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Question
A swimmer wishes to cross a river 500 m wide flowing at a rate u. His speed with respect to still water is V. For this, he makes an angle θ with the vertical to the direction of flow of the river as shown. In order to cross the river in minimum time, the value θ of in degrees should be
A swimmer wishes to cross a river 500 m wide flowing at a rate u. His speed with respect to still water is V. For this, he makes an angle θ with the vertical to the direction of flow of the river as shown. In order to cross the river in minimum time, the value θ of in degrees should be
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Solution
The correct option is A 0∘
The swimmer swims with a velocity V making an angle of θ with the vertical. Hence, the Y component of the velocity =Vy=V cosθ
The distance along the Y direction is the width of the river =d=500 m
The time taken to cross the river =t=dVy=500V cosθ
When θ=0∘,cosθ is maximum, hence t is minimum. As θ increases, the value of cosθ decreases, which in turn increases the value of t. Hence, for time taken t to be minimum, θ should be equal to 0∘.
Hence the correct option is (a)
The swimmer swims with a velocity V making an angle of θ with the vertical. Hence, the Y component of the velocity =Vy=V cosθ
The distance along the Y direction is the width of the river =d=500 m
The time taken to cross the river =t=dVy=500V cosθ
When θ=0∘,cosθ is maximum, hence t is minimum. As θ increases, the value of cosθ decreases, which in turn increases the value of t. Hence, for time taken t to be minimum, θ should be equal to 0∘.
Hence the correct option is (a)
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