A hunter wishes to cross a river that is 1.1 km wide and flows with a speed of 5.0 km/h parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of 9 km/h with respect to the water. What is the minimum time necessary for crossing?
To determine the minimum time necessary for crossing the river, we need to analyze the velocity of both the boat and the river.
Let's break down the velocity components:
1. The boat's velocity with respect to the ground (VB) is the vector sum of its velocity with respect to the water (VW) and the water's velocity (VR):
VB = VW + VR
2. The boat's velocity with respect to the water (VW) is given as 9 km/h.
3. The river's velocity (VR) is given as 5.0 km/h parallel to its banks.
To find the minimum time, we need to determine the angle at which the boat should travel in order to minimize the water displacement.
Let's consider the following:
- Assume the angle between the boat's direction and the river flow is θ.
- The displacement of the boat in the downstream direction (D) would be the product of the boat's speed with respect to the water (VW) and the time taken to cross (t):
D = VW * t
- The displacement of the river flow (DR) is given as 1.1 km (the width of the river).
- The displacement of the boat in the downstream direction (D) is equal to the displacement of the river flow (DR):
D = DR
Using basic trigonometry, we can relate the displacement of the boat with the angle θ:
D = VW * t = DR = VR * t = 5.0 km/h * t
Given that the width of the river is 1.1 km, we can rewrite the equation for D:
D = 1.1 km
Equating the two expressions for D, we get:
VW * t = 5.0 km/h * t
9 km/h * t = 5.0 km/h * t
Simplifying the equation, we find:
9 km/h = 5.0 km/h
Since the velocity of the boat is greater than or equal to the velocity of the river, the boat will always reach the other side of the river. However, we need to minimize the crossing time.
To minimize the time taken to cross, the boat should move perpendicular to the river flow. This means the angle θ should be 90 degrees.
Hence, the minimum time necessary for crossing the river is achieved by moving perpendicular to the river's flow, resulting in a straight line path from one bank to the other.