# Slow motion for me

**Classical Mechanics**Level 5

As soon as new food is acquired (by expanding the inhabited zone), it is divided equally among all existing bacteria so that the food available to a single bacterium is \(c_F = F/N\), where \(F\) is the total amount of uneaten food inside of the inhabited zone.

Finally, the bacteria grow (accumulate mass) according to \[\partial_t m = \lambda_\text{max}m\frac{c_F}{\gamma_F + c_F}\] where \(\gamma_F\) is the mass of food required to build one new bacterium. When food is highly abundant (\(c_F \gg \gamma_F\)), the bacteria accumulate mass at the exponential rate \(\lambda_\text{max}\), but as food becomes scarce, they slow down.

Suppose the colony starts with one newly birthed bacteria (of mass \(m_0\)), what is the mass of the bacterial colony after 1,000 hrs (in terms of \(m_0\))?

**Assumptions and Details**

- Individual bacterium start with mass \(m_0\) (and area \(\delta_A\)), and divide into two when they've grown to \(2m_0\) (and area \(2\delta_A\)).
- The inhabited zone expands by the growing bacteria inside it pushing each other outward from the center.
- \(\lambda_\text{max} = 1\text{ hr}^{-1}\)
- \(\gamma_F = 0.8\times 10^{-15}\) g
- The density of food in the uninhabited zone is \(\rho_F = 1\times 10^{-15}\text{ g}/\delta_A\).

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