Question

A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 57° 40′. A wind is blowing from the south at 27.1 mph. Find the bearing she should fly, and find the plane's ground speed.

Accepted Answer

Identify the vectors involved: the plane's airspeed vector (unknown direction, magnitude 168 mph), and the wind vector (27.1 mph from the south, which means it blows towards the north). The resultant vector is the ground speed vector, which should have a bearing of 57° 40′. Convert the bearing 57° 40′ into decimal degrees for easier calculation: 57 + 40/60 = 57.6667° approximately. This is the direction of the ground speed vector relative to north. Set up a vector diagram where the plane's velocity vector plus the wind vector equals the ground velocity vector. Represent the plane's velocity vector as having magnitude 168 mph and an unknown bearing angle $$ 	heta . $$The wind vector is 27.1 mph towards the north (bearing 0°). Write the components of the vectors in terms of $$ 	heta $$:
- Plane's velocity components: $$ V_p = (168 \sin(	heta), 168 \cos(	heta)) 
- $$Wind velocity components: $$ V_w = (0, 27.1) 
- $$Ground velocity components: $$ V_g = (V_p^x + V_w^x, V_p^y + V_w^y) $$
Since the ground velocity has bearing 57.6667°, its components can be expressed as $$ V_g = (V_g \sin(57.6667°), V_g \cos(57.6667°)) $$, where $$ V_g  is $$the ground speed magnitude. Set up equations equating the components:
$$
168 \sin(	heta) + 0 = V_g \sin(57.6667°) \
168 \cos(	heta) + 27.1 = V_g \cos(57.6667°)
$$
Use these two equations to solve for $$ 	heta  ($$the bearing the pilot should fly) and $$ V_g  ($$the ground speed).

A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 57° 40′. A wind is blowing from the south at 27.1 mph. Find the bearing she should fly, and find the plane's ground speed.

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Key Concepts

Vector Addition in Navigation

Bearing and Angle Measurement

Trigonometric Resolution of Vectors