Airplane Wind Angle Problem

The ground velocity is the vector sum of the airplane's velocity relative to air and the wind velocity:

$\vec{v}_{ground} = \vec{v}_{air} + \vec{v}_{wind}$

Since the pilot points the aircraft directly north, the airplane's velocity relative to air has components:

• $v_{air,x} = 0$
• $v_{air,y} = 293\text{ km/h}$

The wind blows toward the northeast at 45° north of east, so its components are:

• $v_{wind,x} = v_{wind} \cos(45°) = 54 \cdot \frac{\sqrt{2}}{2} = 38.18\text{ km/h}$
• $v_{wind,y} = v_{wind} \sin(45°) = 54 \cdot \frac{\sqrt{2}}{2} = 38.18\text{ km/h}$

Therefore, the ground velocity components are:

• $v_{ground,x} = v_{air,x} + v_{wind,x} = 0 + 38.18 = 38.18\text{ km/h}$
• $v_{ground,y} = v_{air,y} + v_{wind,y} = 293 + 38.18 = 331.18\text{ km/h}$