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Herleitung des Zeitgesetzes

Geschwindigkeitsgesetz:

v = \frac{d x}{d t} = k (a_{1} - x) (a_{2} - x)

Variablentrennung:

\frac{d x}{(a_{1} - x) (a_{2} - x)} = k \cdot d t

Partialbruchzerlegung:

\frac{1}{(a_{1} - x) (a_{2} - x)} = \frac{A}{a_{1} - x} + \frac{B}{a_{2} - x} = \frac{A (a_{2} - x) + B (a_{1} - x)}{(a_{1} - x) (a_{2} - x)}

Rechnung:

\begin{array}{l} A a_{2} - A x + B a_{1} - B x = 1 \\ A a_{2} + B a_{1} - (A + B) x = 1 \\ {\begin{cases} A + B = 0 \\ A a_{2} + B a_{1} = 1 \end{cases}} \Rightarrow A = \frac{- 1}{a_{1} - a_{2}} und B = \frac{1}{a_{1} - a_{2}} \end{array}

oben eingesetzt:

\frac{- 1}{(a_{1} - a_{2}) (a_{1} - x)} + \frac{1}{(a_{1} - a_{2}) (a_{2} - x)} = \frac{1}{a_{1} - a_{2}} (\frac{- 1}{a_{1} - x} + \frac{1}{a_{2} - x})

Integration:

\begin{array}{l} \frac{1}{a_{1} - a_{2}} \cdot \int_{0}^{x} \frac{- 1}{a_{1} - x} + \frac{1}{a_{2} - x} = k \cdot \int_{0}^{t} d t \\ \frac{1}{a_{1} - a_{2}} [- ln \frac{a_{1}}{a_{1} - x} + ln \frac{a_{2}}{a_{2} - x}] = k t \end{array}

\frac{1}{a_{1} - a_{2}} \cdot ln \frac{a_{2} (a_{1} - x)}{a_{1} (a_{2} - x)} = k t

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