Relative Motion:

$$ \vec{v}_{AB} \text{(velocity of A with resprect to B)} = \vec{v}_A - \vec{v}_B $$

$$ \vec{a}_{AB} \text{(acceleration of A with respect to B)} = \vec{a}_A - \vec{a}_B$$

Relative Motion Along Straight Line:

$$ \vec{x_{BA}} = \vec{x_{B}} -\vec{x_{A}} $$

Crossing River:

A boat or man in a river always moves in the direction of resultant velocity of velocity of boat (or man) and velocity of river flow.

(i) Shortest Time :

Velocity along the river, $$v_{x}=v_{R}$$.

Velocity perpendicular to the river, $$v_{f}=v_{m R}$$

The net speed is given by $$v_{m}=\sqrt{v_{m R}^{2}+v_{R}^{2}}$$

(ii) Shortest Path :

velocity along the river, $$\mathrm{v}_{\mathrm{x}}=0$$

and velocity perpendicular to river $$v_{y}=\sqrt{v_{m R}^{2}-v_{R}^{2}}$$

The net speed is given by $$v_{m}=\sqrt{v_{m R}^{2}-v_{R}^{2}}$$

at an angle of $90^{\circ}$ with the river direction.

velocity $v_{y}$ is used only to cross the river, therefore time to cross the river, $$t=\frac{d}{v_{y}}=\frac{d}{\sqrt{v_{m R}^{2}-v_{R}^{2}}}$$

and velocity $\mathrm{v}_{\mathrm{x}}$ is zero, therefore, in this case the drift should be zero.

$$\Rightarrow \quad v_{R}-v_{m R} \sin \theta=0 \quad \text{or} \quad v_{R}=v_{m R} \sin \theta$$

or $$\theta=\sin ^{-1}\left(\frac{v_{R}}{v_{m R}}\right)$$

Rain Problems:

$$ \vec{v}_{Rm} = \vec{v}_R - \vec{v}_m $$

$$ or \quad v_{R m}=\sqrt{v_{R^2} + v_{m^2}}$$