# Day 5: Save the Five Golden Rings!

**Classical Mechanics**Level 5

A nasty elf is determined to ruin Christmas by stealing the five golden rings. However Santa is not so easily beaten; he has a way to stop such thieves.

He has a tank of length \(L\) (with openings of negligible length), filled with a liquid with viscosity \(\mu = 1 \ \mathrm{Nsm}^{-2}\). In order to get away, the elf must swim through the tank in one breath.

Assume that the elf does not sink (so **only consider horizontal forces**), and that the elf does not touch the bottom. Furthermore assume that the elf stays at the top, so that upon reaching the end it can **immediately escape** (without needing to swim upwards).

Now for some more information:

- The elf can hold its breath for a maximum of \(100\) seconds.
- The elf weighs \(7\)kg and it is carrying \(5\) gold rings that each weigh \(0.2\)kg. Apart from this it has no additional weight.
- The elf starts at rest at one end of the tank and it swims forward with a force \(e^{-\alpha t}\)N at time \(0 \leq t \leq 100\), where \(\alpha = 0.001\).

Modelling the elf as a spherical particle of radius \(\frac{4}{3\pi}\)m with the only resistive force being from the viscosity of the liquid **due to Stokes' Law**, find the maximum length of the tank \(L\) **to the nearest cm** for which the elf can escape.

###### This problem is part of the Advent Calendar 2015.

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