A river that is flowing due west is crossed by a boat travelling from the south bank to the north bank. The boat can travel at 30 m/s in still water and the river is moving at 10 m/s.
Determine the direction the boat must point to travel directly across. Express the direction using either the Navigator Method or the Polar Coordinates Method.
Using the Navigator Method, we can break down the boat's velocity into horizontal and vertical components. Let's denote the angle between the boat's velocity and the river's flow as θ.
The boat's velocity in the direction perpendicular to the river's flow is 30 m/s since it is moving directly across. The river's flow speed of 10 m/s also creates a velocity component in the direction perpendicular to the river's flow.
Let's denote the velocity of the boat in the direction perpendicular to the river's flow as Vb and the river's flow velocity as Vr. The boat's velocity perpendicular to the river's flow can be calculated using the Pythagorean theorem:
Vb^2 = 30^2 + 10^2
Vb = √(900 + 100)
Vb = √1000
Vb = 31.62 m/s
Now we can find the angle θ using trigonometry:
sinθ = Vr / Vb
sinθ = 10 / 31.62
θ = sin^(-1)(10 / 31.62)
θ ≈ 18.95 degrees
Therefore, to travel directly across the river, the boat must point in a direction approximately 18.95 degrees north of west.