A boat travelling at 30km/h relative to water is headed away from the bank of a river and downstream. The river is 1/2 km wide and flows at 6km/h. The boat arrives at the opposite bank in 1.25min.

Calculate the component of the boat's velocity directed across the river.
Calculate the total downstream component of the boat's motion.

What does it mean headed downstream? Is it going to be the resultant vector?

I guess we should work in seconds and meters

vb = v boat = 30,000/3600 = 8.33 m/s
c = current = 6,000/3600 = .167 m/s

w = width = 500 m

time to cross = 75 s

so say the boat heads at an angle T to the direction of the opposite shore toward the ocean downstream.
Then the only component across the river is vb cos T = 500/75 = 6.67 m/s
so
500 = 75 * 8.33 cos T
cos T = .800
T = 37 deg
now the component of the boat speed downriver measured on the boat is 8.33 sin T = 5 m/s but we also are drifting with the current downstream at .167 m/s
so total component downstream = 5.167 m/s

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.