A ship's captain wishes to sail his ship northeast. A current is moving his ship with a velocity of 5.0 km/h(S). If the ship has a maximum speed....
....of 30km/h, what is the ship's required heading? (please include the diagrame).
The ship's captain wishes to sail his ship in the northeast direction. The water current is moving the ship with a velocity of 5 km/h towards the South. The maximum speed with which the ship can move relative to the water is 30 km/h. The ship's heading is required.
The ship has to move in the northeast direction, let the heading of the ship be at an angle X to the North towards the East. The component of the final velocity towards the North and the component towards the East have to be equal. The former is equal to 30*cos X - 5. The component towards the East is 30*sin X
30*cos X - 5 = 30*sin X
=> 6*cos X - 1 = 6*sin X
=> 6(cos X - sin X) = 1
=> 36(cos^2X + sin^2X - sin 2X) = 1
=> 36 - 36*sin 2X = 1
=> 35 = 36*sin 2X
=> sin 2X = 35/36
=> 2X = sin^-1(35/36)
=> X = 38.23 degrees
The heading of the ship should be at an angle 38.23 degrees made with the North towards the East.