Monthly Archives: August 2015

my electric cart of the future

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Buxbaum and Sperka cart of future

Buxbaum and Sperka show off the (shopping) cart of future, Oak Park parade July 4, 2015.

A Roman chariot did quite well with only 1 horse-power, while the average US car requires 100 horses. Part of the problem is that our cars weigh more than a chariot and go faster, 80 mph vs of 25 mph. But most city applications don’t need all that weight nor all of that speed. 20-25 mph is fine for round-town errands, and should be particularly suited to use by young drivers and seniors.

To show what can be done with a light vehicle that only has to go 20 mph, I made this modified shopping cart, and fitted it with a small, 1 hp motor. I call it the cart-of the future and paraded around with it at our last 4th of July parade. It’s high off the ground for safety, reasonably wide for stability, and has the shopping cart cage and seat-belts for safety. There is also speed control. We went pretty slow in the parade, but here’s a link to a video of the cart zipping down the street at 17.5 mph.

In the 2 months since this picture was taken, I’ve modified the cart to have a chain drive and a rear-wheel differential — helpful for turning. My next modification, if I get to it, will be to switch to hydrogen power via a fuel cell. One of the main products we make is hydrogen generators, and I’m hoping to use the cart to advertise the advantages of hydrogen power.

Robert E. Buxbaum, August 28, 2015. I’m the one in the beige suit.

Racial symbols: OK or racist

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Washington Redskins logo and symbol. Shows race or racism?

Washington Redskins lost protection of their logo and indian symbol. Symbol of race or racism?

In law, one generally strives for uniformity, as in Leviticus 24:22: “You shall have one manner of law; the same for the home-born as for the stranger,”  but there are problems with putting this into effect when dealing with racism. The law seems to allow each individual group to denigrate itself with words that outsiders are not permitted. This is seen regularly in rap songs but also in advertising.

Roughly a year ago, the US Patent office revoked the copyright protection for the Washington Redskin logo and for the team name causing large financial loss to the Redskins organization. The patent office cited this symbol as the most racist-offensive in sports. I suspect this is bad law, in part because it appears non-uniform, and in part because I’m fairly sure it isn’t the most racist-offensive name or symbol. To pick to punish this team seems (to me) an arbitrary, capricious use of power. I’ll assume there are some who are bothered by the name Redskin, but suspect there are others who take pride in the name and symbol. The image is of a strong, healthy individual, as befits a sports team. If some are offended, is his (or her) opinion enough to deprive the team of its merchandise copyright, and to deprive those who approve?

More racist, in my opinion, is the fighting Irishman of Notre Dame. He looks thick-headed, unfit, and not particularly bright: more like a Leprechaun than a human being. As for offensive, he seems to fit a racial stereotype that Irishmen get drunk and get into fights. Yet the US Patent office protects him for the organization, but not the Washington Redskin. Doesn’t the 14th amendment guarantee “equal protection of the laws;” why does Notre Dame get unequal protection?

Notre Damme Fighting Irish. Is this an offensive stereotype.

Notre Dame’s Fighting Irishman is still a protected symbol. Is he a less-offensive, racist stereotype?

Perhaps what protects the Notre Dame Irishman is that he’s a white man, and we worry more about insulting brown people than white ones. But this too seems unequal: a sort of reverse discrimination. And I’m not sure the protection of the 14th was meant to extend to feelings this way. In either case, I note there are many other indian-named sports teams, e.g. the Indians, Braves, and Chiefs, and some of their mascots seem worse: the Cleveland Indians’ mascot, “Chief Wahoo,” for example.

Chief Wahoo, symbol of the Cleveland Indians. Still protected logo --looks more racist than the Redskin to me.

Chief Wahoo, Still protected symbol of the Cleveland Indians –looks more racist than the Redskin to me.

And then there’s the problem of figuring out how racist is too racist. I’m told that Canadians find the words Indian and Eskimo offensive, and have banned these words in all official forms. I imagine some Americans find them racist too, but we have not. To me it seems that an insult-based law must include a clear standard of  how insulting the racist comment has to be. If there is no standard, there should be no law. In the US, there is a hockey team called the Escanaba (Michigan) Eskimos; their name is protected. There is also an ice-cream sandwich called Eskimo Pie — with an Eskimo on the label. Are these protected because there are relatively fewer Eskimos or because eskimos are assumed to be less-easily insulted? All this seems like an arbitrary distinction, and thus a violation of the “equal protection” clause.

And is no weight given if some people take pride in the symbol: should their pride be allowed balance the offense taken by others? Yankee, originally an insulting term for a colonial New Englander became a sign of pride in the American Revolution. Similarly, Knickerbocker was once an insulting term for a Dutch New Yorker; I don’t think there are many Dutch who are still insulted, but if a few are, can we allow the non-insulted to balance them. Then there’s “The Canucks”, an offensive term for Canadian, and the Boston Celtic, a stereotypical Irishman, but also a mark of pride of how far the Irish have come in Boston society. Tar-heel and Hoosiers are regional terms for white trash, but now accepted. There must be some standard of insult here, but I see none.

The Frito Bandito, ambassador of Frito Lays corn chips.

The Frito Bandito, ambassador of Frito Lays corn chips; still protected, but looks racist to me.

Somehow, things seem to get more acceptable, not less if the racial slur is over the top. This is the case, I guess with the Frito Bandito — as insulting a Mexican as I can imagine, actually worse than Chief Wahoo. I’d think that the law should not allow for an arbitrary distinction like this. What sort of normal person objects to the handsome Redskin Indian, but not to Wahoo or the Bandito? And where does Uncle Ben fit in? The symbol of uncle Ben’s rice appears to me as a handsome, older black man dressed as a high-end waiter. This seems respectable, but I can imagine someone seeing an “uncle tom,” or being insulted that a black man is a waiter. Is this enough offense  to upend the company? Upending a company over that would seem to offend all other waiters: is their job so disgusting that no black man can ever be depicted doing it? I’m not a lawyer or a preacher, but it seems to me that promoting the higher levels of respect and civil society is the job of preachers not of the law. I imagine it’s the job of the law to protect contracts, life, and property. As such the law should be clear, uniform and simple. I can imagine the law removing a symbol to prevent a riot, or to maintain intellectual property rights (e.g. keeping the Atlanta Brave from looking too much like the Cleveland Indian). But I’d think to give people wide berth to choose their brand expression. Still, what do I know?

Robert Buxbaum, August 26, 2015. I hold 12 patents, mostly in hydrogen, and have at least one more pending. I hope they are not revoked on the basis that someone is offended. I’ve also blogged a racist joke about Canadians, and about an Italian funeral.

Winning at Bunker Hill lost America for Britain.

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The greatest single victory of the American Revolution in terms of British soldiers killed or wounded was the battle of Bunker Hill. It was won without global strategy, or any real sense of victory. The British captured the hill but their loss of soldiers and reputation was such that one can echo British General, George Clinton’s comment: “A few more such victories would have shortly put an end to British dominion in America.” How the British came to blunder this way is a lesson in group-think management; it lead to the destruction of an army of the finest soldiers on earth by a band of untrained, leader-less rabble.

A map of metropolitan Boston in May 1775 shows that it was already a major port with far less dry land than it has today. It consisted of a knob-hill peninsula, British-controlled Boston city, cut off from the rest of the colonies except for one narrow road, called “The Neck,” or The Roxbury Neck. The later name was used to distinguish it from a similar neck road that connected the colonies to nearby Charlestown peninsula; Bunker Hill is on Charlestown peninsula. Following the rumpus battles of Lexington and Concord, Boston’s suburbs were over-filled by 15,000, ill-clad, undisciplined colonials who ate, drank, and shot at random stuff in plain view of Boston’s 6000 trained soldiers and four Generals. The Colonials set up barriers and cannon at their end of the Neck road. These were not particularly good barriers, and the British army could leave at any time by the neck to control of the colonies, but only at a cost in men. This potential cost  kept rising as more colonials came to camp around Boston. What would you do?

The British had sea-power that they felt they could use: excellent ships and excellent admirals; the colonies had neither. The British navy could attack anywhere on the American coast, but only at a risk of further alienating the colonials. They thus used their power judicially. E.g., for the attack on Lexington, April ’75, navy ships took the 2000 soldiers from Boston, landing them at Charlestown, at the foot of Breed’s hill. The army then marched out over the Charlestown neck to Lexington and Concord, but not to a smashing success. Many soldiers lost their lives, and not much was captured.

Back in Boston, the four British generals: Gage, Burgoyne, Howe, and Clinton, decided that, to quash the revolt/revolution, they had to break out of Boston and quarter in Massachusetts proper, on some easily defended ground on the mainland, preferably high ground. They needed to establish a base with good connections to the rest of colonies, plus good access to the sea. Looking about, the obvious spot for this base was the heights of Dorchester, a set of hills that overlooked Boston Harbor from the south. Eventually the British would flee Boston because George Washington would capture and hold these heights. The reason the British didn’t capture the heights or at least defend them was the result of group-think ignorance, on the morning of June 17. The British changed their goals, and decided to attack at Charlestown (Breed’s Hill) instead of Dorchester. Capturing Charlestown left Britain with many dead and no good connection to the colonies; it was just another isolated peninsula barely attached to the mainland by an easily defended road.

Why did four, trained Generals attack this worthless spot instead of Dorchester? It was the luck of American disarray and mob-rule. Looking at the Colonials disarray, each of the four generals saw an opportunity for personal glory; the colonials were inept and would never improve. The same British group-think that reawakeneded in the Crimean war with Russia. The American defense of Charleston and Breeds Hill was done so incredibly poorly that the Americans were left as sitting ducks, waiting to be taken. A poor location was chosen for the fort and only 1200 Colonials came to defend it. We’d meant to build a fort on a better location, Bunker Hill, the tall hill overlooking Boston, and we’d meant to build a minor respite foxhole on Breeds hill, but we screwed up.

Our soldiers were digging  at night, fueled with much rum and little or no leadership. When the sun rose on June 17, we found that we’d built next to nothing on Bunker hill, and a vastly too-large, uselessly deep, square hole at Breed’s Hill: a doubtful redoubt. The square was open at back, and too large for the number of soldiers. It was also too deep for people to shoot out of easily. Looking with spyglasses from Boston, the British generals saw that we had no idea what we were doing. Gage and co., thought to show us the consequences of our incompetence. A few thousand British soldiers could easily take this redoubt and its 1200 defenders, and that thought clouded his mind and the minds of his fellows. They forgot that this was not a hill worth taking, and never imagined that we might fix our defenses. Even if Gage could win without a single lost man, he should have realized that a victory would leave him in a worse position than before. His forces would then be divided between two peninsulas both separated from the mainland, and separated from each other by neck-roads. Coordinating an attack from this position would be a logistic nightmare, and any one of the co-Generals should have alerted him to this.

The attack was supposed to work this way: a sea landing at Moulton's hill. two side actions, SA, at the fronts of the Colonial defenses, and a sweeping main attack, MA, at the edge.

The attack was supposed to work this way: a sea landing at Moulton’s hill. two side actions, SA, at the fronts of the Colonial defenses, and a sweeping main attack, MA, at the edge.

But four generals working together were stupider than Gage alone. Their glee at our incompetence made them forget why Dorchester Heights was the right military target. The prospect of personal glory made attacking Charlestown and its hills too tempting to ignore. Their superior force of trained men would land and march forward to an easy victory. They might even do it with bayonets alone, as the Continentals had too few men, no training, and no bayonets. If the Continentals were able to muster together at all (unlikely), they were unlikely to reload fast enough to shoot more than once — that took special guns and training. Under pressure, the colonials would likely miss with most of the only shot they got, and would find themselves over-run before they could reload. The British force could shoot therebels at close range, or they could hold their fire and spear them with bayonets as the rebels tried to reload, or run out of the hole they were in.

It was a perfect plan with only a minor problem, easily addressed: the Americans had a cannon brought to the hill, and a trained cannoneer could kill many with a few follies of grape-shot. Gage and co. thus decided on a complex attack that would avoid the cannon. It included a feint to the front and a side run. This “wheeling motion” was completely unnecessary: the Americans had little powder and no idea what to do with a cannon, the generals didn’t know that.  The plan was to form a single line across from the fort (hole), fake a frontal attack to draw American shots while staying out of range, and then wheel right. That is, on command, every British soldier was to turn right and march, as a column, north to the trench’s right side (the left side if you look as a Colonial). They’d avoid the cannon rifle shots, and take the redoubt from the north side, perhaps without taking a single loss. It should have been a piece of cake, but was not.

Landing the British troops and forming them up took longer than expected, as often happens, and during this time, more Colonials showed up, and some of them took pot-shots at the British officers. What’s more, the rebels began to fix the more-glaring flaws in their defenses. Potshots from Charlestown windows slowed British efforts at mustering into an appropriate line while the Continentals built up the left (north) side of their redoubt — the side the British wished to attack. The colonials added triangular sub forts (Friches) at both sides of their square trench, somewhat in front, and added a wooden fence rail from the hill to the sea somewhat behind. The British naval commander wasted yet more time with a cannon barrage from his ships. He imagined he was softening the defense, but the barrage managed to kill only one colonial, decapitated by a cannon-ball, while providing time for the colonials to build their friches and fence, and allowing for more sniper work. Col. Stark put colonials at the sense with shot markers at 100 feet in front. He then passed the now-famous instruction: don’t shoot till they passed these markers and you see the whites of their eyes.  He needed to preserve ammunition, and assumed that, at 50 to 100 feet, his colonials would not miss, and could fall back. Any British who passed the fence would be taken out by defenders shooting down from Bunker Hill, or up from the hole.

The second attack at Breeds Hill

The second attack at Breeds Hill

At first the British tried the frontal feint attack with a wheel to the north. When this attack failed to heavy losses, they tried again before realizing this attack was ill-suited to the terrain and troops. The British front line was composed of crack Hessians who marched perfectly in step, wearing bright red coats and heavy bear-skin, “Busby hats” to make them look more formidable. It might have worked on even ground, this ground was uneven and mucky, and the hats kept the Hessians from looking down at the brambles and rocks. Their stumbling motion, always aligned, was so slow that the colonials had time to fire and reload. The Hessians who survived the first shots never managed to wheel. Meanwhile, the main British attack, the one at the rail fence, failed because a colonial fired early by mistake. The British force should have ignored it, but instead, stopped and fired back. Hearing the shooting, more Colonials showed up and shot at British soldiers (more or less in range) using the fence to steady their aim. Only a few British got past the fence and these were shot by the retreating Americans and by the garrison on Bunker Hill. The attack was called back, allowing the British to re-muster while the Americans reloaded and repositioned.

Before the second British attack, more colonials wandered onto the peninsula, and built a quick platform in the redoubt so they could shoot better over the top. Some defenders of Bunker Hill — folks who’d seen little action so far — moved forward to get better shots, defending at the fence, and some Colonials wandered off, too. There was still no one in charge. Just everyone doing what seemed right to him.

meanwhile, the generals burnt Charlestown as a way of stopping the snipers, and mustered their men for a simpler attack with a simpler troop arrangement, see map above and picture below. Three ranks of soldiers were set to march straight at the fort without trying to wheel. Those with Busby hats were largely dead or wounded, so the attackers could see where they were stepping. Still, without the wheel, the result was many British dead or wounded, and this second attack was called off.

The second attack: Three ranks and no Busby hats this time, with Charlestown burning in the background. Their's not to question why, their's but to do and die.

The second attack: Three ranks and no Busby hats this time, with the dead strewn around and Charlestown burning in the background. Their’s not to question why; their’s but to do and die. Painting by Pyle.

About at this time, the British should have decided to go home and attack elsewhere (Dorchester), but they persisted, not willing to accept defeat. For the third attack, the soldiers were told to attack as two single, long columns. The generals added some 400 marines (ship-board soldiers) plus some 200 wounded who were now ordered to re-muster. The columns waled straight up to the fort. The folks in front pushed by those behind; the soldiers at the front were killed, but the attack worked, sort of. The British took the fort, but most of the defenders avoided capture. They retreated across the neck and rejoined the main mob. The British captured or killed some 400 at the expense of 1,054 men lost; 226 were killed in the immediate battle, including most of the junior officers, with the rest lost as a result of wounds. The soldiers also lost the sense of invincibility; unorganized colonials could fight, inflicting serious damage at minimal cost.

There arose a myth of the backwoods shooter, but it was largely a myth. The Colonials were able to pick off British officers because the officers dressed to be noticed. It was a mistake the British would keep making. At Bunker Hill, the British lost 1 lieutenant colonel (killed), 5 majors (3 killed), 34 captains (7 killed) 41 lieutenants (9 killed), 57 sergeants (15 killed), and 13 drummers (1 killed). A lesson we learned: don’t dress so fancy. Tactically, the British victory at Bunker hill left their forces divided between two peninsulas. The men defending these peninsulas were unavailable for any attack at Dorchester heights. Thus the British forces lost the opportunity to escape Boston and take positions that could hold the colonies. By January, 1776 Washington Controlled Dorchester heights, and the British left Boston and Charlestown by ship. They would try taking Dorchester again in 1776 and 1777, but by then the continental army was more of an army, less of a rabble with rifles. A life lesson: only fight for something that you really want, a pointless win can be a lost opportunity.

After the battle, the back-stabbing and group think continued among the British generals, while the colonials got a single commander, General Washington. Meanwhile,  a new British general arrived, Burgoyne, who blamed Generals Clinton, Howe, and Gage for the loss of men and opportunity. Burgoyne got to lead an attack from Canada, but stung by Burgoyne’s blame-game, Clinton and Howe did not come his aid at Albany in June 1777. Instead, Clinton left Burgoyne to fend for himself (and be captured) while Howe was sent to attack the continental congress at Philadelphia. Burgoyne lost his army and reputation, and Howe captured Philadelphia, restoring his reputation, sort-of.  The Continental Congress fled Philadelphia ahead of the Brits, and Burgoyne’s defeat led to the French joining in on our side. Burgoyne blamed Clinton and Howe for his defeat, but was really done in by over confidence. He could not see that the chaotic leadership of a rabble was better than a fixed command without real communication, even with the best generals and soldiers.

The colonial chaos was horrible, but workable. The fixed-command mistakes on the British side were not as bad, but disastrous, since they required a coordinated effort that could not be produced. Had there been fewer British generals and a simpler plan, the better-trained British army would have won at Charlestown, or they would have left and attacked at Dorchester and won the war. One on one, General Howe’s forces repeatedly beat Washington, meeting in New York and New Jersey in the summer and fall of 1776. These were the same soldiers who lost at Bunker Hill, but with a simple command structure: one general not four. It was only George Washington’s genius that saved some semblance of an army to keep fighting into 1777.

This is not to say that chaos is good, but that it can work, especially with Americans. We tolerate chaos and fractured leadership better than most, I think, because we are, by nature, chaotic. As Bismarck put it: “God protects children, fools, and the United States of America.”

Robert Buxbaum, August 16, 2015 (edited June, 2024). There were several other howler mistakes of the American Revolution discussed here.  British generals took useless victories while losing opportunities that mattered. Don’t fight readily; only to win something of value that can be won.

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.