Question

A boat moves relative to water with a velocity which is n =2.0 times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?

Answer 1

Let ${\nu }_0$ be the stream velocity and $u^{{}^{\prime }}$ the velocity of boat respect to water. A $\frac{{\nu }_0}{{\nu }^{{}^{\prime }}}=\eta =2>0$, Some drifting of car is invertible.

L:et $\overrightarrow{\nu }$ make an angle $\theta$ with flow direction (Fig) then the time taken to cross to river

$t=\frac{d}{{\nu }^{{}^{\prime }}\sin \theta }$ (where $d$ is the width of the river)

In this time interval, the drifting of the boat

$x=({\nu }^{{}^{\prime }}\cos \theta +{\nu }_0)t$

$=({\nu }^{{}^{\prime }}\cos \theta +{\nu }_0)\frac{d}{{\nu }^{{}^{\prime }}\sin \theta }=(\cot \theta +\eta \csc \theta )d$

For $X_{min}$ (minimum drifting)

$\frac{d}{d\theta }(\cot \theta +\eta \csc \theta )=0$ which yields

$\cos \theta =-\frac{1}{\eta }=-\frac{1}{2}$

Hence, $\theta ={120}^o$