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Question

A man wants to reach point B on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the man have so that he can reach point B?

A
2u
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B
u2
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C
3u
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D
u3
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Solution

The correct option is B u2
Given,
Speed of river =u
Assuming,
Speed of man swimming =v in direction 'θ' anticlock with line AC
On breaking speed of man v into x and y component
vx=vsinθ
vy=vcosθ
Let us suppose vn is the net speed of the man, we know that to reach point B from A, vn should be along AB.
Also, x component of vn, vnx=uvsinθ
y component of vn, vny=vcosθ
In ΔACB,
tan45=vnxvny
1=uvsinθvcosθ
vcosθ=uvsinθ
vcosθ+vsinθ=u
v(cosθ+sinθ)=u
v=ucosθ+sinθ...(1)

Here, to get minimum value of v, value of denominator should be maximum.
Assuming, t=cosθ+sinθ
dtdθ=sinθ+cosθ [Applying concept of maxima and minima]

For maxima or minima,
dtdθ=0sinθ+cosθ=0
cosθ=sinθ
1=tanθ
θ=45

Now, d2tdθ2=cosθsinθ
=(cosθ+sinθ)
=(cos45+sin45)=2
= Negative value
d2tdθ2<0
t has maximum value at θ=45
On putting θ=45 in equation (1)
vmin=ucos45+sin45=u2
vmin=u2

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