'Sup, Nerds?!

In a recent post, I calculated the \(\Delta v\) of an unladen Saturn V rocket and concluded that the maximum speed of such a vehicle, when launching from Earth, is over \(36,000\,\mathrm{mph}\). As I explained then, that is fast, but not fast enough for our purpose! Furthermore, as is typically the case, we want to use our rocket to carry something into space, and the addition of a payload will slow us down even more!

That fact should make intuitive sense, for we all know from experience that it becomes harder and harder to lift heavier and heavier things. This rule applies to rockets as much as it does to people, but we can prove the claim more rigorously using the rocket equation:\begin{equation} \Delta v = I_{sp} \, g_0\ln \left(\frac{m_i}{m_f}\right). \label{081117TreIsp} \end{equation}

When our rocket is "unladen," its initial mass is its wet mass and its final mass is its dry mass:\begin{equation*}
\begin{array}{c}
m_i = m_W,
\\[1.5ex]
m_f = m_D.
\end{array}
\end{equation*}Adding a payload with mass \(m_{PL}\), however, increases the mass of our rocket overall:\begin{equation*}
\begin{array}{c}
m_i = m_W + m_{PL},
\\[1.5ex]
m_f = m_D + m_{PL}.
\end{array}
\end{equation*}

It can be shown that if the numerator and denominator of a fraction are both positive, adding the same positive constant to both of them makes the total ratio closer to \(1\). Therefore, if we increase the initial and final mass of our rocket by the same amount, then \(m_i / m_f\) tends closer to unity. This has an unfortunate consequence on our \(\Delta v\), given the nature of the logarithm:

Figure (1): A plot of the natural logarithm. Notice that \(\mathrm{ln}(1) = 0\). Therefore, as \(m_i/m_f \rightarrow 1,\, \Delta v \rightarrow 0\).

So then, as we add more payload to our rocket, we get less \(\Delta v\) and therefore less maximum speed out of it... assuming our rocket even gets off the ground!

Therein lies our other problem: if we are launching this vehicle from Earth, we have to worry about its thrust-to-weight ratio. The thrust of a rocket is essentially the force produced by its engines, while its weight is the force exerted on it by Earth's surface gravity. If its thrust is less than its weight, then the rocket will not fly. Again, this should be intuitive if you have ever tried and failed to lift something that is really, really heavy. That occurred because the lifting force you exerted upon the object was less than its weight. Lifting objects off the ground means overcoming the pull of gravity, both for you and for rockets!

Figure (2): The arrows labeled T and W represent the thrust and weight of each rocket, respectively. a) The red rocket weighs more than it can lift, and so it will not go to space today. b) The green rocket weighs less than it can lift, and so it wi… — Figure (2): The arrows labeled T and W represent the thrust and weight of each rocket, respectively. a) The red rocket weighs more than it can lift, and so it will not go to space today. b) The green rocket weighs less than it can lift, and so it will go up! Rockets adapted from marauder.

That means there is a limit to the mass of our rocket's payload—it cannot make the vehicle too heavy to get off the ground! Going back to our Saturn V, at its peak, the five F-1 engines of its first stage collectively produced around 7.6 million pounds of thrust at sea level:

Figure (3): Image from NASA via Wikipedia.

Using the latter thrust value in Figure (3), we shall say that our Saturn V produces \(7,610,000\,\mathrm{lb}\) of thrust. Since we calculated it total initial weight to be \(6,363,220\,\mathrm{lb}\), the weight of its payload cannot exceed\begin{equation} 7,610,000\,\mathrm{lb} - 6,363,220\,\mathrm{lb} = 1,246,780\,\mathrm{lb}. \label{081217Mpl} \end{equation}How does this value stack up against a typical Saturn V stack?

Figure (4): The usual payload of a Saturn V en route to the Moon! Image from NASA via Wikipedia.

According to the table we used to calculate the weights of the rocket at different stages, an Apollo spacecraft could weigh just over \(116,000\,\mathrm{lb}\), which is well under the 1.2 million pound limit we calculated in \eqref{081217Mpl} above. That was sufficient for supporting three astronauts on a trip to the Moon and back lasting up to two weeks. However, the mission that I have in mind will require some adjustments... Stay tuned to see what that entails!

May all your rockets go up,

Aaron