In the preceding example, we found that the boat needs to travel in a direction of 11.3 degrees west of north relative to the Earth to move directly across the river.
If the boat is now to travel due north relative to the Earth, we need to find the direction it should travel relative to the water, taking into account the effect of the river's velocity.
Since the boat is traveling with a speed of 20 km/h relative to the water, and the water is flowing with a velocity of 8 km/h in a direction of 60 degrees south of east, we can use vector addition to find the boat's resultant velocity relative to the Earth.
Let's define the x-axis as east and the y-axis as north. The velocity of the water is then:
Vw = 8 km/h at an angle of 240 degrees (60 degrees south of east)
We can convert this to vector form:
Vw = (-4 km/h, -6.93 km/h)
The velocity of the boat relative to the water is:
Vb = 20 km/h at an angle of 0 degrees (due north)
We can convert this to vector form:
Vb = (0 km/h, 20 km/h)
To find the resultant velocity, we can add these two vectors:
Vr = Vb + Vw
Vr = (0 km/h - 4 km/h, 20 km/h - 6.93 km/h)
Vr = (-4 km/h, 13.07 km/h)
The direction of this vector is given by:
tan θ = (13.07 km/h) / (-4 km/h)
θ = -73.74 degrees
Since we defined the x-axis as east, the direction of the boat's velocity relative to the water is 73.74 degrees east of north.
To find the direction the boat should travel relative to the water to move due north, we need to subtract this angle from 90 degrees:
90 degrees - 73.74 degrees = 16.26 degrees
Therefore, the boat should travel in a direction of 16.26 degrees west of north relative to the water to move due north relative to the Earth.