A boat crosses a river of width 120 m in which
the current has a uniform speed of 2.04 m/s.
The pilot maintains a bearing (i.e., the direc-
tion in which the boat points) perpendicular
to the river and a throttle setting to give a
constant speed of 2.58 m/s relative to the wa-
ter.
What is the magnitude of the speed of the
boat relative to a stationary shore observer?
Answer in units of m/s
To find the magnitude of the speed of the boat relative to a stationary shore observer, we need to use vector addition.
Let's break down the boat's motion into two components: the motion across the river and the motion downstream due to the river's current.
1. Motion across the river:
The boat's speed relative to the water is given as 2.58 m/s. Since the boat maintains a bearing perpendicular to the river, the boat's speed across the river will remain constant. Therefore, the motion across the river is simply the boat's speed: 2.58 m/s.
2. Motion downstream due to the current:
The current has a uniform speed of 2.04 m/s. Since the boat is not pointing directly against or with the current, there will be a component of the boat's motion in the downstream direction. This component is given by the current's speed.
Now we can add the two components using vector addition. The magnitude of the speed of the boat relative to a stationary shore observer is the magnitude of the resultant vector.
Using the Pythagorean theorem, we can find the magnitude of the resultant vector:
Resultant speed^2 = (Motion across the river)^2 + (Motion downstream)^2
Resultant speed^2 = 2.58^2 + 2.04^2
Resultant speed^2 = 6.6564 + 4.1616
Resultant speed^2 = 10.818
Resultant speed ≈ √10.818
Resultant speed ≈ 3.29 m/s
Therefore, the magnitude of the speed of the boat relative to a stationary shore observer is approximately 3.29 m/s.