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A man can swim at a speed of 3 km/h in s...

A man can swim at a speed of 3 km/h in still water. He wants t cross a 500 m wide river flowing at 2 km/h. He flow while swimming. A. Find the time he takes to cross the river. b.At what point on the opposite bank will he arrive?

Text Solution

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The situation is shown in figure.
Here `underset(v_(r.g))rarr` velocity of the river with respect to the ground
`underset(v_(m.r))rarr=` velocity of the man with respect to the river

`underset(v_(m.g))rarr=` velocity of the man with respect to the ground
(a) We have, `underset(v_(m.g))rarr = underset(v_(m.r))rarr + underset(v_(r.g))rarr ....(i)`
Hence, the velocity with respect to the ground is along `AC`, Taking `y`-components in equation (i),
`underset(v_(m.g))rarr sin theta = 3km//h cos 30^(@) +2km//h cos 90^(@)`
`= (3sqrt(3))/(2) km//h`
Time taken to cross the river
`= ("displacement along the Y-axis")/("velocity along the Y-axis") = (1//2km)/(3sqrt(3)2km//h) = (1)/(3sqrt(3)) h`.
(b) Taking x-components in equation (i),
`underset(v_(mg))rarr cos theta =-3 km//h sin 30^(@) +2km//h =(1)/(2)km//h`
Displacement along the X-axis as the man crosses the river `= ("velocity along the X-axis") ("time")`
`= ((1km)/(2h)) xx ((1)/(3sqrt(3))h) = (1)/(6sqrt(3))km`.
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