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A man wishes to cross a river of width 1...

A man wishes to cross a river of width 120 m by a motorboat. His rowing speed in still water is `3 m//s` and his maximum walking speed is `1m//s`. The river flows with velocity of `4 m//s`.
(a) Find the path which he should take to get to the point directly opposite to his starting point in the shortest time.
(b) Also, find the time which he takes to reach his destination.

Text Solution

Verified by Experts

First you should check whether the simmer can reach the point b directly opposite to his staring point A along the line AB or not.
The velocity v of swinmmer relative to water is 3m/s which is less than the velocity of river flow ( u= 4 m/s) . In this condition , it is not possible to have any angle `theta` with the AB for which the resultant of ` vecv and vecu` is along AB. As u gt v, the swimmer will reach a point C somewhere to the right of point B for any angle `theta` he choose to start. After reaching that point C, he has to walk along the bank from C to B with walking speed ` v_(0)` ( = 1 m/s) . Let him take a time ` t_(1)` to reach the point C and then a time ` t_(2)` to walk to B from C. Now the angle `theta` he has to choose in such a way that the total time takent `(t_(1) + t_(2))_(0)` to reach B is minmum.
(a) ` t_(1) = d/v_(y) = d/( v cos theta) and tan alpha = ( u - v sin theta)/(v cos theta)`
` t_(2) = BC/v_(0) = (d tan alpha)/v_(0)`
Total time taken ` t= t_(1) + t_(2)`
` Rightarrow t = d/( v cos theta) + d/v_(0) ( ( u - v sin theta)/(v cos theta))`
For minimum value ` t , (dt)/(dtheta)=0 `


` Rightarrow (dt)(dtheta) = d/v sec theta tan theta + ( ud)/( v_(0)v) sec theta tan theta = d/(v_(0)) sec^(2)0=0`
` Rightarrow tan theta ( 1 + u /v_(0)) = v/v_(0) sec theta , or sin theta = v/( u + v_(0))`
` Rightarrow theta = sin^(-1) ( u/(u +v_(0))) = sin^(-1) (3/(4 +1)) = sin^(-1) (3/5)`
` Rightarrow ` the swimmer should start a angle ` , theta = sin(-1) (3/5) ` with AB.
(b) The total time taken is
` t = d/( v cos theta) + d/v_(0) ( (u-vsintheta)/(v cos theta)) = 120/(3 (4/5)) + 120/1 ((4 - 3xx (3/5))/(3xx (4/5))) = 160 s`
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