A boat crosses a river of width 115 m in which

the current has a uniform speed of 1.84 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 1.1 m/s relative to the water.
What is the magnitude of the speed of the
boat relative to a stationary shore observer?
Answer in units of m/s.

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

X = 1.84 m/s = Velocity of the current.

Y = 1.1 m/s = Velocity of the boat.

A. Vb^2 = X^2 + Y^2 = 1.84^2 + 1.1^2 =
4.60
Vb = 2.14 m/s.

B. Tan A = Y/X = 1.1/1.84 = 0.59783
A = 30.9o, CCW = 59.1o E. of N.

Tan59.1 = d/115
d = 115*Tan59.1 = 192 m. Downstream.

To find the magnitude of the speed of the boat relative to a stationary shore observer, we can use the Pythagorean Theorem. The speed of the boat relative to the water is 1.1 m/s, and the current has a speed of 1.84 m/s.

To find the magnitude of the speed of the boat relative to the stationary shore observer, we need to find the resulting velocity vector. This can be found by adding the velocity of the boat relative to the water and the velocity of the current vectorially.

Let's assume the boat's velocity relative to the water is represented by vector V1 and the current velocity is represented by vector V2.

The magnitude of the boat's velocity relative to the stationary shore observer can be found using the formula:

V = sqrt(V1^2 + V2^2)

In this case, V1 = 1.1 m/s and V2 = 1.84 m/s.

Plugging in the values, we get:

V = sqrt((1.1)^2 + (1.84)^2)

V = sqrt(1.21 + 3.3856)

V = sqrt(4.5956)

V ≈ 2.1448 m/s

Therefore, the magnitude of the speed of the boat relative to a stationary shore observer is approximately 2.1448 m/s.

To find the distance downstream from the initial position when the boat reaches the opposite shore, we need to calculate the time it takes for the boat to cross the river.

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

t = d / Vc

where t is the time taken to cross the river, d is the width of the river, and Vc is the velocity of the current.

In this case, d = 115 m and Vc = 1.84 m/s.

Plugging in the values, we get:

t = 115 / 1.84

t ≈ 62.50 seconds

Now, we can calculate the distance downstream using the formula:

d_downstream = Vb * t

where d_downstream is the distance downstream, Vb is the magnitude of the speed of the boat relative to the stationary shore observer, and t is the time taken to cross the river.

In this case, Vb = 2.1448 m/s and t ≈ 62.50 seconds.

Plugging in the values, we get:

d_downstream = 2.1448 * 62.50

d_downstream ≈ 134.05 meters

Therefore, the boat is approximately 134.05 meters downstream from the initial position when it reaches the opposite shore.