Question

Find the true speed and bearing.
Wind vector: N11E (degrees) @55mph
Airspeed: 500mph
Direct fight bearing: N35.2E (degrees)

Answer 1

Find the true speed and bearing. Wind vector: N11E (degrees) 55mph. Airspeed: 500mph. Direct flight bearing: N35.2E (degrees)

$$\alpha = 35,2 - 11 = 24,2{}^\circ$$

To find the true speed use Law of cosine:

$$c = \sqrt{a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos \alpha}$$

$\vec{a}$ – wind vector;

$\vec{b}$ – vector of the direct flight;

$\vec{c}$ – true bearing vector;

$\alpha$ – angle between wind vector and direct flight vector.

$$\vec{c} = \sqrt{55^2 + 500^2 - 2 \cdot 55 \cdot 500 \cdot \cos(180{}^\circ - 24,2{}^\circ)} = \sqrt{253025 - 50166,606} = 550,628$$

To find the true bearing use the Law of sine

$$\frac{55}{\sin \beta} = \frac{500}{\sin(180{}^\circ - \alpha)}$$

$\beta$ – angle between $\vec{c}$ and $\vec{a}$.

$$\beta = \arcsin\left(\frac{55 \cdot \sin(180{}^\circ - \alpha)}{500}\right) = 2,584{}^\circ$$

The true bearing is

$$35,2{}^\circ - 2,584{}^\circ = 32,616{}^\circ$$

The true speed is 550,628 mph, and bearing is $32,616{}^\circ$.