A 200-m-wide river flows due east at a uniform speed of 2.0 m/s. A boat with a speed of 8.0 m/s relative to the water leaves the south bank pointed in a direction west of north. What are the (a) magnitude and (b) direction of the boat’s velocity relative to the ground? (c) How long does the boat take to cross the river?

7.2, 16, and 29

To solve this problem, we can use vector addition to find the boat's velocity relative to the ground.

Let's break down the velocities into their components:

Boat's velocity relative to the water:
Speed = 8.0 m/s
Direction = West of North

River's velocity:
Speed = 2.0 m/s
Direction = Due East

(a) To find the boat's velocity relative to the ground, we need to add the two velocities.

First, let's find the components of the boat's velocity relative to the ground.

Vertical component (north-south direction):
The boat's direction is west of north, which means its angle with respect to the north is 90 degrees - angle west of north.

To find the vertical component, we can use the formula:
Vertical component = speed * sin(angle)

In this case, the angle west of north is not given, so let's assume it's 45 degrees.

Vertical component = 8.0 m/s * sin(45°) = 5.66 m/s

Horizontal component (east-west direction):
Since the boat is moving due west, the horizontal component is 0.

So, the boat's velocity relative to the ground is 5.66 m/s north.

(b) The magnitude of the boat's velocity relative to the ground is the square root of the sum of the squares of its components:

Magnitude = sqrt((vertical component)^2 + (horizontal component)^2)
Magnitude = sqrt((5.66 m/s)^2 + (0 m/s)^2)
Magnitude ≈ 5.66 m/s

So, the magnitude of the boat's velocity relative to the ground is approximately 5.66 m/s.

(c) To find the time it takes for the boat to cross the river, we need to find the horizontal distance it needs to cover.

Horizontal distance = width of the river = 200 m

Since the boat's horizontal component of velocity is 0 m/s (moving directly north or south), it will take the boat 200 m / 5.66 m/s = 35.36 seconds to cross the river.

Therefore, the boat takes approximately 35.36 seconds to cross the river.

To solve this problem, we can break it down into different components. Let's start by finding the velocity of the boat relative to the ground.

(a) To find the magnitude of the boat's velocity relative to the ground, we can use vector addition. The boat's velocity can be split into two components: one parallel to the river's flow and one perpendicular to it.

The component of the boat's velocity parallel to the river's flow is due to the river's flow itself. Since the river is flowing east at a speed of 2.0 m/s, the boat's velocity parallel to the river's flow will be 2.0 m/s east.

The component of the boat's velocity perpendicular to the river's flow is due to the boat's own speed. We can use the Pythagorean theorem to find this component. The boat's speed relative to the water is 8.0 m/s, and the river's width is 200 m. Therefore, the component of the boat's velocity perpendicular to the river's flow can be calculated as:

√((8.0 m/s)^2 - (2.0 m/s)^2) = √(64 - 4) = √60 ≈ 7.75 m/s.

Now we can find the magnitude of the boat's velocity relative to the ground by using vector addition:

Magnitude of the boat's velocity relative to the ground = √((2.0 m/s)^2 + (7.75 m/s)^2) ≈ 8.01 m/s.

Therefore, the magnitude of the boat's velocity relative to the ground is approximately 8.01 m/s.

(b) To find the direction of the boat's velocity relative to the ground, we need to determine the angle it makes with the east direction (opposite to the river's flow). We can use trigonometry to find this angle.

The angle can be calculated as:

θ = arctan((7.75 m/s) / (2.0 m/s)) ≈ 76.94 degrees west of north.

Therefore, the direction of the boat's velocity relative to the ground is approximately 76.94 degrees west of north.

(c) To find how long the boat takes to cross the river, we can use the formula:

Time = Distance / Speed.

The distance to cross the river is the width of the river, which is 200 m. The speed of the boat relative to the water is 8.0 m/s. Therefore, the time it takes for the boat to cross the river can be calculated as:

Time = 200 m / 8.0 m/s = 25 seconds.

Therefore, the boat takes 25 seconds to cross the river.

You know the East component of Velocity, and the N velocity of the Boat. Find the resultant speed of the boat over ground to the other side.

I will be happy to critique your thinking or work on this.