You row a boat, perpendicular to the shore of a river that flows at 3.0 m/s. The velocity of your boat is 4.0 m/s relative to the water.

a. What is the velocity of your boat relative to the shore?
b. What is the component of your velocity parallel to the shore? Perpendicular to it?

a.) 5.0 m/s

35.8 m/s

To find the velocity of the boat relative to the shore, we can use vector addition.

a. The velocity of the boat relative to the shore is the vector sum of the velocity of the boat relative to the water and the velocity of the water. Since the velocity of the water is perpendicular to the shore, we can add the two velocities as follows:

Velocity of boat relative to shore = Velocity of boat relative to water + Velocity of water

Given:
- Velocity of boat relative to water = 4.0 m/s (perpendicular to the shore)
- Velocity of water = 3.0 m/s (along the shore)

Using vector addition, we can calculate the magnitude and direction of the resulting vector:

Magnitude: √((4.0 m/s)^2 + (3.0 m/s)^2)
Direction: arctan(3.0 m/s / 4.0 m/s)

Calculating this:

Magnitude: √(16.0 m^2/s^2 + 9.0 m^2/s^2) = √(25.0 m^2/s^2) = 5.0 m/s
Direction: arctan(3.0 m/s / 4.0 m/s) = arctan(0.75) ≈ 36.87°

Therefore, the velocity of the boat relative to the shore is 5.0 m/s in a direction of approximately 36.87°.

b. The component of the boat's velocity parallel to the shore is equal to the velocity of the water, which is 3.0 m/s. This component represents the boat's speed along the shore.

The component of the boat's velocity perpendicular to the shore is equal to the velocity of the boat relative to the water, which is 4.0 m/s. This component represents the boat's speed across the river.

To find the velocity of the boat relative to the shore, we can use vector addition. The velocity of the boat relative to the shore can also be thought of as the sum of the boat's velocity relative to the water and the water's velocity relative to the shore.

a. Velocity of the boat relative to the shore:
- Boat's velocity relative to the water: 4.0 m/s
- Water's velocity relative to the shore: 3.0 m/s (given)
To find the velocity of the boat relative to the shore, we add these velocities:
4.0 m/s + 3.0 m/s = 7.0 m/s

Therefore, the velocity of the boat relative to the shore is 7.0 m/s.

b. To find the component of the boat's velocity parallel to the shore, we need to consider the angle between the boat's direction of motion and the direction parallel to the shore. Since the boat is rowing perpendicular to the shore, the angle is 90 degrees.

- Boat's velocity relative to the water: 4.0 m/s
The component of the boat's velocity parallel to the shore is given by:
Parallel component = Boat's velocity relative to the water * cosine(angle)
Parallel component = 4.0 m/s * cos(90 degrees)
Parallel component = 0 m/s

Therefore, the component of the boat's velocity parallel to the shore is zero.

The component of the boat's velocity perpendicular to the shore is given by:
Perpendicular component = Boat's velocity relative to the water * sine(angle)
Perpendicular component = 4.0 m/s * sin(90 degrees)
Perpendicular component = 4.0 m/s

Therefore, the component of the boat's velocity perpendicular to the shore is 4.0 m/s.