A boat crosses a river of width 122 m in which the current has a uniform speed of 2.39 m/s. The pilot maintains a bearing (i.e., the direction in which the boat points) perpendicular to the river and a throttle setting to give a constant speed of 2.81 m/s relative to the water.

A.) What is the magnitude of the speed of the boat relative to a stationary shore observer? Answer in units of m/s.

B.)How far downstream from the initial position is the boat when it reaches the opposite shore? Answer in units of m

X = 2.81 m/s. = Speed of boat.

Y = -2.39 m/s. = Speed of current.

A. V^2 = X^2 + Y^2
V^2 = (2.81)^2 + (@.39)^2 = 13.61
V = 3.69 m/s.

B. tanA = Y/X = -2.39/2.81 = -0.85053
A = -40.4o = 40.4o South of East.

d = 122*tan(-40.4) = -103.8 m. = 103.8 m
downstream.

Correction:

X = 2.39 m/s = Speed of current.
Y = 2.81m/s = Speed of boat.

B. Tan A = Y/X = 2.81/2.39 = 1.17573
A = 49.6o = 40.4o E. of N.

Tan40.4 = d/122
d = 122*Tan40.4 = 103.8 m. Downstream.

103.8 m

A.) The magnitude of the speed of the boat relative to a stationary shore observer is the vector sum of the boat's speed relative to the water and the speed of the current.

Using the Pythagorean theorem, we can calculate it as:

speed relative to shore = √((speed relative to water)^2 + (speed of current)^2)
= √((2.81 m/s)^2 + (2.39 m/s)^2)

Calculating this, we get:

speed relative to shore ≈ 3.78 m/s

Therefore, the magnitude of the speed of the boat relative to a stationary shore observer is approximately 3.78 m/s.

B.) To find the distance downstream from the initial position when the boat reaches the opposite shore, we can use the equation:

distance downstream = (speed relative to shore) × (time taken to cross the river)

The time taken to cross the river can be calculated using the formula:

time = distance / speed relative to water
= 122 m / 2.81 m/s

Calculating this, we get:

time ≈ 43.42 seconds

Substituting the values back into the equation, we have:

distance downstream = 3.78 m/s × 43.42 seconds

Calculating this, we get:

distance downstream ≈ 163.94 meters

Therefore, the boat is approximately 163.94 meters downstream from the initial position when it reaches the opposite shore.

A.) To find the magnitude of the speed of the boat relative to a stationary shore observer, we need to consider the boat's velocity vector in relation to the current.

The boat's velocity relative to the water is given as 2.81 m/s. The current has a uniform speed of 2.39 m/s.

Since the boat maintains a bearing perpendicular to the river, the boat's velocity perpendicular to the current is the same as its velocity relative to the shore observer.

To get the magnitude of the boat's velocity relative to the shore observer, we can use the Pythagorean theorem. The magnitude is given by:

v = √(vboat^2 + vcurrent^2)

Where:
- v is the magnitude of the boat's velocity relative to the shore observer.
- vboat is the velocity of the boat relative to the water (2.81 m/s in this case).
- vcurrent is the velocity of the current (2.39 m/s in this case).

Plugging in the values:

v = √(2.81^2 + 2.39^2)
v ≈ √(7.8961 + 5.7121)
v ≈ √13.6082
v ≈ 3.69 m/s

Therefore, the magnitude of the speed of the boat relative to a stationary shore observer is approximately 3.69 m/s.

B.) To calculate how far downstream from the initial position the boat is when it reaches the opposite shore, we need to determine the time it takes for the boat to cross the river.

We can use the time to calculate the distance traveled downstream.

The time taken to cross the river can be found by dividing the width of the river by the component of the boat's velocity perpendicular to the river.

time = width / vperpendicular

Where:
- time is the time taken to cross the river.
- width is the width of the river (122 m in this case).
- vperpendicular is the component of the boat's velocity perpendicular to the river.

Since the boat's velocity relative to the water is perpendicular to the river, vperpendicular is equal to the boat's velocity relative to the water (2.81 m/s in this case).

Plugging in the values:

time = 122 m / 2.81 m/s
time ≈ 43.42 s

Now that we have the time taken to cross the river, we can calculate the distance traveled downstream using:

distance = vcurrent * time

Where:
- distance is the distance traveled downstream.
- vcurrent is the velocity of the current (2.39 m/s in this case).
- time is the time taken to cross the river.

Plugging in the values:

distance = 2.39 m/s * 43.42 s
distance ≈ 103.17 m

Therefore, the boat is approximately 103.17 m downstream from the initial position when it reaches the opposite shore.

360