A ferry is crossing a river. The ferry is headed
due north with a speed of 1.2 m/s relative to
the water and the river’s velocity is 3.5 m/s
to the east.
Find the direction in which the ferry is
moving (measured from due east, with counterclockwise positive).
Answer in units of ◦
tan(Θ) = 1.2 / 3.5
direction is Θº north of east
To find the direction in which the ferry is moving, we can use vector addition.
The velocity of the ferry relative to the ground is the vector sum of its velocity relative to the water and the velocity of the river.
Let's call the velocity of the ferry relative to the ground Vf, the velocity of the ferry relative to the water Vw, and the velocity of the river Vr.
Given:
Vw = 1.2 m/s (due north)
Vr = 3.5 m/s (to the east)
To find Vf, we can use the Pythagorean theorem:
|Vf|^2 = |Vw|^2 + |Vr|^2
|Vf|^2 = (1.2 m/s)^2 + (3.5 m/s)^2
|Vf|^2 = 1.44 m^2/s^2 + 12.25 m^2/s^2
|Vf|^2 = 13.69 m^2/s^2
|Vf| = √(13.69 m^2/s^2)
|Vf| ≈ 3.7 m/s
So, the speed of the ferry relative to the ground is approximately 3.7 m/s.
To find the direction of Vf (measured from due east, with counterclockwise positive), we can use trigonometry.
Let θ be the angle between Vf and the east direction.
tan(θ) = |Vr| / |Vw|
tan(θ) = 3.5 m/s / 1.2 m/s
tan(θ) ≈ 2.92
θ ≈ arctan(2.92)
θ ≈ 71.2°
Therefore, the direction in which the ferry is moving, measured from due east with counterclockwise positive, is approximately 71.2°.
To find the direction in which the ferry is moving, we can use vector addition. The ferry's velocity relative to the Earth is the vector sum of its velocity relative to the water and the velocity of the river.
Given:
Ferry's velocity relative to the water (north direction): 1.2 m/s
Velocity of the river (east direction): 3.5 m/s
To find the direction angle, we can use the inverse tangent function:
θ = arctan(Vferry_relative_to_water / Vriver)
First, let's find the magnitudes of the velocity vectors:
Vferry_relative_to_water = 1.2 m/s
Vriver = 3.5 m/s
Now, we can calculate the direction angle:
θ = arctan(1.2 / 3.5)
Using a calculator, we find:
θ ≈ 18.32 degrees (rounded to two decimal places)
Therefore, the ferry is moving at an angle of approximately 18.32 degrees counterclockwise from due east.