BK 16

A man in a rowboat must get from point A on one bank to a point B on the opposite bank of a river. The distance BC = a. The width of the river AC = b. At what minimum speed $ v_b $ relative to the water should the boat travel to reach point B?
The current velocity is $ v_r $

$v_b \, cos \,\theta + v_r) \,t = a \,\,\color {white} {\text (1)}$

$v_b \: sin \,\theta \, t = b \,\,\color {white} {\text (2)}$

${{v_b \, cos \,\theta + v_r} \over {v_b\, sin \,\theta}} = {a \over b}$

$b\;(v_b \, cos \,\theta+v_r)=a\,{v_b \,sin \,\theta}\;\color {white}{\text (3)}$

$ \text For\; minimum\; of \; v_b $ ${{dv_b} \over {d\theta} }= 0$

$b({dv_b\over{d\theta}}\, cos \,\theta\,-\,v_b\,sin\,\theta) = a(v_b\,cos \,\theta + {{dv_b}\over{d\theta}}\, sin \,\theta) \;\;\color {white}{\text (4)}$

$ -b\,v_b \, sin \,\theta = a\,v_b\,cos \,\theta \;\;\;\text or \;\;\; tan \,\theta =-{a \over b} $

$ b({v_b}{{-b} \over \sqrt{a^2+b^2}}+v_r) = {{av_b} {a \over {\sqrt{a^2+b^2}}}} $

${b\,v_r}=v_b{a^2 \over \sqrt{a^2+b^2}} + v_b{b^2 \over \sqrt{a^2+b^2}}$ $= v_b{{a^2+b^2} \over \sqrt{a^2+b^2}}$

$ b\,v_r = v_b{\sqrt{a^2+b^2}} $ $ \;\;\;\text or \;\;\; \color {white} v_b={{b\,v_r} / \sqrt{a^2+b^2}} $

$ (v_b \, cos \, \beta +v_r\, cos \,\alpha)t = \sqrt{a^2+b^2} $

$ v_b \, sin \, \beta - v_r\, sin \,\alpha = 0 $

$ v_b = { {v_r\, sin \,\alpha} \over { sin \,\beta}} $

$v_b \; \text is \;minimum \;when \; \beta = 90° $

$ v_b = {v_r\, sin \,\alpha} $

$v_b = {{b\,v_r} \over \sqrt{a^2+b^2}} $

$$ v_b = v_r\, sin \,\alpha $$

$$ v_b = {{b\,v_r} \over \sqrt{a^2+b^2}}$$