Question

An aeroplane flies horizontally at 80m/s in still air. If the aviator wishes to fly due south and the wind is blowing from the south-east at 30m/s.
1. What course must he steer.
2. How long will it take him to arrive at his destination 200km away.

Answer 1

$\overline{v}_{r}$ − resulting velocity

$\overline{v}_{a}$ − aeroplane velocity

$\overline{v}_{w}$ − wind velocity

$v_{rx}$, $v_{ax}$, $v_{wx}$ − x-components of the $\overline{v}_{r}$, $\overline{v}_{a}$ and $\overline{v}_{w}$, respectively

$v_{ry}$, $v_{ay}$, $v_{wy}$ − y-components of the $\overline{v}_{r}$, $\overline{v}_{a}$ and $\overline{v}_{w}$, respectively

$\lvert\overline{v}_{r}\rvert, \lvert\overline{v}_{a}\rvert, \lvert\overline{v}_{w}\rvert$ − absolute values of resulting velocity, aeroplane velocity and wind velocity, respectively

1.

$\overline{v}_{a}+\overline{v}_{w}=\overline{v}_{r}$

$v_{ax}-v_{wx}=0$

$\lvert\overline{v}_{a}\rvert\cdot\sin\beta-\lvert\overline{v}_{w}\rvert\cdot\sin\alpha=0$

$80\cdot\sin\beta-30\cdot\frac{\sqrt{2}}{2}=0$

$sin\beta\approx0.265$

$\beta\approx15.4^\circ$

2.

$S=v\cdot t=\lvert\overline{v}_{r}\rvert\cdot t$

$t=\frac{S}{\lvert\overline{v}_{r}\rvert}$

$\lvert\overline{v}_{r}\rvert=\sqrt{v_{rx}^2+v_{ry}^2}=\sqrt{0+v_{ry}^2}=v_{ry}$

$v_{ay}-v_{wy}=v_{ry}$

$\lvert\overline{v}_{a}\rvert\cdot\cos\beta-\lvert\overline{v}_{w}\rvert\cdot\cos\alpha=v_{ry}$

$\lvert\overline{v}_{r}\rvert=v_{ry}=80\cdot0.964-30\cdot\frac{\sqrt{2}}{2}\approx55.9\;m/s$

$v[km/h]=\frac{v[m/s]\cdot1000}{3600}$

$\lvert\overline{v}_{r}\rvert=\frac{55.9\cdot1000}{3600}\approx15.5\;km/h$

$t=\frac{200}{15.5}\approx12.9\;h$