Final answer:
The boat should head 24.6° south of west at a speed of 3.0 m/s in order to cross the river directly. The velocity of the boat relative to the shore will be 3.4 m/s in a direction 24.6° south of west.
Step-by-step explanation:
When a boat is being affected by a river current, it needs to head in a direction that compensates for the current in order to cross directly. In this case, since the current is flowing towards the north, the boat needs to head slightly south of west. To determine the angle at which the boat should head, we can use the trigonometric equation:
tan(theta) = velocity of the current / velocity of the boat
Theta is the angle between the desired direction (west) and the actual direction the boat needs to travel. Plugging in the values, we get:
tan(theta) = 1.4 / 3.0
Solving for theta gives theta ≈ 24.6° south of west.
ii) The velocity of the boat relative to the shore can be determined by using vector addition. The magnitude of the boat's velocity relative to the shore is the square root of the sum of the squares of the boat's velocity in the x-direction (west) and the boat's velocity in the y-direction (north). The direction of the boat's velocity relative to the shore can be determined using the inverse tangent function:
Velocity relative to the shore = sqrt(velocity of the boat in the x-direction^2 + velocity of the boat in the y-direction^2)
Velocity relative to the shore = sqrt(3.0^2 + (-1.4)^2) ≈ 3.4 m/s
The direction of the boat's velocity relative to the shore can be found using the inverse tangent function:
Direction = tan^-1(-1.4 / 3.0) ≈ -24.6° south of west