A bird is flying towards north with a 40 ${\mathrm{kmh}}^{-1}$ and a train is moving with velocity 40 ${\mathrm{kmh}}^{-1}$ towards east. What is the velocity of the bird noted by a man in the train?

To find the relative velocity of bird w.r.t. train, superimpose velocity $-{\overrightarrow{v}}_T$ on both the objects. Now as a result of it, the train is at rest, while the bird possesses two velocities, ${\overrightarrow{v}}_B$ towards north and $-{\overrightarrow{v}}_T$ along west.

$\begin{array}{l}\left|{\overrightarrow{v}}_{BT}\right|=\sqrt{{\left|{\overrightarrow{v}}_B\right|}^2+{\left|-{\overrightarrow{v}}_T\right|}^2}\left[\text{ By formula, }\theta ={90}^{\circ }\right]\\ =\sqrt{{40}^2+{40}^2}=40\sqrt{2}{\mathrm{kmh}}^{-1}\text{ north-west }\end{array}$