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Question

A launch travels across a river from a point A to a point B of the opposite bank along the line AB forming angle α with the bank. The flag on the mast of the launch makes an angle β with its a direction of motion. Determine the speed of the launch w.r.t. the bank. The velocity of wind is u perpendicular to the stream.
985172_1beb30ccde1c482e8298f022d325d497.png

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Solution

Flag on a launch flies in the direction of the wind relative to the launch. Therefore, β is the angle made by relative velocity of the wind in relation to the launch.
¯vwind,Earth=¯u=u¯j
¯vLaunch,Earth=¯v=vcosα¯i+vsinα^j
¯VWind,Launch=¯u¯v=vcosα^i+(uvsinα)^j
tan(α+β)=uvsinαvcosα
vtan(α+β)=uvsinαvcosα
v[sinαcosαtan(α+β)]=u
v=ucos(α+β)sinβ=usin(α+βπ/2)sinβ

1047644_985172_ans_1011284eabb9460cab747c43d549c383.png

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