Application of vectors in problem solving

Question

An aircraft cruises at a speed 300 km/h in still air. If the wind is blowing from the east at 100 km/h, in what direction should the aircraft head in order to fly in a straight line from city P to city Q, 400 km north-northeast of P? How long will the trip take?

The magnitude of the green vector is what I wan't to figure out, and the angle v created by the red vector (v is theta).

How do I solve this?

Answer 1

First, draw an appropriate picture that represents the situation at hand.

Then solve the following equations for $\theta$ and ground speed $S$. Let $\alpha$ be the angle of $22{1\over 2}^\circ$.

$\sin \theta - S {\sin \alpha \over 300} = {100 \over 300}$, $\cos \theta - S {\cos \alpha \over 300} = 0$.

Explicitly:

Multiply the first by $\cos \alpha$ and the second by $- \sin \alpha$ and add the equations to get $\sin ( \theta-\alpha) = {100 \over 300} \cos \alpha$.

Computing numbers:

Substituting numbers gives $\theta \approx 40.4^\circ$ and $S \approx 247$ km/hr.

Given the ground speed the time taken can be easily computed.