Final answer:
To moor as far upstream as possible, the captain needs to compensate for the river flow by setting the boat at a specific angle θ. The angle is calculated using vector addition and trigonometry considering the boat's velocity and the river's velocity profile.
Step-by-step explanation:
The problem involves a river with a velocity profile dependent on position and a ship that aims to moor upstream. The captain needs to compensate for the river's flow direction and speed as well as the limitation of the ship's velocity. To maximize the upstream distance on the opposite bank, the captain should set an angle θ relative to the x-direction, considering the ship's power output remains constant.
Without a specific calculation, the direction should be such that the resultant velocity of the vessel is oriented as directly across the stream as possible. The strategy involves choosing a coordinate system with the x-axis parallel to the river flow, and then the angle can be found by using the resultant velocity vector, combining the ship's velocity and the river's velocity vector.
If the speed of the river were equal to the velocity of the boat, the angle would be 45°. However, since the river's velocity profile suggests that it can potentially be greater than the boat's velocity, the resultant velocity will have an angle of less than 45° to the x-axis. Essentially, the velocity vector of the boat and river will be added, considering components based on the coordinate system, using vector addition and trigonometry to find the direction and magnitude of the total velocity.