Question

Show that a boat must move at an angle with respect to river water in order to cross the river in minimum time?

Answer 1

If the boat has to move along the line AB. The resultant velocity of $V_{BE}$ should be directed along boat reaches the point B directly for this to happen , the boat velocity with respect to water $V_{BW}$ should be directed such that it makes an angle $\alpha$ with the line AB upstream as shown in figure.

$\frac{V_{WE}}{V_{BW}}=sin\alpha$

$\implies\alpha=sin^{-1}\left(\frac{V_{WE}}{V_{BW}}\right)$

And $V_{BE}=\sqrt{V_{BW}^2-V_{WE}^2}$
The minimum time taken by the the boat to cross the river , t = $\frac{AB}{V_{BE}}$

or, t = $\frac{AB}{\sqrt{V_{BW}^2-V_{WE}^2}}$

Answer 2

For any arbitrary motion in space, which of the following relations are true: a. $V_{average}= (\frac{1}{2})[v(t_1)+v(t_2)]$ b. $V_{average}=\frac {[r(t_2)-r(t_1)]} {(t_2-t_1)}$ c. v (t) = v (0) + at d. r (t) = r (0) + v (0) t + (1/2) at² e. $a_{average}=\frac {[v(t_2)-v(t_1)]} {(t_2-t_1)}$ (The 'average' stands for average of the quantity over the time interval $t_1 to t_2$)