A hunter wishes to cross a river that is 2.0 km wide and that flows with a speed of 5.1 km/h. The hunter uses a small powerboat that moves at a maximum speed of 12.3 km/h with respect to the water. What is the minimum time necessary for crossing?
Does it matter which portion of the opposite shore he arrives at?
It makes a difference. Aiming across the shore while drifting downstream is the quickest way to cross, and takes
2.0/12.3 = 0.1626 h = 9.76 minutes
To find the minimum time necessary for crossing the river, the hunter needs to take into account the speed of the river and the maximum speed of the powerboat. Here's how you can calculate it:
Step 1: Determine the velocity of the river:
The velocity of the river is given as 5.1 km/h.
Step 2: Calculate the resultant velocity of the powerboat:
The resultant velocity of the powerboat is the vector sum of the velocity of the boat in still water and the velocity of the river.
Since the powerboat moves at a maximum speed of 12.3 km/h with respect to the water, and the river flows at a velocity of 5.1 km/h, the resultant velocity can be found by subtracting the velocity of the river from the maximum speed of the powerboat:
Resultant velocity = Powerboat speed - River speed = 12.3 km/h - 5.1 km/h = 7.2 km/h
Step 3: Calculate the time to cross the river:
The time to cross the river can be calculated using the equation:
Time = Distance / Velocity
The distance across the river is given as 2.0 km, and the resultant velocity of the powerboat is 7.2 km/h.
Plugging in these values into the equation, we get:
Time = 2.0 km / 7.2 km/h ≈ 0.278 hours
To convert hours to minutes, multiply by 60:
Time ≈ 0.278 hours * 60 minutes/hour ≈ 16.67 minutes
Therefore, the minimum time necessary for crossing the river is approximately 16.67 minutes.