A river flows due east at 2.00 m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 11.0 m/s due north relative to the water.
An airplane flies 200 km due west from city A to city B and then 270 km in the direction of 31.0° north of west from city B to city C.
(a) In straight-line distance, how far is city C from city A?
(b) Relative to city A, in what direction is city C?
To find the boat's actual velocity, we need to determine its resultant velocity, which takes into account both the velocity of the river and the boat's velocity relative to the river.
First, let's break down the velocities in terms of their components.
The river's velocity is solely in the eastward direction, so we can represent it as Vr = 2.00 m/s due east.
The boat's velocity relative to the water, Vb (water), is given as 11.0 m/s due north.
To find the resultant velocity, we'll use vector addition. We need to add the river's velocity vector to the boat's velocity vector.
Let's represent the resultant velocity as Vr+b.
The eastward component of the resultant velocity, Ve (resultant), is equal to the eastward component of the river's velocity, Ve (river), since the boat's velocity relative to the water does not contribute to eastward movement.
Ve (resultant) = Ve (river) = 2.00 m/s
The northward component of the resultant velocity, Vn (resultant), is the sum of the northward components of both velocities:
Vn (resultant) = Vn (river) + Vn (boat)
Since the boat's velocity is given as due north, Vn (boat) = Vb (water).
Vn (resultant) = 0 + 11.0 m/s = 11.0 m/s
Therefore, the resultant velocity, Vr+b, is a vector with a magnitude of 11.0 m/s and an angle of 90 degrees north of east.
In summary, the boat's actual velocity is 11.0 m/s due north relative to the water, but its resultant velocity is 11.0 m/s with a direction 90 degrees north of east, taking into account the river's velocity.