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.

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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+(u−vsinα)^j

tan(α+β)=u−vsinα−vcosα

−vtan(α+β)=u−vsinα−vcosα

v[sinα−−cosαtan(α+β)]=u

v=−ucos(α+β)sinβ=usin(α+β−π/2)sinβ

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