Day 5: Save the Five Golden Rings!

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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