Question

an airplane has a speed of 600km/hr with respect to air.It has to fly a distance of 800km northward.In what direction should it fly if a steady wind is blowing at a speed of 120km/hr from the west?what will its speed with respect to the ground be?Draw the relevant diagram for your calculations.

Answer 1

Let vectors $\vec{w}$ , $\vec{v}$ and $\vec{u}$ represent velocities of wind, airplane with respect to air and airplane with respect to the ground respectively. Airplane's velocity is the resultant of sum of vectors $\vec{v}$ and $\vec{w}$ . So, we have $\vec{u} = \vec{v} + \vec{w}$ . As the wind is blowing from the west and airplane needs to fly northward, we get a right triangle with hypotenuse $\vec{v}$ (see the diagram below).

Using the Pythagorean theorem, we get the equation for the speed of airplane with respect to the ground:

$$| \vec {u} | = \sqrt {| \vec {v} | ^ {2} - | \vec {w} | ^ {2}} = \sqrt {6 0 0 ^ {2} - 1 2 0 ^ {2}} = \sqrt {3 4 5 6 0 0} \approx 5 8 8 k m / h$$

The direction is determined by angle $\alpha$ . It can be found from:

$$\cos \alpha = \frac {| \vec {w} |}{| \vec {v} |} = \frac {1 2 0}{6 0 0} = 0. 2.$$ $$\alpha = \mathrm {a c o s} 0. 2 \approx 7 8. 5 {}^ {\circ}.$$

So, the plane must fly 78.5 degrees from west to north.