91Ó°ÊÓ

The relative velocity $\left({\stackrel{→}{v}}_{dg}\right)$of the flock of the ducks relative to the ground is equal to vector sum of the velocity $\left({\stackrel{→}{v}}_{dv}\right)$of the flock of the ducks relative to the wind and velocity $\left({\stackrel{→}{v}}_{\text{wg}}\right)$of the wind relative to the ground.

The figure given below represents the direction air blow, the direction of flock of the ducks:

In the figure, $\mathrm{θ}$is the angle made by the direction heading of the flock of the ducks with south direction.

From the figure, the trigonometry ratio $\sin \mathrm{θ}$can be written as follows:

$\sin \mathrm{θ}=\frac{v_{wg}}{v_{dw}}$

Here, ${\stackrel{→}{v}}_{dw}$is the velocity of the flock of the ducks relative to the wind and ${\stackrel{→}{v}}_{ug}$is the velocity of the wind relative to the ground.

Rearrange the equation for $\mathrm{θ}$.

$\mathrm{θ}={\sin }^{−1}\left(\frac{v_{ug}}{v_{dw}}\right)$

Substitute $6.0\text{m}/\text{s}$for $v_{\text{vg}}$and $8.0\text{m}/\text{s}$for $v_{dw}$.

$\mathrm{θ}={\sin }^{−1}\left(\frac{6.0\text{m}/\text{s}}{8.0\text{m}/\text{s}}\right)≃{49}^{°}$

Therefore, the flock of the ducks must fly at an angle ${49}^{°}$west of south to go directly south.