NEET Class neet Answered
a boat having a speed of 5km/hr . in still water crossing a river of width 1km long the shortest possible paths in 15 min the speed of the river in km/ hr
Asked by kishorantarkar26 | 05 Sep, 2023, 02:58: AM
![](https://images.topperlearning.com/topper/tinymce/imagemanager/files/de33a172285124f92f25860a973c133664f6da8a1c6833.43263065g1.png)
Vector diagram showing velocity of boat and velocity of river are shown above.
u is velocity of boat so that direction of velocit of boat u makes angle θ with the line parallel to width of river.
ur is velocity of river.
magnitude of velocity of boat , u = 5 km / hr = 5 × ( 5 / 18) m/s = ( 25/18 ) m/s
velocity component along the line parallel to width of river = u cosθ .
If boat travels along shortest path to cross the river and time taken to cross the river is 15 min ,
then we have
![begin mathsize 14px style u space cos theta space cross times space 15 space cross times space 60 space equals space 1000 space m end style](https://images.topperlearning.com/topper/tinymce/cache/840ab17458bec5cd3db9fd00d777a7af.png)
If we substitute u = ( 25/18 ) m/s , then we get
![begin mathsize 14px style 25 over 18 cross times space cos theta cross times 900 space equals space 1000 space m end style](https://images.topperlearning.com/topper/tinymce/cache/2561e6bab2aa3c9fc7fa1bbe22678c17.png)
From above expression we get , cosθ = 0.8
Hence , sinθ = ![begin mathsize 14px style square root of 1 space minus space left parenthesis 0.8 right parenthesis squared end root space equals space 0.6 end style](https://images.topperlearning.com/topper/tinymce/cache/bb51a2cbca32686350153f186ea26000.png)
![begin mathsize 14px style square root of 1 space minus space left parenthesis 0.8 right parenthesis squared end root space equals space 0.6 end style](https://images.topperlearning.com/topper/tinymce/cache/bb51a2cbca32686350153f186ea26000.png)
From vector diagram, we see that
![begin mathsize 14px style u subscript r subscript space end subscript space equals space u space sin theta space equals space 25 over 18 cross times 0.6 space m divided by s space equals space 5 over 6 space m divided by s end style](https://images.topperlearning.com/topper/tinymce/cache/d81d3c2557d7d6ff90faa6a0cf75e4c4.png)
Hence, speed of river = (5/6) × (18/5) km/hr = 3 km/hr
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