Answer to Question #110611 in Calculus for Chinmoy Kumar Bera

Question #110611
two boat each move a velocity v relative to the water and both cross a river of breadth a running with uniform velocity V.they start together one boat crossing by shortest path and the otherin shortest time.find the difference between the time of arrival.
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Expert's answer
2020-04-23T18:58:28-0400

Shortest path

Relative Velocity of boat is in a direction at an angle θ\theta with the vertical

vsinθ=Vvsin\theta=V

cosθ=v2V2v2cos\theta=\sqrt{\frac{v^2-V^2}{v^2}}

Time=DistancespeedTime=\frac{Distance}{speed}

t1=d/(vcosθ)t_1=d/(vcos\theta)

t1=dv2V2t_1=\frac{d}{\sqrt{v^2-V^2}}

Shortest time

Relative velocity of boat is perpendicular to the velocity of river.

t2=d/vt_2=d/v

t2t1=d[1v1v2V2]t_2-t_1=d[\frac1 v-\frac{1}{\sqrt{v^2-V^2}}]


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