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Question

A man wants to swim in a river from A to B and back from B to A always following line AB. The distance between points A and B is S. The velocity of the river current v is constant over the entire width of the river. A The line AB makes an angle α with the direction of current. With what velocity u and at what angle β to the line AB should the man swim to cover distance AB and back in time t?
983388_3d521f3554474127a1d38892700a6ad9.png

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Solution

In this problem, we choose axis along AB and normal to it.
Since the man moves along AB, the velocity components of current and man must cancel out, i.e.,
vsinβ=vsinα
When the man moves from A to B, his resultant velocity along
AB=(ucosβ+vcosα)
Hence, S=(ucosβ+vcosα)t1
while for motion from B to AS=(ucosβvcosα)t2
From the condition of the problem, t1+t2=t
Sucosβ+vcosα+Sucosβvcosα=t
S[ucosβvcosα+ucosβ+vcosαu2cos2β+v2cos2α]=t
S(2ucosβ)u2cos2β+v2cos2α=t
1027087_983388_ans_f3770e9155a14e90a82c37168d038dba.png

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