The Flying Roadster and the Tyranny of the Rocket Equation

Many readers of this post will have watched the video demonstration of Elon Musk and Tesla's so-called "Flying Roadster." Of course, the invitees to the demonstration were 100% hard-core Tesla/Musk faithful, no skepticism was apparent. And, frankly, it was an impressive feat of engineering, regardless of whether, in the end, it has anything to do with a production Roadster.

Anyway, a YouTube video (linked below) turned up in my feed this week titled "Why the Flying Tesla Roadster Is Physically Impossible (The 1963 Proof)." As anyone who's followed me knows, I can't resist anything involving airplanes, test pilots, and a little thermodynamics. So I watched it, read the transcript, and spent an evening checking the numbers, in full transparency, with some help from Claude.

The premise is that the promised cold-gas thruster package for the new Roadster is a rehash of the reaction controls Lockheed put on the NF-104A Starfighter in 1963. The video says Cold War engineers already learned the hard way that it can't work. The conclusion is right, but a surprising amount of the supporting story is wrong.

Start with what's right, because it's the important part. A wing flies by pushing a lot of air down slowly. A thruster does the opposite: it throws a small amount of gas out very fast. Thrust is just mass flow times exhaust velocity, ~F = \dot{m} v_e~, but the power needed goes as ~\tfrac{1}{2}\dot{m} v_e^2~. Fast and small is the expensive way to make thrust. That's why helicopters have big rotors.

Let's toss in some numbers. Rocket people (of which I am conspicuously not one) rate efficiency by specific impulse, ~I_{sp}~: how many seconds one pound of propellant produces one pound of thrust. Kerosene and oxygen gets around 300 seconds. Cold compressed air gets about 60. To hover a 4,500 lb car, you need 4,500 lb of thrust, so:

~\dot{m} = \frac{F}{I_{sp}} = \frac{4{,}500\text{ lb}}{60\text{ s}} = 75\text{ lb/s}~

The video gets this right. Now the tank. They posit a 300-liter carbon-fiber pressure vessel at 10,000 psi where the back seat used to be. At that pressure air is about 550 kg/m³ (it isn't an ideal gas anymore), so the tank holds roughly 165 kg, or about 360 lb. The video says 180–200 lb, which is low by almost half. Allowing for the air you can't get out, call it 290 lb usable:

~t = \frac{290\text{ lb}}{75\text{ lb/s}} \approx 4\text{ seconds}~

So the video's "two to three seconds" should be more like four. Either way, that's a hop, not a flight.

Refilling is where it gets ugly. The minimum work to compress that much air to 10,000 psi is about 25 kWh, closer to 35 - 40 kWh with real compressors. That's a big bite out of even a large EV battery, and a compressor small enough to ride along would take hours. All for four more seconds of hover. The video is correct here and, if anything, understates the case. Running some quick numbers:

Average battery power used for compression and cooling	Refill time
3 kW	About 17 hours
5 kW	About 10 hours
10 kW	About 5 hours

Of course, one wouldn't always need to refill an empty tank to completely full, but you can see the issue, I'm sure.

Now the parts I had trouble with:

Early on we're told 10,000 psi is "triple the chamber pressure of a military sniper rifle." It's the other way around. A .308 runs about 62,000 psi, so 10,000 psi is about one-sixth. Later, the thruster exhaust at 2,000 feet per second is "the exact muzzle velocity of an M16." An M16 round leaves the barrel north of 3,000 ft/s. These are easy things to look up.

The bigger problem is the hook itself - the video relies heavily on NASA's experimention in the 1960s. You can read all about it here. The NF-104A did not use cold-gas thrusters. Its reaction controls ran on hydrogen peroxide decomposed over a silver catalyst: hot steam, not cold air. The video admits this around the nine minute mark, which contradicts its own opening. So the "1963 proof" isn't really about the technology in question.

Then there's the long buildup about Joule-Thomson cooling freezing valves at –100°F. Valve and regulator icing is real (ask any diver who's had a regulator free-flow in cold water). But simple throttling through a valve cools the gas by tens of degrees, not hundreds. The deep cold happens in the nozzle, where heat becomes velocity, and in the tank as it empties. And moisture icing is a known problem with known fixes: dry gas, heaters, the right seals.

That buildup leads into Chuck Yeager's December 10, 1963 crash in NF-104A 56-0762. The video gives a dramatic and mostly accurate account: rocket lit at 35,000 feet, a steep zoom past Mach 2, a flat spin, and an ejection with terrible burns when the seat's rocket motor struck his helmet. Most accounts put that flight's peak near 101,600 feet rather than the 108,700 quoted. That higher altitude was achieved on Yeager's first flight of the day. The crash occurred on his second.

But here's the thing. By the video's own telling, the crash happened because the nose thrusters didn't have the authority to stop a pitch-up into a spin. No valve froze. Ten minutes of setup about frozen valves, and then the crash it builds to has nothing to do with them.

The road-hazard section has the same problem. Ground "suck-down" under a wide, flat body is a real phenomenon from the VTOL era, and the Harrier description is accurate. But "a Claymore mine aimed at every windshield within 50 yards" is dramatized, and I could find nothing on Soviet T-55 tanks with rockets vaulting trenches.

The video also leaves out two points that cut the other way. First, 10,000 psi carbon tanks are already on public roads. Hydrogen fuel cell cars like the Toyota Mirai carry them, crash-certified. Having spent a career around testing and inspection, I'd say a COPV (Composite Overwrapped Pressure Vessel) behind the seat is an engineering problem, not a death sentence. Second, as originally pitched, the thrusters were mainly about acceleration, braking, and cornering, not hovering. That takes a fraction of the car's weight in thrust, and the air lasts much longer.

So the verdict? The Roadster won't fly in any meaningful sense, it will only hop. The video is right about that. Compressed gas is magnificent for nudging a satellite and pretty lousy for holding up two tons of electric car. But the video gets there with a misleading history, a couple of flatly wrong comparisons, and a lot of scary adjectives. The physics was enough on its own. The drama wasn't needed, though, in this age of content creators looking for eyeballs, I guess it's understandable.

Ps: If anyone has a good source on the T-55 claim, send it my way. I'd love to be wrong about that one.

Labels: aviation, engineering, physics, Tesla, Elon Musk

More $Trump memecoin

 The way a meme coin typically works is:

1. Someone achieves some notoriety for something, or exploits some notoriety they already have. They seek to monetize this notoriety.
2. They use one of any number of meme coin generators and choose a blockchain.
3. They generate a pot of, say, ten million coins, and release, say, four million onto the block chain with a LOT of fanfare and a lot of, basically, b.s. about building a community, adding value, etc. They keep the other six million coins. The reputable thing to do is to "lock up" most or all of the six million coins for some period. If they simply sell into the initial wave of buyers and walk away, this is known as a "rug pull" and, while it's considered unseemly at best and results in a ton of howling and shedding of crocodile tears, this is often what happens. See &Hawk for a good example.
4. As the lockup period's end approaches, the originators have to create some additional fanfare to drive the price up or back up.
5. The originators sell into the market at the increased price.
6. End of story - a bunch of people have a bunch of worthless meme coins, less money, and a sad story to tell. The originators have a bunch of money and can move on to the next scam event.

In essence, this is the story of the $Trump con coin. There is a lockup period end coming, the price of the coins has declined, so the Trump organization needs some razzle dazzle around the coins. So, to boost demand, Trump will be hosting an "intimate private dinner," for the top "220 Special $Trump Meme Coin Holders" (as shown on a "leader board,") that is, in order to have dinner with the President, you must have or purchase sufficient $Trump coins to be in the top 220 holders. For the top 25, there is a reception and White House tour. Clearly. a great way for the Trump organization to boost the price into which they sell their locked up coins, and a great way for Trump sycophants supporters to bribe him throw cash his way. Note that the coin rose some 60% in "value" after the dinner was announced.

In very small print, we see "This Tour is being arranged by Fight Fight Fight LLC. President Trump is appearing at the dinner as a guest and not soliciting any funds for it." No, he needn't do so since the increase in value of the coins his organization holds will provide plenty of cash.

It's always been obvious that Trump is completely lacking in any moral compass or sense of right and wrong (at least where his own behavior is concerned, he's certainly willing to lash out at others). I wouldn't have thought, though, that he could continuously reach new depths of self-serving behavior.

As I quoted someone in my previous $Trump coin blog post, we voted for corruption, and Trump is delivering. As MAGAVoice constantly prattles on X, née Twitter, "Promises made, promises kept."

More on "epsilon" in the "Reciprocal Tariffs" equation

 

In the equation published by the Office of the U.S. Trade Representative* (~\Delta\tau_{i}=\frac{x_{i}-m_{i}}{\varepsilon*\varphi*m_{i}}~), we see the Greek letter ~\varepsilon~, epsilon. This number is stated to be the elasticity of import demand with respect to import prices. An economist would call this the "price elasticity of demand (PED)", a number which represents the rising and falling of demand with the falling and rising of prices.

The actual definition is that the PED at a given point on the supply and demand curve  is the percentage change in quantity demanded divided by the percentage change in price (~\frac{\%\Delta\text{Q}_{d}}{\%\Delta\text{P}}~).

Let's toss in some numbers and see what this means. Suppose that, at an existing aggregate price for widgets of $100,000,000.00 there are 1,000,000 widgets demanded. Next, suppose the PED is -4. Then ~\frac{\%\Delta\text{Q}_{d}}{\%\Delta\text{P}}=-4~. Then suppose, as a result of a price increase due to tariffs, the price of widgets increases by 10%. Arranging the equation above, ~\%\Delta\text{Q}_{d}=-4\%\Delta\text{P}=-40\%~. That is, a 10% increase in price will result in a 40% decrease in demand.

In thinking about whether this is reasonable, we need to consider the actual items being evaluated. For example, a Type 1 diabetic relies on insulin to stay alive, and hence must pay whatever the cost is for insulin. This is referred to as "demand being inelastic." On the other hand, it's likely that a significant increase in the price of, for example, iPhones will reduce demand, since new iPhones are not a necessity. This is referred to as "demand being elastic."

There are many considerations that go into PED. These include individual choices, availability of substitutes, the length of time a price increase has been in place (more time gives a greater time to search for substitutes), and others.

Whether it's stated as such or not, the "Reciprocal Tariffs" equation is intended specifically to reduce trade deficits. A larger ~\varepsilon~ in the equation will reduce the magnitude of the fraction and imply smaller tariffs, a smaller ~\varepsilon~ will do the opposite.

However, a "one size fits all" policy doesn't make a lot of sense. Some of our imports could, in theory, be replaced with domestic products (e.g., automobiles, microchips, etc.) while others could not (e.g., coffee, cobalt, bananas). Going back to Malaysia from my previous post, top U.S. imports are electrical and electronic equipment and machinery, nuclear reactors, and boilers. These items total about 71% of our imports and are among the categories that could, in principle, be produced domestically, though at considerably higher cost.

As examples, the PED for coffee in the U.S. is estimated to be 0.25 (quite inelastic, gotta have that caffeine!), that of fresh tomatoes is estimated to be 4.6 (very elastic). Note that economists typically utilize the absolute value of PED, that is, only showing its magnitude.

With this information, it's quite clear that, on top of the analysis of non-tariff trade barriers shown on the Trump Reciprocal Tariff chart and the rationalization on the U.S. Trade Representative's web page*, the numbers have little, if any, relation to the actual trade situation or to any deep analysis of price elasticity.

Ps: I guess I do have to thank Trump for getting me reengaged with blogging!

*As noted in my previous post, the page with the equation and the explanation has disappeared, or at least been moved to where I can't find it. If someone can find it and post a link in the comments, that would be great. Also as I alluded to, given all the mockery and derision the chart and the equation have engendered, I'd be embarrassed and ashamed as well.

Reciprocal Tariffs?

I'm wrapping my head around the equation stated by the Office of the President of the United States Trade Representative for calculating the so-called "Reciprocal Tariffs." * From that site, we see the following:~\Delta\tau_{i}=\frac{x_{i}-m_{i}}{\varepsilon*\varphi*m_{i}}~. We're told that ~\Delta\tau_{i}~ is the "change in the tariff rate"; ~x_{i}~ is "total exports to country i" (that is, U.S. exports); ~m_{i}~ is "total imports from country i" (that is, U.S. imports); ~\varepsilon~ is the "elasticity of imports with respect to import prices" (an economics concept measuring how much demand goes up or down when prices go down or up, but it can be ignored here because the next term cancels it out); and ~\varphi~ is the pass through from tariffs to import prices (this is the term that cancels out the elasticity term since the latter is stated to be 4 and the former is stated to be 0.25, thus totaling 1). Finally, since "we are a kind people," the derived number is divided by two to determine the final "reciprocal tariff."

Multiple sites characterize this as "trade deficit divided by total imports." It took me a moment to understand this, because trade deficit is imports minus exports, whereas the numerator of the fraction is exports minus imports, the negative of the deficit. But the site states that "~\varepsilon\lt0~," so the denominator is negative and, lo and behold, it all works out as described.

The chart above has three columns: Country (more on that later); "Tariffs Charged to the U.S.A. Including Currency Manipulation and Trade Barriers"; and "U.S.A. Discounted Reciprocal Tariffs" (calculated as above). One might reasonably infer from these column labels that, for each country, an analysis was done of the tariffs on U.S. exports in place along with currency manipulation and trade barriers to determine the number, but no. The site linked above states that:

While individually computing the trade deficit effects of tens of thousands of tariff, regulatory, tax and other policies in each country is complex, if not impossible, their combined effects can be proxied by computing the tariff level consistent with driving bilateral trade deficits to zero. If trade deficits are persistent because of tariff and non-tariff policies and fundamentals, then the tariff rate consistent with offsetting these policies and fundamentals is reciprocal and fair.

That is, in lieu of the above analysis, the fraction calculated in accordance with the equation above is used as a proxy. Note the "If." If my mother had wheels, she'd be a bus.

Now, with the background out of the way, let's stipulate that the general concept of there being a strong relationship between trade barriers imposed by countries receiving U.S. exports and the trade deficit with such countries as a fraction of U.S. imports from those countries is not completely irrational. In essence, this is a theory that trade is only fair with any given country if the balance of trade with that county is $0.00 and that "reciprocal tariffs" are the fair way to force this equilibration. I do not agree with this stipulation, but let's proceed nevertheless. Does the fraction calculated per the above reasonably capture this?

Let's start with an easy one. We'll take Malaysia as an example. In 2024, the U.S. imported $52.5B in goods from Malaysia and exported $27.7B in goods to Malaysia. Using the equation, the imputed "tariffs" would then be 47.2% and we "kindly" will only reciprocate with a 24% tariff. But Malaysia's population is 35.13 million, that of the U.S. is 340.1 million, almost 10 times as large. So, to "balance trade," each Malaysian would have to purchase ~\frac{340.1}{35.13}=\$9.68~ for every dollar of Malaysian goods purchased by Americans. Clearly, this makes zero sense.

Instead then, let's see how things change if we use per capita numbers. Then we have ~\frac{\$52.5\text{ Billion}}{340.1\text{ million}}=\$154~ dollars of U.S. imports per capita and ~\frac{\$27.7\text{ Billion}}{35.13\text{ million}}=\$788~ in Malaysian imports per capita (that is, U.S. exports). That is, each Malaysian (on average) imports $788 of U.S. goods per year, while each member of the U.S. population imports a measly $154 of Malaysian goods per year. What happens if we use the per capita numbers in the equation? Then we have ~\frac{154-788}{154}=-431\%~. Trump constantly rails about the U.S. being "ripped off" by our trading partners. But, using the silly logic described in the block quoted section, I have to ask "who's ripping off whom?"

Another aspect of the naive calculation as presented is the fact that, for example, the median annual household income in the U.S. (2023 figure) is $78,538. That of a Malaysian is $1,419. Is it reasonable to expect a Malaysian to purchase as much in U.S. goods as a U.S. resident purchases Malaysian goods? I don't think anyone could justify a "yes" answer (with the possible exception of Peter Navarro). But can we consider an adjustment to the equation to capture this as we did with the per capita trade quantities as above?

We could consider dividing the U.S. import number by the U.S. household income, that would yield imports per dollar of income. Similarly, we could divide Malaysia's import of U.S. goods by Malaysian household income. This would have even a more dramatic effect on the fraction. Taking both population and household income together would show that we're absolutely pillaging Malaysia!

There are other categories of countries and entities as well. Let's think about some countries with more in common with the U.S. If we consider the so-called "developed" countries, e.g., European Union, United Kingdom, Japan, South Korea, Norway, Australia, New Zealand, etc. and normalize them all similarly, what do we see?

I ran the per capita numbers for these countries and the results were:

Note that, on a per capita basis, every country (or entity in the case of the EU) shows a negative number, with the exception of the EU. That is, each nominal resident of the foreign country purchases, on average, more U.S. goods than a nominal U.S. resident does of theirs. The EU's positive number is a bit higher than calculated by Trump, et al, on a per capita basis. The EU's population significantly exceeds ours. But again, who is ripping off whom?

Were I to go through and utilize household income, the results would be even more stark, but this post is already excessively long. But I do want to say a bit more about Peter Navarro. Navarro, who is now the Senior White House Counselor for Trade and Manufacturing in the Trump Administration is, at best, a fringe economist. Among other hilarities, in books and articles, he quotes "Harvard economist Ron Vara" in support of his policies. You'll note that Ron Vara is an anagram of Navarro, and Navarro has conceded that Vara is purely fictional.

I want to cite a couple of sites. Matt Parker, of Stand Up Maths, explains the calculations and some of their faults and, finally, Coyote Blog expresses a viewpoint very much aligned with my own with respect to what I'll call the "philosophy of tariffs."

* Apparently, The Office of the U.S. Trade Representative has decided that it's best not to try to explain the so-called "Reciprocal Tariffs." The equation and its purported explanation are no longer there, at least that I can find. I'd have been embarrassed too. 

Howard Lutnick and Econ 101

Our esteemed Commerce Secretary, Howard Lutnick, stated in an interview on CBS that “You have to sort of understand how things work. When tariffs come into place, foreign goods may become a little more expensive, but domestic goods do not.”

Interesting. I’m no economist, but I have taken a couple of econ courses and read a bit. And, in studying the fundamental aspects of supply and demand, I encountered supply and demand curves. Searching the deep recesses of my memory and then scribbling on my phone, I produced the following:

Crude at best, but the point is valid. The solid lines represent price vs. quantity demanded and quantity supplied. They cross at a point referred to as the "equilibrium point," that is, the point at which the quantity demanded by buyers at that price matches the quantity willing to be supplied by producers. If the quantity demanded (the "demand curve") shifts to the right, the equilibrium price rises along with the quantity supplied.

Here, we're looking at domestic goods. Because the prices of imported goods are expected to rise significantly due to tariffs (Trump's claim that these will be paid by the exporting countries notwithstanding), demand for domestic goods can be expected to rise as buyers look for substitutes (another fairly basic concept from Econ 101). That is, the demand curve will shift to the right. This inevitably leads to increased production of domestic goods (which Trump, Lutnick, et al champion) but it does not do so without also increasing the price of those goods.

Now, Lutnick graduated from Haverford College in 1983 with a degree in economics. I don't imagine such a degree is conferred without learning about this concept. From that, we can infer a couple of things: 1) Lutnick is quite willing to lie; and 2) He likes his position as Secretary of Commerce. Saying Trump is incorrect while occupying a position he appointed is not a way to keep that position.

Clearly, there are vast complexities beyond the scope of this very simple analysis, but just because something is simple does not mean that it's wrong.

Just cut 'em loose, just send 'em packing!

 

So, co-Presidents Trump and Musk propose to eliminate at least $1T (one trillion U.S.D.) from the federal budget. Let's do some off the cuff estimating. For workplaces that have not undergone automation, it's often the case that, to first order, about half of the expenses are for employee compensation and half are for materials (could be hard goods, soft goods, software, etc.).

Using this heuristic, we'd eliminate $500B in employee compensation costs. Again, to first order, I'll estimate that each employee's compensation totals $100K. Thus, something like five million former government employees would be looking for work. It's very difficult to see where these five million would find such work. I suspect that the co-Presidents would say "we'll deport at least that many undocumented aliens, the former government employees can take the jobs left vacant by such departures, and our tariffs will result in a huge influx of manufacturing jobs being on-shored or reshored."

I have my doubts. It's an intriguing thought experiment, but I'm not so sure that the skill set of, say, a NOAA meteorologist or an air traffic controller adapts well to lawn maintenance, housing construction, or fruit harvesting. Could they learn to do such work? No doubt. Would it be an efficient use of the skill set that such people have spent years developing? No. And that's emphatically NOT to say that the skill sets of lawn maintenance, housing construction, or fruit harvesting are any less valuable (though in many cases, not so time consuming to acquire) as those of the terminated federal workers.

So, do we then provide unemployment benefits for such people? Can we deport undocumented aliens as fast as the co-Presidents can fire federal workers?We certainly do live in interesting times, I hope not to come to the attention of important people!

Pay to play

 

With some exceptions, I've mostly stayed away from politics. I do have a few "embarrassed to be a conservative" posts though, and it's time for another. While there are many aspects of Joe Biden's policies with which I disagreed, and several policies professed by Donald Trump with which I concur (though far from all), it's horrifying to me to see the "in your face" grift exemplified by the Trump meme coin ($Trump).

It's true that, to within epsilon, 100% of meme coins are worthless (other than in the sense of the "greater fool" exploit) and only benefit the initial promoters. And I've felt little sorrow for those who've lost money in, for example, the $Hawk coin promoted by the Hawk Tuah Girl. Yes, she and her team did the prototypical rug pull and buyers of the coin mostly just gave away their money to her and her team. But, what did they actually expect to happen? Come on...

But to see Trump, days away from swearing in as our 47th President, promote a meme coin where he is in control of the large majority (80%) of the coins and watch as his supporters purchase the coins is despicable. Trump and his supporters often speak of the machinations of the Deep State, wherein malign agents, embedded deep within the government, work in secret to undermine freedom and advance a New World Order agenda.

But Trump can't be bothered with working in the dark to promote himself and his personal enrichment, one might call Trump and his ilk the Shallow State. His self-aggrandizement and self-enrichment is out in front, for all to see.

And, not to be outdone, next came the $Melania coin from the First Lady. Both coins, initially and unsurprisingly, soared in value as the zealots rushed to funnel their money to their heroes. Prices have since fallen back somewhat, so some have already lost money. Now, it is true that you cannot take the number of coins held by the Trump group, multiply by the current posted value, and determine that Trump has made billions overnight through this obviously corrupt grift. Were he to dump all of his coins, the market would tank.

Nevertheless, he could sell them slowly into the market. My bet would be that, when some wild Trump event takes place, the value of the coins will rise and he'll make small sales into that market. But, as I type, the price of a Trump coin is $29.53. The total supply of coins is 1 billion, and Trump owns 80% of them. Keep in mind, they were created out of thin air, a few minutes of programming was all it took.

That's our President. As someone noted, we voted for corruption, and Trump is delivering.