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two boats A and B move away from a buoy anchored at the middle of a river along mutually perpendicular straight lines. The boat A moves along the stream and the boat B across the river. After moving off an equal distance from the buot, both the boats returned to their original position. Calculate the ratio of the times taken by boat A to that taken by boat B if the velocity of each boat with respect to still water is 2 times the stream velocity

two boats A and B move away from a buoy anchored at the middle of a river along mutually perpendicular straight lines. The boat A moves along the stream and the boat B across the river. After moving off an equal distance from the buot, both the boats returned to their original position. Calculate the ratio of the times taken by boat A to that taken by boat B if the velocity of each boat with respect to still water is 2 times the stream velocity

Grade:11

1 Answers

Aman Bansal
592 Points
9 years ago

Dear Aditi,

et u be the stream velocity, then velocity of each boat(with respect to water) is v = 1.2 u.

Moving away from buoy : If A moves away a distance d in time {t_A} and B in {t_B}, then

{T_B} = \frac{d}{{{v_B}}} = \frac{d}{{v\cos \theta }} = \frac{d}{{v.\frac{{\sqrt {{v^2} - {u^2}} }}{v}}}

 

\;\,\, = \frac{d}{{\sqrt {{v^2} - {u^2}} }}\left[ {{\bf{since}}{\rm{ }}\;v\sin \theta  = u\,\;\;\;{\rm{i}}{\rm{.e}}{\rm{.}}\;\;\sin \theta  = u{\rm{/}}v} \right]
and\;\;\;\;\;{t_A} = \frac{d}{{v + u}}

Coming back to buoy: If A takes a time of {t_A}' seconds and B takes {t_B}' seconds in coming back to buoy, then+

t_B^' = \frac{d}{{\sqrt {{v^2} - {u^2}} }}
and\;\;\;\;\;t_A^' = \frac{d}{{v - u}}

Therefore, total time of motion of B is

Hence\;\;\;\;\;\frac{{{T_A}}}{{{T_B}}} = \frac{{1.2}}{{\sqrt {{{(1.2)}^2} - 1} }} = 1.8

Plz Approve the answer...!!!!

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Thanks

Aman Bansal

Askiitian Expert

 

{T_B} = {t_B} + {t_B}' = 2{t_B} = \frac{{2d}}{{\sqrt {{v^2} - {u^2}} }}

and that for A is

{T_A} = {t_A} + {t_A}' = \frac{d}{{v + u}} + \frac{d}{{v - u}}
\;\,\, = \frac{{2vd}}{{{v^2} - {u^2}}}

 

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