A river has a 7.3 m/s current due East. With respect to the water, the boat's velocity is 16 m/s north of east. what is the velocity of the boat relative to the earth?
To find the velocity of the boat relative to the earth, we need to combine the effect of both the river's current and the boat's velocity.
First, let's consider the river's current. The current is flowing due East at 7.3 m/s. This means that the river is moving towards the East, and we can represent this velocity as (7.3 m/s)East.
Next, we need to consider the boat's velocity relative to the water. Given that it is 16 m/s north of east, we can represent its velocity as (16 m/s)North of East.
To find the overall velocity of the boat relative to the earth, we need to combine these two velocities. We can do this using vector addition.
Since the velocities have different directions, we will use the Pythagorean theorem to find the magnitude of the overall velocity. The direction can be determined using trigonometry.
Using the Pythagorean theorem, we can find the magnitude of the overall velocity as:
Magnitude = √[(7.3 m/s)^2 + (16 m/s)^2]
= √[53.29 + 256]
≈ √309.29
≈ 17.59 m/s
Now let's find the direction. We can use trigonometry to find the angle between the overall velocity vector and the positive x-axis.
tan θ = Opposite/Adjacent
tan θ = (16 m/s)/(7.3 m/s)
θ = arctan[(16 m/s)/(7.3 m/s)]
θ ≈ 64.11°
Therefore, the velocity of the boat relative to the earth is approximately 17.59 m/s at an angle of 64.11° north of east.