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From the data in .7 you can find the space shuttle's

Chapter 3, Problem 3.9

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QUESTION:

From the data in Problem 3.7 you can find the space shuttle's initial mass and the rate of ejecting mass \(-\dot{m}\) (which you may assume is constant). What is the minimum exhaust speed \(v_{\mathrm{ex}}\) for which the shuttle would just begin to lift as soon as burn is fully underway? [Hint: The thrust must at least balance the shuttle's weight.]

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QUESTION:

From the data in Problem 3.7 you can find the space shuttle's initial mass and the rate of ejecting mass \(-\dot{m}\) (which you may assume is constant). What is the minimum exhaust speed \(v_{\mathrm{ex}}\) for which the shuttle would just begin to lift as soon as burn is fully underway? [Hint: The thrust must at least balance the shuttle's weight.]

ANSWER:

Step 1 of 2

The maximum exhaust speed is the speed of exhaust required to just start the rocket. In this case, the weight of the rocket is balanced by thrust.

\(\begin{array}{l}
m g=-m v_{\text {ex }} \\
v_{\text {ex }}=\frac{m g}{-m}
\end{array}\)

Find the mass \(\dot m\) as follows.

\(\begin{aligned}
\dot m & =\frac{\left(1 \times 10^{6} \mathrm{~kg}\right)-\left(2 \times 10^{6} \mathrm{~kg}\right)}{(2 \mathrm{~min})\left(\frac{60 \mathrm{~s}}{1 \mathrm{~min}}\right)} \\
& =-8333 \mathrm{~kg} / \mathrm{s}
\end{aligned}\)

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