'Sup Nerds?!

For an upcoming blog post, I need to know the maximum speed that can be achieved by a Saturn V rocket—for SCIENCE!

Figure \((1)\): Behold—THE BEST rocket—from back when America went places... Image Credit: NASA

Unfortunately, this does not seem to be something that one can merely look up online, though there are some good estimates out there. However, for my purposes, I need a slightly deeper understanding of what is going on and control over things like payload mass and launch configurations. Alas, this means I must do the calculations myself!

Fortunately, this is actually pretty easy. Despite the popular conception of rocket science being synonymous with "really difficult," rockets themselves are fundamentally simple objects: they throw stuff out the back as fast as they can, and through the conservation of linear momentum, this pushes them forward. In fact, way back in my first term of freshman physics as an undergraduate, we used momentum conservation and a little bit of calculus to drive the Tsiolkovsky rocket equation, which tells you how much a rocket's speed will change (\(\Delta v\)) after it has thrown a certain amount of mass (\(m_f - m_i\)) out of its nozzles at a particular exhaust speed (\(v_e\)), assuming no external forces act upon it:\begin{equation}
\Delta v = v_e \ln \left(\frac{m_i}{m_f}\right).
\label{080717Tre} \end{equation}

Often, this equation is written not in terms of exhaust speed out the nozzle, but instead using a quantity called specific impulse, \(I_{sp}\). These two quantities are related via\begin{equation} v_e = I_{sp} \, g_0, \label{080717Isp} \end{equation} where \(g_0\) is the standard gravitational acceleration at Earth's surface, \(9.8\,\mathrm{m}/\mathrm{s}^2\). Upon combining equations \eqref{080717Tre} and \eqref{080717Isp}, we find\begin{equation}
\Delta v = I_{sp} \, g_0\ln \left(\frac{m_i}{m_f}\right).
\label{080717TreIsp} \end{equation}

So then, we must now find the wet and dry masses (with fuel and without fuel, respectively) of the three Saturn V stages, their specific impulses, and the mass of the rocket's payload. Fortunately, the Internet readily provides all of this information:

Figure \((2)\): Saturn V stage diagram. Image from the National Air and Space Museum.

The table above contains all of the mass information we need for our \(\Delta v\) calculations. Due to the variety of figures for different missions, I am averaging together the relevant values from Apollos 8–17, all of which flew on Saturn V's. (Apollo 7 launched aboard a Saturn IB.) I am also including the interstage masses with the dry and total masses of the stages located beneath them, since these adapters are jettisoned after those stages but before ignition of the stages above them. Similarly, the instrument unit and spacecraft adapter contribute to the dry and total masses of the S-IVB stage. With these values and specific impulses from Wikipedia, we obtain\begin{equation}
\begin{array}{l|rrl}
\text{Stage} & \text{Wet Mass (lbs)} & \text{Dry Mass (lbs)} & I_{sp} \, \text{(s)} \\ \hline
\text{S-IC} & 5,017,760 & 301,633 & 263^* \\
\text{S-II} & 1,077,830 & 89,391 & 421 \\
\text{S-IVB} & 267,628 & 29,648 & 421
\end{array}
\label{080717TwdI} \end{equation}

*This is the specific impulse of the S-IC stage at sea level; all other \(I_{sp}\)'s are reported in vacuum.

Assuming our Saturn V is not lofting an Apollo spacecraft to the Moon (and is, therefore, "unladen"), we can calculate its maximum speed as follows:\begin{equation}
\begin{array}{l|rrlrr}
\text{Stage} & m_i\,\mathrm{(lbs)} & m_f\,\mathrm{(lbs)} & I_{sp}\,\mathrm{(s)} & \text{Stage}\,\Delta v\,(\mathrm{m/s}) & \text{Total}\,\Delta v\,(\mathrm{m/s}) \\ \hline
\text{I} & 6,363,220 & 1,647,100 & 263^* & 3,483& 3,483 \\
\text{II} & 1,345,460 & 357,019 & 421 & 5,474 & 8,957 \\
\text{III} & 267,628 & 29,648 & 421 & 9,078 & 18,035
\end{array}
\label{070817TDvU}\end{equation}

According to table \eqref{070817TDvU}, the maximum speed of an unladen Saturn V is just over \(18\,\mathrm{km/s}\). However, this figure ignores the fact that the rocket must climb out of Earth's gravity well on its way up! That's not even to mention the drag of Earth's atmosphere, which should not be ignored if we are using the \(I_{sp}\) of the first stage at sea level. The problem is, \(\Delta v\) loses due to gravity and air resistance depend upon the specific characteristics of the rocket itself and are not easily found online. Fortunately, the Saturn V is famous enough that somebody went and simulated one's launch in very minute detail! According to him:

Figure \((4)\): Thank you, Robert A. Braeunig! — Figure \((4)\): Thank you, Robert A. Braeunig!

Accounting for these losses, our final \(\Delta v\) becomes\begin{equation}
18,035\,\mathrm{m/s} - 1,791\,\mathrm{m/s} = 16,244\,\mathrm{m/s}.
\label{080817Dvf}\end{equation}At the end of the day, an unladen Saturn V launching from Earth winds up traveling just over \(16.2\,\mathrm{km/s}\), or \(10\,\mathrm{mi/s} = 36,336\,\mathrm{mph}\) for 'Mericans. That is fast, but not fast enough for what I have in mind, so we're gonna need a bigger rocket...

Stay tuned!

Aaron