A river is one mile wide and the water flows at the rate 3 miles/hour. If you can paddle a canoe 5 miles/hour in still water, then how long will it take you to paddle across the river and back? How long would it take you to travel one mile upstream (measured along the bank) and back?
To go straight across, you would have to paddle in an upstream direction with respect to the water. The velocity component across the river would then be sqrt(5^2 - 3^2) = 4 miles/h. The time to go across and back is then
T1 = 2 mile/4 mph = 0.5 hours = 30 minutes
To go 1 mile upstreeam and then down requires
T2 = 1/(5-3) + 1/(5+3) = 1/2 + 1/8 = 5/8 hr = 37.5 minutes.
To find the time it takes to paddle across the river and back, we need to consider the speed of the canoe and the speed of the river's current.
Let's first look at paddling across the river and back:
Paddling across the river:
Since the river is one mile wide and the water flows at a rate of 3 miles/hour, the current will push the canoe downstream.
To determine the effective speed of the canoe while crossing the river, we need to subtract the speed of the river's current from the canoe's speed in still water.
Effective speed across the river = Canoe's speed in still water - Speed of the river's current
Effective speed across the river = 5 miles/hour - 3 miles/hour = 2 miles/hour
Therefore, it would take (1 mile across) / (2 miles/hour) = 0.5 hours or 30 minutes to paddle across the river.
Paddling back across the river:
When paddling back across the river, we need to consider that now the river's current will work against the canoe's motion. So the effective speed of the canoe will be the sum of the canoe's speed in still water and the speed of the river's current.
Effective speed back across the river = Canoe's speed in still water + Speed of the river's current
Effective speed back across the river = 5 miles/hour + 3 miles/hour = 8 miles/hour
Therefore, it would take (1 mile across) / (8 miles/hour) = 0.125 hours or 7.5 minutes to paddle back across the river.
Finally, to calculate the total time spent paddling across the river and back:
Total time = Time to cross the river + Time to paddle back
Total time = 0.5 hours + 0.125 hours = 0.625 hours or 37.5 minutes.
Now let's consider the time it would take to travel one mile upstream (measured along the bank) and back:
As we are going upstream, we would need to subtract the speed of the river's current from the canoe's speed in still water to determine the effective speed.
Effective speed upstream = Canoe's speed in still water - Speed of the river's current
Effective speed upstream = 5 miles/hour - 3 miles/hour = 2 miles/hour
Therefore, it would take (1 mile upstream) / (2 miles/hour) = 0.5 hours or 30 minutes to travel one mile upstream.
Coming back downstream, the speed of the river's current is now in the same direction as the canoe's motion. So the effective speed of the canoe will be the sum of the canoe's speed in still water and the speed of the river's current.
Effective speed downstream = Canoe's speed in still water + Speed of the river's current
Effective speed downstream = 5 miles/hour + 3 miles/hour = 8 miles/hour
Therefore, it would take (1 mile downstream) / (8 miles/hour) = 0.125 hours or 7.5 minutes to travel one mile downstream.
To calculate the total time spent traveling one mile upstream and back:
Total time = Time to go upstream + Time to go downstream
Total time = 0.5 hours + 0.125 hours = 0.625 hours or 37.5 minutes.