🧲 Physics T-Shirt 🥼: How long will it take a log to float from the grove to the dam?

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:

$� = (�_{�} - �_{�}) \cdot �_{1}$ (1) $� = (�_{�} + �_{�}) \cdot �_{2}$ (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 $� = \frac{�}{�_{�}}$ .

From equations (1) and (2), we can solve for $�_{�}$ and $�$ :

$�_{�} = \frac{�}{�_{2}} - �_{�} = \frac{�}{�_{1}} + �_{�}$ $2 �_{�} = \frac{�}{�_{2}} - \frac{�}{�_{1}}$ $�_{�} = \frac{�}{2} (\frac{1}{�_{2}} - \frac{1}{�_{1}})$

Substituting $�_{�}$ back into equation (1):

$� = (\frac{�}{2} (\frac{1}{�_{2}} - \frac{1}{�_{1}}) - �_{�}) \cdot �_{1}$ $2 � = � (\frac{�_{1}}{�_{2}} - 1 - \frac{�_{�} �_{1}}{�})$ $�_{�} = \frac{�}{�_{1}} - \frac{�}{2} (\frac{1}{�_{2}} + \frac{1}{�_{1}})$ $�_{�} = \frac{2 �}{�_{1} �_{2}} \cdot \frac{�_{2} - �_{1}}{�_{2} + �_{1}}$

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

$� = \frac{�}{�_{�}} = \frac{�_{1} �_{2}}{�_{2} - �_{1}} \cdot \frac{�_{2} + �_{1}}{2} = \frac{�_{1} �_{2}}{2} (\frac{1}{�_{2} - �_{1}}) (\frac{�_{2} + �_{1}}{�_{1} �_{2}})$ $� = \frac{�_{1} �_{2}}{�_{2} - �_{1}}$

So, the time it takes for a log to float from the grove to the dam is $� = \frac{�_{1} �_{2}}{�_{2} - �_{1}}$ .

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Saturday, February 17, 2024

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

Physics Problem

Solution

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