Home » Blog » Find the distance a rocket travels in a given time

Find the distance a rocket travels in a given time

A rocket of weight w pounds starts at rest. It moves in a straight line, consuming fuel at a constant rate of k pounds per second. Material is discharged directly backward at a constant speed of c feet per second relative to the rocket. Find the distance the rocket travels in time t.


We are given the following information

    \begin{align*}  m(t) &= \frac{w-kt}{g}, \\  c(t) &= -c, \\  F(t) &= 0. \end{align*}

This implies

    \begin{align*}  \left( \frac{w-kt}{g} \right) r''(t) = \frac{kc}{g} && \implies && r''(t) &= \frac{kc}{w-kt} \\  && \implies && r'(t) &= -c \log |w -kt| + A. \end{align*}

Since r'(0) = 0 we then have A = c \log |w|. Therefore,

    \begin{align*}  && r'(t) &= -c \log \left| \frac{w-kt}{w} \right| \\  \implies && r(t) &= -\frac{c(kt-w)}{k} \cdot \log \left( \frac{w-kt}{w} \right) + \frac{c(kt-w)}{k} + B. \end{align*}

Since r(0) = 0 we have B = \frac{cw}{k}. So,

    \begin{align*}  r(t) &= - \frac{c(k-w)}{k} \cdot \log \left| 1 - \frac{kt}{w} \right| + ct \\  &= ct + c \left( \frac{w}{k} - t \right) \log \left( 1 - \frac{kt}{w} \right). \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):