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A man wishes to row across a river flowing to the right with speed of 2m//s . If the velocity that the boat can have V_(B)=4 m//s how should the man row so as to reach across in (a) shortest path (b) shortest time |
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Answer» Solution :(a) Shortest path If the MAN has to MOVE along the shotest path , then the path of his boat should be AB (perpendicular to river) If the man rows vertically, the river will carry him to right, hence the man should incline his boat to the left . When his velocity is inclined it has two components `V_(x) "and" V_(y)`[compare with inclined velocity of cat] The horizontal component `V_(x)` should cancel the speed of the river. Thus, `V_(x)=V_(b)cos theta =V_(R)` `:. 4 cos theta =2 , theta 60^(@)` Alternatively , we can use resultant velocity that should lie along th evertical as in cat mouse problem (Method3)  . (b) Shortest time Nothe that when the man rows along shortest distance, he DOES NOT cross the river in shortest time, because time taken to cross the river =`("width of river" )/("vertical velocity" )`  As long as the boat is inclined at angle `theta` , vertical velocity is `V_(B) sin theta ` which is less than `V_(B)`. To row in shortest time , the entire velocity of boat should be USED in vertical direction i.e. Man should row perpendicular to stream . When the man rows in shortest time, he does not reach point B but instead point B . shortest distance is ANALOGOUS to reaching a point on opposite bank (say house) whereas shortest time is linkedwith only reaching opposite bank. 
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