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{\bf Question}

A rocket emits mass (fuel) at a rate $\alpha$ with a speed $u$
relative to the rocket.
\begin{description}
\item[(a)] Show that the thrust exerted on the rocket by its
engines is $u\alpha$.

\item[(b)] If the rocket has initial mass on the launch pad of
$m_0$ what is the minimum exhaust velocity that will allow it to
lift off \emph{immediately?}

\end{description}

\vspace{.25in}

{\bf Answer}

\begin{description}
\item[(a)]
Thrust is the rate of change of momentum of the rocket.

$\ds T = m\frac{dv}{dt} = -u \frac{dm}{dt}$ (as $m dv = u dm$)

$\ds \Rightarrow T = u\alpha$ as $\ds \frac{dm}{dt} = -\alpha$

\item[(b)]

\begin{center}
\begin{tabular}{c}
\epsfig{file=215-4-1.eps, width=30mm}
\end{tabular}
\ \
\begin{tabular}{c}
For immediate start the\\
acceleration of the rocket must be $\geq
0$\\
Therefore $\ds T - m_0g \geq 0$ \\
$\Rightarrow u \geq \frac{m_0g}{\alpha}$
\end{tabular}
\end{center}

\end{description}



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