When Neil was 2 miles upstream from camp on a canoe trip, he passed a log floating downstream with the current. He paddled upstream for 1 more hour and then returned to camp just as the log arrived. What was the rate of the current?
Neil: 2 mi/hr
current: 1 mi/hr
Check:
In 1 hour, Neil rowed 1 mi upstream.
In 1 more hour, he rowed 3 mi downstream back to camp
In those 2 hours, the log floated 2 miles downstream to camp
To solve this problem, we need to use the concept of relative speed. Let's break down the information given:
1. Neil paddled upstream for 1 more hour after passing the log.
2. Neil returned to camp just as the log arrived.
First, let's consider Neil's speed relative to the log when he is paddling upstream. If Neil paddled for 1 hour after passing the log, and during this time the log floated downstream with the current, that means Neil covered the same distance as the log and met it when he turned back. Let's denote Neil's speed as Vn and the log's speed as Vl.
When Neil is paddling upstream relative to the log:
Vn - Vl = Distance / Time
Similarly, when Neil is traveling downstream relative to the log:
Vn + Vl = Distance / Time
We know Neil paddled for an additional hour, so the time spent by Neil while paddling upstream is 1 hour more than the time taken by the log to reach camp. Therefore, the relationship becomes:
(Vn - Vl) * (Time + 1) = Distance
At the same time, we also know that Neil reached the camp just as the log arrived, implying they covered the same distance. So we have:
(Vn + Vl) * Time = Distance
Since we are interested in finding the rate of the current, we can solve these two equations to determine the value of Vl.
By rearranging the first equation, we can express Distance as follows:
Distance = (Vn - Vl) * (Time + 1)
Substituting this expression for Distance in the second equation:
(Vn + Vl) * Time = (Vn - Vl) * (Time + 1)
Expanding and simplifying the equation:
Vn * Time + Vl * Time = Vn * Time - Vl * Time + Vn - Vl
Simplifying further:
Vl * Time + Vl = Vn
Finally, we can solve for Vl by dividing both sides by Time + 1:
Vl = Vn / (Time + 1)
Therefore, the rate of the current (Vl) is equal to Neil's speed (Vn) divided by the time taken by Neil paddling upstream (Time + 1).