A boat is crossing a river. The velocity of the boat relative to the water is 4.0 m/s directed perpendicular to the river's current. The velocity of the current relative to the shore is 2.0 m/s. How long will it take for the boat to cross the river?

To find out how long it will take for the boat to cross the river, we need to consider the velocity of the boat relative to the ground.

First, let's break down the velocities involved:

1. Velocity of the boat relative to the water (vb): 4.0 m/s directed perpendicular to the river's current
2. Velocity of the current relative to the shore (vc): 2.0 m/s

To find the velocity of the boat relative to the ground (vbg), we can use vector addition:

vbg = √(vb^2 + vc^2)

Given that vb = 4.0 m/s and vc = 2.0 m/s, we can calculate vbg:

vbg = √(4.0^2 + 2.0^2)
vbg = √(16.0 + 4.0)
vbg = √20.0
vbg ≈ 4.47 m/s

Therefore, the velocity of the boat relative to the ground is approximately 4.47 m/s.

To determine the time it will take for the boat to cross the river, we need to know the width of the river (d) or the distance the boat needs to travel.

If we know the width of the river, we can divide it by the velocity of the boat relative to the ground:

Time = Distance / Velocity

For example, if the width of the river is 100 meters:

Time = 100 m / 4.47 m/s
Time ≈ 22.39 seconds

So, it will take approximately 22.39 seconds for the boat to cross the river if the width is 100 meters.

We need to know the width of the river.