# Saturn S-IVB math

https://en.m.wikipedia.org/wiki/Saturn_V

https://physics.stackexchange.com/questions/88145/why-are-rockets-so-big/88163#88163

User Asad posted a graph based on rocket science.

https://i.stack.imgur.com/hZ8Tn.png

His graph indicates that the dry mass of a rocket cannot exceed 100 tons. Is this correct? This only applies to rockets which reach escape velocity.

I am asking because Saturn V had a significantly larger dry mass relatively. So this result would suggest that the rocket was incompetently designed or fake.

According to Wikipedia the Saturn V third stage had an exhaust velocity of 4km/s and burn rate of 0.2 tons/sec. These together make the math over 10x worse than the estimates Asad used and reduce the maximal dry mass to less than 10 tons. SIVB was over 40 tons with payload. This is with the SIVB orbiting at 191 km and 8km/s relative to earth when it attempted to leave orbit.

• The equations that Asad is using in the QA refer specifically to vertical liftoff from Earth's surface (hyperphysics.phy-astr.gsu.edu/hbase/rocket2.html#c1), where indeed an S-IVB performs rather poorly, as it doesn't have enough thrust to lift its own weight. The analysis doesn't hold for a stage already safely in orbit. – Russell Borogove Jun 8 '18 at 0:33
• I think you are talking about this answer from @AsadSaeeduddin but I think the maximum $m_r$ (rocket mass or dry mass) is for the specified burn rate of 8640 kg/s which sets the scale. A different burn rate would define a different rocket mass. – uhoh Jun 8 '18 at 0:39
• The linked answer appears to describe only a single-stage-to-orbit configuration. The third stage of a multi-stage rocket has no very obvious relationship to this conclusion, therefore. – Nathan Tuggy Jun 8 '18 at 1:02
• By the way, here's another graph based on rocket science: thespacereview.com/archive/3277d.jpg – Nathan Tuggy Jun 8 '18 at 1:04
• "So this result would suggest that the rocket was incompetently designed or fake." History suggests otherwise. Have a downvote. – Organic Marble Jun 8 '18 at 1:19

The graph posted was for a single stage rocket launching vertically, and showed that there comes a point where adding fuel stops increasing the final speed.

The author concludes, that for an engine of 440 MN, the limit is "somewhere around $4 \cdot 10^6$ kg of fuel mass"

This point corresponds to the point at which the takeoff mass exceeds the thrust of the rocket. The red line curve only curves down because his equation allowed negative acceleration, whereas in the real world the pad would hold the rocket until it had burnt off enough fuel to lift off - so the red line would become horizontal instead of curving down. (the author accepted this in the chat)

In the real world, most rockets take off and quickly start flying at an angle from vertical, to reduce the gravity loss term $- g\left(\frac{m_f}{\dot m}\right)$.

To answer your question, if the SIVB had been designed to launch vertically from earth and reach escape velocity, then it was an incompetent design. If somebody showed it doing that, it must have been faked. Anybody trying to claim this is what NASA did has either severely misunderstood, or is being deliberately misleading, probably earning YouTube revenue for all the adverts targeted at the gullible who would believe it.

The plot in that answer shows the initial conditions:

$$v_e=4000 m/s$$

$$v_esc=11200 m/s$$

$$burnrate=8640 kg/s$$

According to Wikipedia the Saturn V 1st stage had a fuel mass of 2,160,000 kg and a burn time of 168 seconds, suggesting an average burn rate of 12,860 kg/s. I believe the initial burn rate had to be higher than that since there was usually an engine shutdown during first stage ascent to lower acceleration once some of the fuel burned off.

Also the exhaust velocity $v_e$ of 4000 m/s represents an Isp of 4000/9.81 = 408 seconds, which is not at all the Isp given for the first stage of 263 seconds.

So I think the 100 ton limit is not really representative of a Saturn V. It is only representative of a rocket who's initial burn rate is 8640 kg and who's exhaust velocity is 4000 m/s. I wonder if it might have been better to just leave a comment for @AsadSaeeduddin under that question and ask for the source of the 8640 kg/s and the 4000 m/s?