A hunter wishes to cross a river that is 1.0 km wide and that flows with a speed of 4.9 km/h. The hunter uses a small powerboat that moves at a maximum speed of 9.6 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 components involved and calculate the resulting velocity.
Let's start by breaking down the velocity components:
1. Velocity of the river: The river flows with a speed of 4.9 km/h.
2. Velocity of the powerboat: The powerboat moves at a maximum speed of 9.6 km/h with respect to the water.
When the boat crosses the river, the boat's velocity must have a horizontal component that cancels out the river's velocity, which is perpendicular to the shoreline. This will ensure that the boat reaches the opposite shore directly.
Now, let's calculate the magnitude of the boat's velocity relative to the shore. We can use the Pythagorean theorem to find the resultant velocity:
v_ship^2 = v_river^2 + v_boat^2
where:
v_ship = resultant velocity of the boat relative to the shore
v_river = velocity of the river = 4.9 km/h
v_boat = velocity of the powerboat = 9.6 km/h
Plugging in the values, we have:
v_ship^2 = (4.9 km/h)^2 + (9.6 km/h)^2
v_ship^2 = 24.01 km^2/h^2 + 92.16 km^2/h^2
v_ship^2 = 116.17 km^2/h^2
Now, take the square root to find the magnitude of the boat's velocity relative to the shore:
v_ship = √116.17 km^2/h^2
v_ship ≈ 10.78 km/h
Since the boat's velocity relative to the shore is approximately 10.78 km/h, the minimum time necessary for crossing the river can be calculated using the distance formula:
Time = Distance / Velocity
Time = 1.0 km / 10.78 km/h
Time ≈ 0.09 hours
Therefore, the minimum time necessary for crossing the river is approximately 0.09 hours, which is equivalent to 5.4 minutes.