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Comparing Artemis SLS to Saturn V and Falcon heavy

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This week, the Artemis I, Orion capsule splashed down to general applause after circling the moon with mannequins. The launch cost $4.1 Billion, and the project, $50 Billion so far, of $93 Billion expected. Artemis II will carry people around the moon, and Artemis III is expected to land the first woman and person of color. The goal isn’t one I find inspiring, and I feel even less inspired by the technology. I see few advances in Artemis compared to the Saturn V of 50 years ago. And in several ways, it looks like a step backwards.

The graphic below compares the Artemis I SLS (Space Launch System) to the Saturn V. The SLS is 10% lighter, but the payload is lighter, too. It can carry 27 tons to the moon, while the Saturn V sent 50 tons to the moon. I’d expect more weight by now. We have carbon fiber and aramids, and they did not. Add to this that the cost per flight is higher, $4.1 B, versus $1.49 B in 2022 dollars for a Saturn V ($185 million in 1969 dollars). What’s more there was no new engine development or production, so the flight numbers are limited: Each SLS launch throws away five, space shuttle engines. When they are all gone, the project ends. We have no plans or ability to make more engines.

Comparison of Apollo Saturn V and Artemis SLS. The SLS has less lift weight and costs more per launch.

As it happens, there was a better alternative available, the Falcon heavy from SpaceX. The Falcon heavy has been flying for 5 years now, and costs only $141 million per launch, about 1/30 as much as an Artemus launch. The rocket is largely reusable, with 3D printed engines, and boosters that land on their tails. Each SLS is expensive because it’s essentially a new airplane built specially for each flight. Every part but the capsule is thrown away. Adding to the cost of SLS launches is the fuel; hydrogen, the same fuel as the space shuttle. Per energy it’s very expensive. The energy cost for the SLS boosters is high too, and the efficiency is low; each SLS booster costs $290M, more than the cost of two Falcon heavy launches. Falcon launches are cheap, in part because the engines burn kerosine, as did the Saturn V at low altitude. Beyond cost hydrogen has low thrust per flow (low momentum), and is hard to handle; hydrogen leaks caused two Artemis scrubs, and numerous Shuttle delays. I discussed the physics of rocket engines in a post seven years ago.

This graph of $/kg to low earth orbit is mostly from futureblind.com. I added the data for Artemis SLS. Saturn V and Falcon use cheaper fuel and a leaner management team.

It might be argued that Artemis SLS is an inspirational advance because it can lift an entire moon project in one shot, but the Saturn V lifted that and more, all of Skylab. Besides, there is no need to lift everything on one launch. Elon Musk has proposed lifting in two stages, sending the moon rocket and moon lander to low earth orbit with one launch, then lifting fuel and the astronauts on a second launch. Given the low cost of a Falcon heavy launch, Musk’s approach is sure to save money. It also helps develop space refueling, an important technology.

Musk’s Falcon may still reach the moon because NASA still needs a moon lander. NASA has awarded the lander contract to three companies for now, Jeff Bezos’s Blue Origin, Dynetics-Aerodyne makers of the Saturn V, and Musk’s SpaceX. If the SpaceX version wins, a modified Falcon will be sent to the moon on a Falcon heavy along with a space station. Artemis III will rendezvous with them, astronauts will descend to the moon on the lander, and will use the lander to ascend. They’ll then transfer to an Orion capsule for the return journey. NASA has also contracted with Bezos’s Blue origin for planetary, Earth observation, and exploration plans. I suspect that Musk’s lander will win, if only because of reliability. There have been 59 Falcon launches this year, all of them with safe landings. By contrast, no Blue Origin or Dynetics rocket has landed, and Blue Origin does not expect to achieve orbital velocity till 2025.

As best I can tell, the reason we’re using the Artemis SLS with its old engines is inspiration. The Artemis program director, Charlie Blackwell-Thompson is female, and an expert in space shuttle engines. Previous directors were male. Previous astronauts too were mostly male. Musk is not only male, but his products suffer from him being considered a horrible person, a toxic male, in the Tony Stark (Iron Man) mold. Even Jeff Bezos and Richard Branson are considered better, though their technology is worse. See my comparison of SpaceX, Virgin Blue, and Blue Origin.

To me, the biggest blocks to NASA’s inspirational aims, in my opinion, are the program directors who gave us the moon landing. These were two Nazi SS commanders (SS Sturmbannführers), Arthur Rudolph and Wernher Von Braun. Not only were they male and white, they were barely Americanized Nazis, elevated to their role at NASA after killing off virtually all of their 20,000, mostly Jewish, slave workers making rockets for Hitler. Here’s a song about Von Braun, by Tom Lehrer. Among those killed was Von Braun’s professor. In his autobiography, Von Braun showed no sign of regret for any of this, nor does he take blame. The slave labor camp they ran, Dora-Mittelbau, had the highest death rate of all slave labor camps, and when some workers suggested that they could work better if they were fed, the directors, Rudolph and Von Braun had 80 machine gunned to death. Still, Von Braun got us to the moon, and his inspirational comments line the walls at NASA, Kennedy. Blackwell-Thompson and Bezos are surely more inspirational, but their designs seem like dead ends. We may still have to use Musk’s SpaceX if we want a lander or a moon program after the space shuttle’s engines are used up. As Von Braun liked to point out, “Sacrifices have to be made.”

Robert Buxbaum, December 21, 2022. Here’s a bit more about Rudolph, von Braun, the Peenemünda rocket facility, and the Dora-Mittelbau slave labor camp. I may post photos of Von Braun with Hitler and Himmler in SS regalia, but feel uncomfortable doing so at the moment. I feel similarly about posting links to Von Braun’s inspirational interviews.

Branson’s virgin spaceplane in context.

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Virgin Galactic Unity 22, landing.

Branson’s Virgin Space Ship (VSS) Unity was cheered as a revolutionary milestone today (July 10) after taking Branson, three friends and two pilots on a three minute ride to the edge of space, an altitude of 53.5 miles or 283,000 feet. I’d like to put that achievement into contest, both with previous space planes, like the Concorde and X-15 (the 1960s space plane), and also in context with the offerings of Elon Musk’s Space-X and Bezos’s, Blue Horizon.

To start with, the VSS Unity launched from a sub-sonic mother ship, as the X-15 had before it. This saves a lot in fuel weight and safety equipment, but it makes scale up problematic. In this case, the mother-ship was named Eve. Unity launched from Eve at 46,000 feet, about 9 miles up, and at Mach 0.5; it took Eve nearly 90 minutes to get to altitude and position. It was only after separation, that Unity began a one minute, 3 G rocket burn that brought it to its top speed, Mach 3, at about 16 miles up. What followed was a 3 minute, unpowered glide to 53.5 miles and down. Everyone seems to have enjoyed the three minutes of weightlessness, and it should be remembered that there is a lot of difference between Mach 3 and orbital speed, Mach 31. Also there is a lot of difference between a sub-orbital and orbital.

Concorde SST landing in Farnborough.

By comparison, consider the Concorde SSTs that first flew in 1976. It reached about 2/3 the speed of Unity, Mach 2.1, but carried 120 commercial passengers. It took off from the ground and maintained this speed for 4500 miles, going from London to Houston in 4.5 hours. While the Concorde only reached an altitude of 60,000 feet, it is far more impressive going at Mach 2.1 for 4.5 hours than going at Mach 3 for three minutes. And there is a lot of difference between 120 passengers and 4. There is also the advantage of taking off from the ground. A three minute ride in a space plane should not require a 90 minute ascent on a mother ship.

X-15 landing, 1962.

Next consider the X-15 rocket plane of the 1960s. This was a test platform devoted to engine and maneuverability tests; it turns out that maneuverability is very difficult. The X-15 hit a maximum altitude of 354,200 ft, 67 miles, and a maximum speed of Mach 6.72, or 4520 mph. That’s significantly higher than Branson’s VSS, and double the maximum speed. As an aside, the X-15 project involved the development of a new nickel alloy that I use today, Inconel X-750. I use this as a support for my hydrogen membranes. If any new materials were developed for VSS, none were mentioned.

The Air Force’s X-37B Orbital Test Vehicle at Kennedy Space Center, May 7, 2017.

Continuing with the history of NASA’s X-program, we move to the X-41, a air-breathing scramjet of the 1980s and 90s. It reached 95,000 feet, and a maximum speed of Mach 9.64. That’s about three times as fast as Virgin’s VSS. The current X-plane is called X-37B, it is a rocket-plane like the X-15 and VSS, but faster and maneuverable at high speed and altitude. It’s the heart of Trump’s new, US Space force. In several tests over the past 5 years, it has hit orbital speed, 17,426 mph, Mach 31, and orbital altitudes, about 100 miles, after being launched by a Atlas V or a Falcon 9 booster. The details are classified. Apparently it has maneuverability. While the X-37B is unmanned, a larger, manned version, is being built, the X-37C. It is supposed to carry as many as six.

Reaching orbital speed or Mach 31 implies roughly 100 times as much kinetic energy per mass as reaching the Mach 3.1 of Virgin’s VSS. In this sense, the space shuttle, and the current X-plane are 100 times more impressive than Virgin’s VSS. There is also a lot to be said for maneuverability and for a longer flight duration– more than a few minutes. Not that I require Branson to beat NASA’s current offerings, but I anyone claiming cutting edge genius and visionary status should at least beat NASA’s offerings of the 1960s, and the Concorde planes of 1976.

Bezos’s Blue Origin, and the New Shepard launcher.

And that bring’s us to the current batch of non-governmental, space cadets. Elon Musk stands out to me as a head above the rest, at least. Eight years ago, his Grasshopper rocket premiered the first practical, example of vertical take off and landing booster. Today, his Falcon 9 boosters send packages into earth orbit, and beyond, launching Israel’s moon project, as one example. That implies speeds of Mach 31 and higher, at least at the payload. It’s impressive, even compared to X-37, very impressive.

Bezos’ offering, the Blue Origin Shepherd, seems to me like a poor imitation of the SpaceX Falcon. Like Falcon, it’s a reusable, vertical takeoff and landing platform, that launches directly from earth, and like Falcon it carries a usable payload, but it only reaches speeds of Mach 3 and altitudes about 65 miles. Besides, the capsule lands by way of parachutes, not using wings like the space shuttle, or the X-37B, and there is no reusable booster like Falcon. Blue Origin started carrying payloads only in 2019, five yers after SpaceX. There is nothing here that’s cutting edge, IMHO, and I don’t imagine it will be cheaper either.

Branson has something that the other rocket men do not have, quite: a compelling look: personal marketing, a personal story, and a political slant that the press loves and I find hypocritical and hokey. The press, and our politicians, managed to present this flight as more than an energy wasting, joy ride for rich folks. Instead, this is accepted as Branson’s personal fight against climate change. Presented this way, it should qualify as a tax-dodge. I don’t see it getting folks to stop polluting and commit to small cars, but the press is impressed, or claims to be. The powers have committed themselves to this type of Tartuffe, and the press goes along. You’d think that, before giving Branson public adoration for his technology or environmentalism, he should have cutting technology and have been required to save energy, or pollute less. At least beat the specs of the X-15. Just my opinion.

Robert Buxbaum, July 12, 2021

Let’s visit an earth-like planet: Trappist-1d

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According to Star Trek, Vulcans and Humans meet for the first time on April 5, 2063, near the town of Bozeman, Montana. It seems that Vulcan is a relatively nearby, earth-like planet with strongly humanoid inhabitants. It’s worthwhile to speculate why they are humanoid (alternatively, how likely is it that they are), and also worthwhile to figure out which planets we’d like to visit assuming we’re the ones who do the visiting.

First things first: It’s always assumed that life evolved on earth from scratch, as it were, but it is reasonably plausible that life was seeded here by some space-traveling species. Perhaps they came, looked around and left behind (intentionally or not) some blue-green algae, or perhaps some more advanced cells, or an insect or two. A billion or so years later, we’ve evolved into something that is reasonably similar to the visiting life-form. Alternately, perhaps we’d like to do the exploring, and even perhaps the settling. The Israelis are in the process of showing that low-cost space travel is a thing. Where do we want to go this century?

As it happens we know there are thousands of stars with planets nearby, but only one that we know that has reasonably earth-like planets reasonably near. This one planet circling star is Trappist-1, or more properly Trappist 1A. We don’t know which of the seven planets that orbit Trappist-1A is most earth-like, but we do know that there are at least seven planets, that they are all roughly earth size, that several have earth-like temperatures, and that all of these have water. We know all of this because the planetary paths of this star are aligned so that seven planets cross the star as seen from earth. We know their distances from their orbital times, and we know the latter from the shadows made as the planets transit. The radiation spectrum tells us there is water.

Trappist 1A is smaller than the sun, and colder than the sun, and 1 billion years older. It’s what is known as an ultra-cool dwarf. I’d be an ultra cool dwarf too, but I’m too tall. We can estimate the mass of the star and can measure its brightness. We then can calculate the temperatures on the planets based their distance from the star, something we determine as follows:

The gravitational force of a star, mass M, on a planet of mass, m,  is MmG/r2, where G is the gravitational constant, and r is the distance from the star to the planet. Since force = mass times acceleration, and the acceleration of a circular orbit is v2/r, we can say that, for these orbits (they look circular),

MmG/r2 = mv2/r = mω2r.

Here, v is the velocity of the planet and ω is its rotational velocity, ω = v/r. Eliminating m, we find that

r3 = MG/ω2.

Since we know G and ω, and we can estimate M (it’s 0.006 solar masses, we think), we have a can make good estimates of the distances of all seven planets from their various rotation speeds around the star, ω. We find that all of these planets are much closer to their star than we are to ours, so the their years are only a few days or weeks long.

We know that three planets have a temperatures reasonably close to earths, and we know that these three also have water based on observation of the absorption of light from their atmosphere as they pass in front of their star. To tell the temperature, we use our knowledge of how bright the star is (0.0052 times Sol), and our knowledge of the distance. As best we can tell, the following three of the Trappist-1 planets should have liquid surface water: Trappist 1c, d and e, the 2nd, 3rd and 4th planets from the star. With three planets to choose from, we can be fairly sure that at least one will be inhabitable by man somewhere in the planet.

The seven orbital times are in small-number ratios, suggesting that the orbits are linked into a so-called Laplace resonance-chain. For every two orbits of the outermost planet, the next one in completes three orbits, the next one completes four, followed by 6, 9 ,15, and 24. The simple whole number relationships between the periods are similar to the ratios between musical notes that produce pleasant and harmonic sounds as I discussed here. In the case of planets, resonant ratios keep the system stable. The most earth-like of the Trappist-1 planets is likely Trappist-1d, the third planet from the star. It’s iron-core, like earth, with water and a radius 1.043 times earth’s. It has an estimated average temperature of 19°C or 66°F. If there is oxygen, and if there is life there could well be, this planet will be very, very earth-like.

The temperature of the planet one in from this, Trappist-1c, is much warmer, we think on average, 62°C (143°F). Still, this is cool enough to have liquid water, and some plants live in volcanic pools on earth that are warmer than this. Besides this is an average, and we might the planet quite comfortable at the poles. The average temperature of the planet one out from this, Trappist-1e, is ice cold, -27°C (-17°F), an ice planet, it seems. Still, life can find a way. There is life on the poles of earth, and perhaps the plant was once warmer. Thus, any of these three might be the home to life, even humanoid life, or three-eyed, green men.

Visiting Trappist-1A won’t be easy, but it won’t be out-of hand impossible. The system is located about 39 light years away, which is far, but we already have a space ship heading out of the solar system, and we are developing better, and cheaper options all the time. The Israeli’s have a low cost, rocket heading to the moon. That is part of the minimal technology we’d want to visit a nearby star. You’d want to add enough rocket power to reach relativistic speeds. For a typical rocket this requires a fuel whose latent energy is on the order mc2. That turns out to be about 1 GeV/atomic mass. The only fuel that has such high power density is matter-antimatter annihilation, a propulsion system that might have time-reversal issues. A better option, I’d suggest is ion-propulsion with hydrogen atoms taken in during the journey, and ejected behind the rocket at 100 MeV energies by a cyclotron or bevatron. This system should work if the energy for the cyclotron comes from solar power. Perhaps this is the ion-drive of Star-Trek fame. To meet the Star-Trek’s made-up history, we’d have to meet up by April, 2063: forty-four years from now. If we leave today and reach near light speed by constant acceleration for a few of years, we could get there by then, but only as time is measured on the space-ship. At high speeds, time moves slower and space shrinks.

This planetary system is named Trappist-1 after the telescope used to discover it. It was the first system discovered by the 24 inch, 60 cm aperture, TRAnsiting Planets and PlanetesImals Small Telescope. This telescope is operated by The University of Liége, Belgium, and is located in Morocco. The reason most people have not heard of this work, I think, has to do with it being European science. Our news media does an awful job covering science, in my opinion, and a worse job covering Europe, or most anything outside the US. Finally, like the Israeli moon shot, this is a low-budget project, the work to date cost less than €2 million, or about US $2.3 million. Our media seems committed to the idea that only billions of dollars (or trillions) will do anything, and that the only people worth discussing are politicians. NASA’s budget today is about $6 billion, and its existence is barely mentioned.

The Trappist system appears to be about 1 billion years older than ours, by the way, so life there might be more advanced than ours, or it might have died out. And, for all we know, we’ll discover that the Trappist folks discover space travel, went on to colonize earth, and then died out. The star is located, just about exactly on the ecliptic, in the constellation Aquarius. This is an astrological sign associated with an expansion of human consciousness, and a revelation of truths. Let us hope that, in visiting Trappist, “peace will guide the planets and love will steer the stars”.

Robert Buxbaum, April 3, 2019. Science sources are: http://www.trappist.one. I was alerted to this star’s existence by an article in the Irish Times.

It’s rocket science

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Here are six or so rocket science insights, some simple, some advanced. It’s a fun area of engineering that touches many areas of science and politics. Besides, some people seem to think I’m a rocket scientist.

A basic question I get asked by kids is how a rocket goes up. My answer is it does not go up. That’s mostly an illusion. The majority of the rocket — the fuel — goes down, and only the light shell goes up. People imagine they are seeing the rocket go up. Taken as a whole, fuel and shell, they both go down at 1 G: 9.8 m/s2, 32 ft/sec2.

Because 1 G ofupward acceleration is always lost to gravity, you need more thrust from the rocket engine than the weight of rocket and fuel. This can be difficult at the beginning when the rocket is heaviest. If your engine provides less thrust than the weight of your rocket, your rocket sits on the launch pad, burning. If your thrust is merely twice the weight of the rocket, you waste half of your fuel doing nothing useful, just fighting gravity. The upward acceleration you’ll see, a = F/m -1G where F is the force of the engine, and m is the mass of the rocket shell + whatever fuel is in it. 1G = 9.8m/s is the upward acceleration lost to gravity.  For model rocketry, you want to design a rocket engine so that the upward acceleration, a, is in the range 5-10 G. This range avoids wasting lots of fuel without requiring you to build the rocket too sturdy.

For NASA moon rockets, a = 0.2G approximately at liftoff increasing as fuel was used. The Saturn V rose, rather majestically, into the sky with a rocket structure that had to be only strong enough to support 1.2 times the rocket weight. Higher initial accelerations would have required more structure and bigger engines. As it was the Saturn V was the size of a skyscraper. You want the structure to be light so that the majority of weight is fuel. What makes it tricky is that the acceleration weight has to sit on an engine that gimbals (slants) and runs really hot, about 3000°C. Most engineering projects have fewer constraints than this, and are thus “not rocket science.”

Basic force balance on a rocket going up.

Basic force balance on a rocket going up.

A space rocket has to reach very high, orbital speed if the rocket is to stay up indefinitely, or nearly orbital speed for long-range, military uses. You can calculate the orbital speed by balancing the acceleration of gravity, 9.8 m/s2, against the orbital acceleration of going around the earth, a sphere of 40,000 km in circumference (that’s how the meter was defined). Orbital acceleration, a = v2/r, and r = 40,000,000 m/2π = 6,366,000m. Thus, the speed you need to stay up indefinitely is v=√(6,366,000 x 9.8) = 7900 m/s = 17,800 mph. That’s roughly Mach 35, or 35 times the speed of sound at sea level, (343 m/s). You need some altitude too, just to keep air friction from killing you, but for most missions, the main thing you need is velocity, kinetic energy, not potential energy, as I’ll show below. If your speed exceeds 17,800 m/s, you go higher up, but the stable orbital velocity is lower. The gravity force is lower higher up, and the radius to the earth higher too, but you’re balancing this lower gravity force against v2/r, so v2 has to be reduced to stay stable high up, but higher to get there. This all makes docking space-ships tricky, as I’ll explain also. Rockets are the only way practical to reach Mach 35 or anything near it. No current cannon or gun gets close.

Kinetic energy is a lot more important than potential energy for sending an object into orbit. To get a sense of the comparison, consider a one kg mass at orbital speed, 7900 m/s, and 200 km altitude. For these conditions, the kinetic energy, 1/2mv2 is 31,205 kJ, while the potential energy, mgh, is only 1,960 kJ . The potential energy is thus only 1/16 the kinetic energy.

Not that it’s easy to reach 200 miles altitude, but you can do it with a sophisticated cannon. The Germans did it with “simple”, one stage, V2-style rockets. To reach orbit, you generally need multiple stages. As a way to see this, consider that the energy content of gasoline + oxygen is about 10.5 MJ/kg (10,500 kJ/kg); this is only 1/3 of the kinetic energy of the orbital rocket, but it’s 5 times the potential energy. A fairly efficient gasoline + oxygen powered cannon could not provide orbital kinetic energy since the bullet can move no faster than the explosive vapor. In a rocket this is not a constraint since most of the mass is ejected.

A shell fired at a 45° angle that reaches 200 km altitude would go about 800 km — the distance between North Korea and Japan, or between Iran and Israel. That would require twice as much energy as a shell fired straight up, about 4000 kJ/kg. This is still within the range for a (very large) cannon or a single-stage rocket. For Russia or China to hit the US would take much more: orbital, or near orbital rocketry. To reach the moon, you need more total energy, but less kinetic energy. Moon rockets have taken the approach of first going into orbit, and only later going on. While most of the kinetic energy isn’t lost, it’s likely not the best trajectory in terms of energy use.

The force produced by a rocket is equal to the rate of mass shot out times its velocity. F = ∆(mv). To get a lot of force for each bit of fuel, you want the gas exit velocity to be as fast as possible. A typical maximum is about 2,500 m/s. Mach 10, for a gasoline – oxygen engine. The acceleration of the rocket itself is this ∆mv force divided by the total remaining mass in the rocket (rocket shell plus remaining fuel) minus 1 (gravity). Thus, if the exhaust from a rocket leaves at 2,500 m/s, and you want the rocket to accelerate upward at an average of 10 G, you must exhaust fast enough to develop 10 G, 98 m/s2. The rate of mass exhaust is the average mass of the rocket times 98/2500 = .0392/second. That is, about 3.92% of the rocket mass must be ejected each second. Assuming that the fuel for your first stage engine is less than 80% of the total mass, the first stage will flare-out in about 20 seconds. Typically, the acceleration at the end of the 20 burn is much greater than at the beginning since the rocket gets lighter as fuel is burnt. This was the case with the Apollo missions. The Saturn V started up at 0.5G but reached a maximum of 4G by the time most of the fuel was used.

If you have a good math background, you can develop a differential equation for the relation between fuel consumption and altitude or final speed. This is readily done if you know calculous, or reasonably done if you use differential methods. By either method, it turns out that, for no air friction or gravity resistance, you will reach the same speed as the exhaust when 64% of the rocket mass is exhausted. In the real world, your rocket will have to exhaust 75 or 80% of its mass as first stage fuel to reach a final speed of 2,500 m/s. This is less than 1/3 orbital speed, and reaching it requires that the rest of your rocket mass: the engine, 2nd stage, payload, and any spare fuel to handle descent (Elon Musk’s approach) must weigh less than 20-25% of the original weight of the rocket on the launch pad. This gasoline and oxygen is expensive, but not horribly so if you can reuse the rocket; that’s the motivation for NASA’s and SpaceX’s work on reusable rockets. Most orbital rocket designs require three stages to accelerate to the 7900 m/s orbital speed calculated above. The second stage is dropped from high altitude and almost invariably lost. If you can set-up and solve the differential equation above, a career in science may be for you.

Now, you might wonder about the exhaust speed I’ve been using, 2500 m/s. You’ll typically want a speed at lest this high as it’s associated with a high value of thrust-seconds per weight of fuel. Thrust seconds pre weight is called specific impulse, SI, SI = lb-seconds of thrust/lb of fuel. This approximately equals speed of exhaust (m/s) divided by 9.8 m/s2. For a high molecular weight burn it’s not easy to reach gas speed much above 2500, or values of SI much above 250, but you can get high thrust since thrust is related to momentum transfer. High thrust is why US and Russian engines typically use gasoline + oxygen. The heat of combustion of gasoline is 42 MJ/kg, but burning a kg of gasoline requires roughly 2.5 kg of oxygen. Thus, for a rocket fueled by gasoline + oxygen, the heat of combustion per kg is 42/3.5 = 12,000,000 J/kg. A typical rocket engine is 30% efficient (V2 efficiency was lower, Saturn V higher). Per corrected unit of fuel+oxygen mass, 1/2 v2 = .3 x 12,000,000; v =√7,200,000 = 2680 m/s. Adding some mass for the engine and fuel tanks, the specific impulse for this engine will be, about 250 s. This is fairly typical. Higher exhaust speeds have been achieved with hydrogen fuel, it has a higher combustion energy per weight. It is also possible to increase the engine efficiency; the Saturn V, stage 2 efficiency was nearly 50%, but the thrust was low. The sources of inefficiency include inefficiencies in compression, incomplete combustion, friction flows in the engine, and back-pressure of the atmosphere. If you can make a reliable, high efficiency engine with good lift, a career in engineering may be for you. A yet bigger challenge is doing this at a reasonable cost.

At an average acceleration of 5G = 49 m/s2 and a first stage that reaches 2500 m/s, you’ll find that the first stage burns out after 51 seconds. If the rocket were going straight up (bad idea), you’d find you are at an altitude of about 63.7 km. A better idea would be an average trajectory of 30°, leaving you at an altitude of 32 km or so. At that altitude you can expect to have far less air friction, and you can expect the second stage engine to be more efficient. It seems to me, you may want to wait another 10 seconds before firing the second stage: you’ll be 12 km higher up and it seems to me that the benefit of this will be significant. I notice that space launches wait a few seconds before firing their second stage.

As a final bit, I’d mentioned that docking a rocket with a space station is difficult, in part, because docking requires an increase in angular speed, w, but this generally goes along with a decrease in altitude; a counter-intuitive outcome. Setting the acceleration due to gravity equal to the angular acceleration, we find GM/r2 = w2r, where G is the gravitational constant, and M is the mass or the earth. Rearranging, we find that w2  = GM/r3. For high angular speed, you need small r: a low altitude. When we first went to dock a space-ship, in the early 60s, we had not realized this. When the astronauts fired the engines to dock, they found that they’d accelerate in velocity, but not in angular speed: v = wr. The faster they went, the higher up they went, but the lower the angular speed got: the fewer the orbits per day. Eventually they realized that, to dock with another ship or a space-station that is in front of you, you do not accelerate, but decelerate. When you decelerate you lose altitude and gain angular speed: you catch up with the station, but at a lower altitude. Your next step is to angle your ship near-radially to the earth, and accelerate by firing engines to the side till you dock. Like much of orbital rocketry, it’s simple, but not intuitive or easy.

Robert Buxbaum, August 12, 2015. A cannon that could reach from North Korea to Japan, say, would have to be on the order of 10 km long, running along the slope of a mountain. Even at that length, the shell would have to fire at 450 G, or so, and reach a speed about 3000 m/s, or 1/3 orbital.