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Find the distance a rocket travels in a given time

by RoRi

Solve the previous exercise (Section 8.19, Exercise #11) if we are given that the initial speed of the rocket is v_0 and if the discharged material is fired so that it remains at rest in space.

We are given the equations

    \[ F(t) = 0, \quad c(t) = -v(t) = -r'(t), \quad m(t) = \frac{w-kt}{g} \implies m'(t) = -\frac{k}{g}. \]

Therefore,

    \begin{align*} &&\left( \frac{w-kt}{g} \right) r''(t) &= \frac{k}{g} r'(t) \\ \implies && \frac{r''(t)}{r'(t)} &= \frac{k}{w-kt} \\ \implies && \log r'(t) &= \int \frac{k}{w-kt} \, dt \\ \implies && \log r'(t) &= -\log (w - kt) + A_0 \\ \implies && r'(t) &= A \left( \frac{1}{w-kt} \right). \end{align*}

(where A = e^{A_0} is a constant). Since r'(0) = v_0 we have A = w v_0. Therefore,

    \begin{align*} &&r'(t) &= \frac{wv_0}{w-kt} \\ \implies && r(t) &= \int \frac{wv_0}{w-kt} \, dt \\ &&&= (wv_0) \left( \frac{-1}{k} \right) \log (w-kt) + B. \end{align*}

Since r(0) = 0 we have B = \frac{wv_0}{k} \log w so

    \begin{align*} r(t) &= -\frac{wv_0}{k} \log (w-kt) + \frac{wv_0}{k} \log w \\ &= \frac{wv_0}{k} \log \left( \frac{w}{w-kt} \right). \end{align*}

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