Saturday, February 17, 2024

How long will it take a log to float from the grove to the dam?

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 1. When the beaver swims back from the grove to the dam (downstream), its effective velocity with the current is +, and the time taken is 2.

Let be the distance between the dam and the grove. We can write:

=()1 (1) =(+)2 (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 :

=2=1+ 2=21 =2(1211)

Substituting back into equation (1):

=(2(1211))1 2=(1211) =12(12+11) =212212+1

Finally, the time it takes for a log to float from the grove to the dam is:

==12212+12=122(121)(2+112)