Question

How should a man cross the river in order that he takes

(i) shortest time and,

(ii) shortest path?

Answer 1

\(V_\vec r\)= Absolute velocity of river

\(V_\vec b\)= Absolute velocity of boat or w.r.t. ground

\(V_{b\vec r}\)= velocity of boat with respect to river or in still water

Time to reach,

speed = \(\frac{distance}{time}\)

\(V_{br} cos\theta = \frac{d}{t}\)

t' = \(\frac{d}{V_{br}\,cos\,\theta}\)

displacement in x-axis = speed × time

= \((V_r-V_{br}sin\, \theta)t\)

= \(\frac{(V_r-V_br\,sin\,\theta)d}{V_{br}\,cos\theta}\)

(i) Shortest time :-

For t_min cos θ must be maximum

So cos θ = 1

Or θ = 0

\(t_{min}=\frac{d}{V_{br}}\)

(ii) Shortest path i.e., drift = 0

\(\frac{(V_r-V_br\,sin\,\theta)d}{V_{br}\,cos\,\theta}\)  ∵ sin θ = ( \(\frac{V_r}{V_{br}}\) )

When we want to cross the river via shortest path, we should cross the river at an angle of

 θ = \(\sin^{-2}(\frac{V_r}{V_{br}})\)