The captain of a boat wants to travel directly across a river that flows due east with a speed of 1.09 m/s. He starts from the south bank of the river and wants to reach the north bank by travelling straight across the river. The boat has a speed of 6.44 m/s with respect to the water. What direction (in degrees) should the captain steer the boat? Note that 90° is east, 180° is south, 270° is west, and 360° is north.
To find the direction the captain should steer the boat, we can use the concept of vector addition. The velocity of the boat can be broken down into its components: the horizontal velocity (Vx, parallel to the river) and the vertical velocity (Vy, perpendicular to the river).
Given:
Speed of the river (Vr) = 1.09 m/s (east)
Speed of the boat (Vb) = 6.44 m/s (with respect to the water)
Now, let's find the vector components of the boat's velocity relative to the water:
Vx = Vb * cosθ
Vy = Vb * sinθ
We need to find the angle (θ) at which the boat should steer. Considering that the boat wants to move directly across the river, the vertical component of the boat's velocity should be equal to the speed of the river.
Vy = Vr
Substituting the values:
Vb * sinθ = 1.09
6.44 * sinθ = 1.09
Now, solve for θ:
sinθ = 1.09 / 6.44
sinθ ≈ 0.1695
θ ≈ arcsin(0.1695)
θ ≈ 9.78° (rounded to two decimal places)
So, the captain should steer the boat at approximately 9.78° with respect to east (90°) to travel directly across the river.
To determine the direction the captain should steer the boat, we need to consider the relative velocities of the boat and the river.
Let's break down the velocities involved:
1. The velocity of the river: The river flows due east with a speed of 1.09 m/s. Since the captain is starting from the south bank, the direction of the river's velocity is perpendicular to the boat's path.
2. The velocity of the boat: The boat has a speed of 6.44 m/s with respect to the water. The captain wants to travel directly across the river, so the boat's velocity will have two components: one perpendicular to the river's flow and one parallel to the river's flow.
Now, let's find the direction the captain should steer the boat:
1. Find the angle between the boat's velocity and the river's velocity: Since the boat's velocity has two components (perpendicular and parallel to the river), we want to find the angle at which both velocities align. This angle can be determined using the arctangent function with the ratio of the perpendicular component to the parallel component.
tan(θ) = perpendicular component / parallel component
tan(θ) = 1.09 m/s / 6.44 m/s
θ = arctan(1.09 m/s / 6.44 m/s)
2. Convert the angle to degrees: Use the arctan function to calculate the angle between the boat's velocity and the river's velocity, and then convert it from radians to degrees.
θ (in degrees) = arctan(1.09 m/s / 6.44 m/s) * (180° / π)
Therefore, the captain should steer the boat at the angle (in degrees) calculated above.
Vbw = Vb + Vw = 6.44i
Vb + 1.09 = 6.44i
Vb = -1.09 + 6.44i, Q2.
Tan A = Vbw/Vw = 6.44//-1.09 = -5.90826
A = -80.4o = 80.4o N. of W. = 99.6o CCW = 9.6o W. of N. = The direction.