break down the wind's velocity into its x and y components.
Given: Velocity of the wind = 106.7 m/s at 21.8° E of N.
Velocity of the wind in the x-direction:
Vwx = v * sin(θ)
Vwx = 106.7 m/s * sin(21.8°) ≈ 39.07 m/s
Velocity of the wind in the y-direction:
Vwy = v * cos(θ)
Vwy = 106.7 m/s * cos(21.8°) ≈ 98.68 m/s
Step 2: Add the x and y components separately.
Vx_total = Vx + Vwx
Vx_total = 467.46 m/s + 39.07 m/s ≈ 506.53 m/s
Vy_total = Vy + Vwy
Vy_total = 548.35 m/s + 98.68 m/s ≈ 647.03 m/s
Step 3: Find the magnitude (speed) of the resultant vector.
The speed of the airplane relative to the ground can be calculated using the Pythagorean theorem:
Speed = √(Vx_total^2 + Vy_total^2)
Speed = √(506.53^2 + 647.03^2) ≈ 819.61 m/s
Therefore, the speed of the airplane relative to the ground is approximately 819.61 m/s.
Vp = 725m/s[W.48.8oN] + 106.7[E68.2N] =
-725*Cos48.8+725*sin48.8 + 106.7*Cos68.2
+106.7*sin68.2 = -478+546i + 39.6+99.1i = -438.4 + 645.1i = 780m/s[W55.8oN] =
780m/s[N34.2oW].