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 relative velocities of the boat and the river.
First, let's break down the velocities involved:
1. The velocity of the river flow: 5.0 km/h parallel to the banks.
2. The maximum speed of the boat with respect to the water: 9 km/h.
Now, to calculate the minimum time necessary, we need to find the angle at which the boat must navigate to minimize the influence of the river flow.
Let's use some trigonometry:
1. Draw a diagram depicting the river flow direction parallel to the banks.
2. Let the angle between the boat's heading and the perpendicular to the banks be θ. (This angle represents the angle at which the boat aims to cross the river.)
3. The component of the boat's velocity perpendicular to the river flow is given by: v⊥ = v * sin(θ), where v is the boat's velocity with respect to the water.
4. The component of the boat's velocity parallel to the river flow is given by: v∥ = v * cos(θ).
5. The magnitude of the river flow's velocity is 5.0 km/h, and it acts perpendicular to the banks, so v_river = 5.0 km/h.
6. The effective velocity of the boat along the perpendicular direction is the vector sum of v∥ and v_river.
v_effective = v⊥ - v_river = v * sin(θ) - 5.0 km/h.
To minimize the time taken to cross the river, the boat should aim for v_effective to be zero, or in other words:
v * sin(θ) - 5.0 km/h = 0.
Now, we solve for the angle θ:
sin(θ) = (5.0 km/h) / v.
Since we know the maximum speed of the boat with respect to the water is 9 km/h, we can substitute this value into the equation:
sin(θ) = (5.0 km/h) / 9.0 km/h.
Using a scientific calculator or a trigonometric table, we can find the value of θ:
θ ≈ 28.07 degrees.
Now we have the angle at which the boat should navigate to minimize the influence of the river flow.
To find the minimum time necessary for crossing the river, we need to calculate the time it takes for the boat to travel the distance of 1.1 km at its maximum speed along the perpendicular direction.
The time taken is given by:
time = distance / velocity = (1.1 km) / (v * cos(θ)).
Substituting the maximum speed of the boat, we have:
time = (1.1 km) / (9.0 km/h * cos(28.07 degrees)).
Evaluating this expression, we can determine the minimum time necessary for crossing the river.