Direction is usually specified as bearing, however, with regards to vectors direction is specified relative to the positive, x-axis
The correct option for the ground speed and direction of the plane is option B
B) Ground speed: 639 mph; direction 161°
The given parameters are;
Direction of airplane = 165° (vector, measured from the horizontal x-axis)
Airspeed of airplane = 650 mph
The speed of the wind blowing = 50 mph
The direction of the wind = N30°E
Required:
The ground speed of the airplane
Solution:
The velocity of the airplane in vector form is found as follows;
[tex]\overset \longrightarrow P[/tex] = 650 × cos(165)·i + 650 × sin(165)·j ≈ -628·i + 168·j
The velocity of the wind in vector form is found as follows;
[tex]\overset \longrightarrow W[/tex] = 50 × cos(60)·i + 50 × sin(60)·j ≈ 25·i + 43·j
The sum of the vectors, [tex]\overset \longrightarrow R[/tex] = The ground speed, therefore;
[tex]\overset \longrightarrow R[/tex] = [tex]\overset \longrightarrow P[/tex] + [tex]\overset \longrightarrow W[/tex]
[tex]\overset \longrightarrow R[/tex] = (25 - 628)·i + (43 + 168)·j = -603·i + 211·j
The magnitude of the ground speed = √((-603)² + 211²) ≈ 639
The magnitude of the ground speed ≈ 639 mph
The direction, Φ ≈ arctan(211/(-603)) ≈ -19°
Φ is measured from the negative x-axis in the clockwise direction, therefore;
The direction of the airplane from the positive x-axis, in the anticlockwise direction, θ = 180° + Φ
θ = 180° + (-19) = 161°
The direction of the airplane measured from the positive x-axis, θ ≈ 161°
Therefore, the correct option is ground speed; 639 mph; direction 161°
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