An aeroplane has to go from a point A to another point B, 500 km away due 30° east of north. A wind is blowing due north at a speed of 20 m/s. The air-speed of the plane is 150 m/s. Find the time taken by the plane to go from A to B.

#### Solution

Given:

Distance between points A and B = 500 km

B from A is 30˚ east of north.

Speed of wind due north, v_{w} = 20 m/s

Airspeed of the plane, v_{a} = 150 m/s

Let \[\vec{R}\] be the resultant direction of the plane to reach point B.

Time taken by the plane to reach point B from point A: \[\sin^{- 1} \left( \frac{1}{15} \right) = 3^\circ 48'\]

⇒ 30° + 3°48' = 33°48

\[R = \sqrt{A^2 + B^2 + 2AB\cos\theta}\]

\[R = \sqrt{150 + 20 + 2\left( 150 \right)\left( 20 \right) \cos\left( 33^\circ48' \right)}\]

\[ = \sqrt{27886} = 167 \text{ m/s } \]

\[\text{ Time } = \frac{s}{v} = \frac{500000}{167}\]

\[ = 2994 s = 49 . 0 \approx 50 \text{ minutes } \]