Question

an airplane heads west at 400 miles per hour in a 50 mile per hour northwest wind. Find the ground speed and the bearing of the airplane

Answer 1

Answer on Question#37682-Physics-Mechanics

An airplane heads west at 400 miles per hour in a 50 mile per hour northwest wind. Find the ground speed and the bearing of the airplane.

Solution

The angle between west and northwest directions is $\alpha=45{}^{\circ}$. From hence, according the cosine theorem of triangle the ground speed of airplane is

$$\begin{array}{l} v_{g}^{2} = v_{a}^{2} + v_{w}^{2} - 2 v_{a} v_{w} \cos(180{}^{\circ} - \alpha) \Rightarrow \\ v_{g}^{2} = v_{a}^{2} + v_{w}^{2} + 2 v_{a} v_{w} \cos(\alpha) \Rightarrow \\ v_{g} = 436.9 \frac{km}{h} \\ \end{array}$$

The bearing of the airplane we can find from sine law

$$\begin{array}{l} \frac{v_{w}}{\sin(\theta)} = \frac{v_{g}}{\sin(180{}^{\circ} - \alpha)} \\ \sin \theta = \frac{v_{w} \sin \alpha}{v_{g}} = 0.08 \\ \theta \approx 4.6{}^{\circ} \\ \end{array}$$

Answer:

$$\begin{array}{l} v_{g} = 436.9 \frac{km}{h} \\ \theta \approx 4.6{}^{\circ} \\ \end{array}$$