To calculate the component of the boat's velocity directed across the river, you need to break it down into its horizontal and vertical components. Let's call the horizontal component "Vx" and the vertical component "Vy".
First, let's calculate the component of the boat's velocity directed across the river (Vx). Since the boat is travelling at 30 km/h relative to the water and the river is flowing at 6 km/h, the boat's velocity relative to the ground (Vg) would be the vector sum of the boat's velocity relative to the water (Vw) and the river's velocity (Vr):
Vg = Vw + Vr
Vw = 30 km/h (since the boat's velocity relative to the water is 30 km/h)
Vr = -6 km/h (since the river's velocity is in the opposite direction to the boat's motion)
Now, let's calculate the horizontal component (Vx). Since the component of the boat's velocity directed across the river is perpendicular to the river's flow, we can use Pythagoras' theorem:
Vx^2 + Vy^2 = Vg^2
Vy = 0 km/h (since the boat is not moving vertically)
Therefore, Vx^2 = Vg^2 - Vy^2
= (Vw + Vr)^2
= (30 km/h - 6 km/h)^2
= (24 km/h)^2
= 576 km^2/h^2
Vx = sqrt(576 km^2/h^2)
= 24 km/h
So, the component of the boat's velocity directed across the river is 24 km/h.
To calculate the total downstream component of the boat's motion, we need to calculate the component of the boat's velocity parallel to the river's flow. Since the boat is heading downstream, the component of the boat's velocity parallel to the river's flow would be the same as the river's velocity (Vr = -6 km/h).
Therefore, the total downstream component of the boat's motion is -6 km/h.
"Headed downstream" means that the boat is moving in the same direction as the river's flow. In this case, it means that the boat is traveling in the direction of the downstream component of its motion, which is the vector sum of the boat's velocity and the river's velocity. The resultant vector represents the boat's overall motion relative to the ground.