A boat crosses a river of width 123 m in which
the current has a uniform speed of 1.49 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 2.66 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.
017 (part 2 of 2) 10.0 points
How far downstream from the initial position
is the boat when it reaches the opposite shore?
Answer in units of m.
To find the magnitude of the speed of the boat relative to a stationary shore observer, we can use the concept of vector addition.
Let's break down the velocities involved in this situation:
1. Speed of the boat relative to the water (vboat): 2.66 m/s
2. Speed of the river current (vcurrent): 1.49 m/s
The boat's velocity relative to the shore observer is the vector sum of the boat's velocity relative to the water and the velocity of the river current. Since the boat maintains a bearing perpendicular to the river, the two velocities are at right angles to each other.
To find the magnitude of the boat's velocity relative to the shore observer, we can use the Pythagorean theorem:
vboat_relative = sqrt(vboat^2 + vcurrent^2)
vboat_relative = sqrt(2.66^2 + 1.49^2)
vboat_relative = sqrt(7.0756 + 2.2201)
vboat_relative = sqrt(9.2957)
vboat_relative ≈ 3.05 m/s
Therefore, the magnitude of the speed of the boat relative to a stationary shore observer is approximately 3.05 m/s.
Moving on to the second part of the question: To find how far downstream the boat is when it reaches the opposite shore, we need to calculate the time it takes for the boat to cross the river.
Given:
Width of the river (d): 123 m
Speed of the boat relative to the water (vboat): 2.66 m/s
The time taken to cross the river (t) can be calculated using the formula:
t = d / vboat
Substituting the given values:
t = 123 / 2.66
t ≈ 46.24 s
Now, the distance the boat drifts downstream during this time can be calculated using the formula:
Distance downstream (ds) = vcurrent * t
Substituting the given values:
ds = 1.49 * 46.24
ds ≈ 68.82 m
Therefore, the boat is approximately 68.82 m downstream from its initial position when it reaches the opposite shore.