A ferry is crossing a river. The ferry is headed due north with a speed of 1.1 m/s relative to the water and the river’s velocity is 4.2 m/s to the east.Find the direction in which the ferry is moving (measured from due east, with coun- terclockwise positive)
To find the direction in which the ferry is moving, measured from due east with counterclockwise positive, we can use vector addition.
We have two vectors: the velocity of the ferry relative to the water (1.1 m/s due north) and the velocity of the river (4.2 m/s due east).
To find the total velocity of the ferry, we add these two vectors.
Given that the motion is happening in two perpendicular directions, we can use the Pythagorean theorem to find the magnitude of the total velocity:
Total velocity^2 = (velocity of ferry relative to water)^2 + (velocity of river)^2
Total velocity^2 = (1.1 m/s)^2 + (4.2 m/s)^2
Total velocity^2 = 1.21 m^2/s^2 + 17.64 m^2/s^2
Total velocity^2 = 18.85 m^2/s^2
Taking the square root, we find the total velocity:
Total velocity = √(18.85 m^2/s^2)
Total velocity ≈ 4.34 m/s
Now, to find the direction, we can use trigonometry. The angle can be found using the tangent function:
tan(angle) = (velocity of river) / (velocity of ferry relative to water)
tan(angle) = 4.2 m/s / 1.1 m/s
tan(angle) ≈ 3.82
Taking the inverse tangent, we find the angle:
angle ≈ tan^(-1)(3.82)
angle ≈ 76.4°
Since the angle is measured counterclockwise from due east, the direction in which the ferry is moving is approximately 76.4° counterclockwise from due east.