A boat travels at 3.8 m/s and heads straight across a river 240m wide at an angle of 145'. The river flows at 1.6 m/s.

a. What is the boat's resultant speed with respect to the river bank?

b. How long does ti take the boat to cross the river?

c. How far downstream or upstream is the bot when it reaches the other side?

To solve this problem, we can break it down into different components and apply vector addition.

a. First, let's find the boat's velocity components. The boat is traveling at 3.8 m/s, and we can break it down into two components: one parallel to the river's flow and one perpendicular to it. The component parallel to the river's flow is the same as the boat's velocity, which is 3.8 m/s. The component perpendicular to the river's flow is zero because the boat is traveling straight across the river.

Next, let's find the river's velocity component. The river is flowing at 1.6 m/s, and this velocity is entirely parallel to the river's flow.

Now, we can add these vectors using vector addition. Since the boat's motion is perpendicular to the river's flow, we only need to consider the magnitudes of the velocities. The resultant velocity with respect to the river bank is given by the Pythagorean theorem:

resultant velocity^2 = (boat's velocity)^2 + (river's velocity)^2

Plugging in the values:

resultant velocity^2 = (3.8 m/s)^2 + (1.6 m/s)^2
resultant velocity^2 = 14.44 m^2/s^2 + 2.56 m^2/s^2
resultant velocity^2 = 17 m^2/s^2

Taking the square root of both sides, we find:

resultant velocity ≈ √17 ≈ 4.12 m/s

So, the boat's resultant speed with respect to the river bank is approximately 4.12 m/s.

b. To find the time it takes for the boat to cross the river, we can use the formula:

time = distance / velocity

The distance the boat needs to cross is the width of the river, which is 240 m. The velocity to consider when calculating the time is the resultant velocity, which is 4.12 m/s.

time = 240 m / 4.12 m/s
time ≈ 58.25 s

So, it takes approximately 58.25 seconds for the boat to cross the river.

c. To find how far downstream or upstream the boat is when it reaches the other side, we can use the formula:

distance = time * velocity

The time taken to cross the river is 58.25 s, and the river's velocity is 1.6 m/s.

distance = 58.25 s * 1.6 m/s
distance ≈ 93.2 m

The boat is approximately 93.2 meters downstream from its starting point when it reaches the other side of the river.

To calculate the answers to these questions, we can use components and vector addition. Let's break down the boat's velocity into its horizontal and vertical components.

Given:
Boat speed (Vb) = 3.8 m/s
River speed (Vr) = 1.6 m/s
Angle with respect to the river (Θ) = 145 degrees
Width of the river (d) = 240 m

a. To find the boat's resultant speed with respect to the river bank, we can use the Pythagorean theorem:

Resultant speed (V) = √(Vb² + Vr²)

V = √((3.8 m/s)² + (1.6 m/s)²)
V ≈ √(14.44 + 2.56)
V ≈ √17
V ≈ 4.123 m/s

Therefore, the boat's resultant speed with respect to the river bank is approximately 4.123 m/s.

b. To calculate how long it takes for the boat to cross the river, we can use the formula:

Time (t) = Distance (d) / Velocity (v)

Time (t) = 240 m / 3.8 m/s
t ≈ 63.1579 s

Therefore, it takes approximately 63.1579 seconds for the boat to cross the river.

c. To find how far downstream or upstream the boat is when it reaches the other side, we can calculate the horizontal component of the boat's velocity:

Horizontal component = Resultant speed × cos(Θ)

Horizontal component = 4.123 m/s × cos(145 degrees)
Horizontal component = 4.123 m/s × (-0.5736)
Horizontal component ≈ -2.366 m/s

Since the horizontal component is negative, it means the boat crosses the river in the upstream direction.

To find the distance downstream or upstream, we can use the formula:

Distance = Velocity × Time

Distance = -2.366 m/s × 63.1579 s
Distance ≈ -149.51 m

Therefore, the boat is approximately 149.51 meters upstream when it reaches the other side of the river.