Question

A plane has an airspeed of 111 km/h. It is flying on a bearing of 79 degrees while there is a 25 km/h wind out of the northeast​ (bearing 225 degrees​). What are the ground speed and the bearing of the​ plane?

Answer 1

Answer: Ground Speed = 91 km/hr,   Bearing = 189°

Step-by-step explanation:

Step 1: Draw a picture (see attached) to determine the angle between the given vectors.  Notice that I moved the wind vector 180° so the head of the wind vector would line up with the tail of the plane vector. This created an angle of 34° between the plane and wind vectors. Why?

• the dashed line is 45°
• 79° (plane) - 45° (wind) = 34°

Step 2: Solve for the length of the resultant vector using Law of Cosines

c² = a² + b² - ab cos C

c² = (111)² + (25)² - (111)(25) cos 34°

c² = 12,946 - 4601

c² = 8345

c = 91

Ground speed is 91 km/hr

Step 3: Solve for the bearing of the resultant vector using Law of Sines

$\dfrac{sin\ A}{a}=\dfrac{sin\ C}{c}$

$\dfrac{sin\ A}{25}=\dfrac{sin\ 34}{91}$

$sin\ A=\dfrac{25\ sin\ 34}{91}$

$A=sin^{-1}\bigg(\dfrac{25\ sin\ 34}{91}\bigg)$

A = 9°

Reminder that we moved the wind vector 180° to create the resultant vector so we need to add 180° to our answer.

Bearing = A + 180°

              =  9° + 180°

              = 189°