Physics Problem
The beavers noticed that swimming from the dam to the grove along the river takes time t₁, and swimming back along the river from the grove to the dam takes time t₂, and t₁ < t₂.
How long will it take a log to float from the grove to the dam?
Solution
Let's assume the river has a constant current velocity , and the beaver swims with a constant velocity relative to the water. When the beaver swims from the dam to the grove (upstream), its effective velocity against the current is , and the time taken is . When the beaver swims back from the grove to the dam (downstream), its effective velocity with the current is , and the time taken is .
Let be the distance between the dam and the grove. We can write:
(1) (2)
We want to find the time it takes for a log to float from the grove to the dam. Since the log is carried only by the current, its velocity is , and the time taken is .
From equations (1) and (2), we can solve for and :
Substituting back into equation (1):
Finally, the time it takes for a log to float from the grove to the dam is:
So, the time it takes for a log to float from the grove to the dam is .