A fisherman wishes to cross a river that is 1.5 km wide and flows with a speed of 5.0 km/h parallel to its banks.
The fisherman uses a small power boat that moves at a maximum speed of 12 km/h with respect to the water.
(a) What direction should the fisherman head in order to end up directly across the river? (b) What is the time
necessary for crossing? 25º upstream, 0.14 h or 8.4 min
To end up directly across the river, the fisherman should head at some angle upstream. Let's call this angle θ.
(a) To determine the direction the fisherman should head, we can use the concept of relative velocity. The fisherman's boat will have a velocity relative to the water, and the water has a velocity due to its flow parallel to the banks. The resultant velocity of the boat with respect to the banks should be directly across the river.
Let Vb be the velocity of the boat with respect to the banks, Vw be the velocity of the water, and Vr be the resultant velocity of the boat with respect to the banks.
We can use the Pythagorean theorem to find Vr:
Vr² = Vb² + Vw²
Since the fisherman wants Vr to be directly across the river, Vb will be the horizontal component of Vr, and Vw will be the vertical component of Vr.
Vb = Vr * cos(θ)
Vw = Vr * sin(θ)
Substituting these values into the Pythagorean equation:
Vr² = (Vr * cos(θ))² + (Vr * sin(θ))²
Vr² = Vr² * cos²(θ) + Vr² * sin²(θ)
1 = cos²(θ) + sin²(θ)
Since cos²(θ) + sin²(θ) = 1 for any angle θ, the equation is satisfied.
Therefore, the direction the fisherman should head is directly upstream, opposite to the flow of the river.
(b) To determine the time necessary for crossing, we need to find the time it takes for the boat to travel across the river.
The time needed to cross the river can be given by:
time = distance / velocity
The distance to be crossed is 1.5 km, and the maximum speed of the boat with respect to the water is 12 km/h.
Therefore, the time necessary for crossing is:
time = 1.5 km / 12 km/h
time = 0.125 hours
To convert 0.125 hours to minutes, we multiply by 60:
time = 0.125 hours * 60 minutes/hour
time ≈ 7.5 minutes
Therefore, the time necessary for crossing is approximately 7.5 minutes.