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CBSE Class 12-science Answered

River speed varies as v = t/2 and a man wants to reach directly opposite bank of river .his velocity in still water is 5 and river width is 48. a. Find directikn which msn should swim and time he takes to cross B. Find trajectory Ans is 37 degree and 53 degree and time is 12 ,16 s  
Asked by cudio1212 | 30 Oct, 2020, 11:27: PM
answered-by-expert Expert Answer
Figure shows a river in which water flow speed v = t2 m/s  , where t is time
 
Let swimmer who swims 5 m/s in still water swimming in the direction that makes angle θ with horizontal bank as shown in figure.
 
It is given that swimmer reaches the opposite bank exactly the same horizontal location , then we have
 
begin mathsize 14px style integral subscript 0 superscript tau open parentheses negative space 5 space cos theta space plus space t over 2 close parentheses space d t space equals space minus space 5 space cos theta space tau space plus space 1 fourth tau squared space equals space 0 end style  ........................... (1)
where τ is time to cross the river , τ = 48 / ( 5 sinθ ) = 9.6 / sinθ  ................(2)
 
From eqn.(1) , we get ,  5 cosθ = 2.4 / sinθ  ................(3)
 
from eqn.(2) and (3) , we get,  sinθ cosθ = 0.48     or   sin2θ = 0.96  ..................... (4)
 
Hence  2θ = sin-1 ( 0.96 ) = 74o  or   θ = 37o  
 
River crossing time , τ = 9.6 / sin37  ≈ 16 s
 
We know that , sin (90-α) = sin (90+α) 
 
We get from eqn.(4) ,  2θ = 90-α = 74o    or  α = 16o 
 
Hence other angle (90+α ) that satisfies eqn.(4)  , 2θ  = 90+α = 106o  or  θ = 53o 
 
For this angle θ = 53o , river crossing time , τ = 9.6 / sin53  ≈ 12 s
--------------------------------------------
 
Horizontal displacement x of swimmer in time t ,   
 
begin mathsize 14px style x space equals space integral subscript 0 superscript t left parenthesis negative u cos theta space plus space fraction numerator t apostrophe over denominator 2 end fraction right parenthesis d t apostrophe space equals space minus u space cos theta space t space plus space 1 fourth t squared end style
 
x  = ( - u cosθ ) t + (t2/4) )  ..................(5)
 
where u is swimmer's speed in still water and θ is the direction of angle made by swimmer with horizontal bank.
 
during this time t , vertical displacement y = ( u sinθ ) t   or  t = y / ( u sinθ )..............(6)
 
using eqn.(6) , we rewrite eqn.(5) as ,  x  = { (-u cosθ) [ y / ( u sinθ ) ]  + [ (1/4) y2 ] / ( u2 sin2 θ ) }
 
Above equation can be simplified as,  4 u2 sin2θ x = - ( 2 u2 sin2θ ) y + y2 
 
If we substitute u = 5 m/s  and θ = 37o in above eqn., we get trajectory as ,  y2 - 48 y = 36 x
 
If we substitute u = 5 m/s  and θ = 53o in above eqn., we get trajectory as ,  y2 - 48 y = 64 x
Answered by Thiyagarajan K | 31 Oct, 2020, 04:31: PM
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