Today I'd like to remind my readers of the dreadful mathematical blunder that Hoagland point-blank refused to discuss with Howard Hughes on 2nd June. I covered this rigorously back on 25th August 2008, but this time I'm going to use a more narrative format. So here's an account of how badly Hoagland botched the rocket equation—but without the equation.
Altitude and velocity
To achieve Earth orbit, you have to accomplish two separate tasks. First, you have to raise your satellite high enough that it's above the atmosphere. Secondly, you have to propel it horizontally fast enough that it's unable to fall back to Earth under the influence of gravity. The minimums are roughly 120 km and 7 km/sec respectively. Explorer 1, America's first satellite, did much better than the minimums. It went into orbit at an altitude of 357 km and a velocity of 8.215 km/sec. It achieved this with a four-stage rocket. The first stage, a liquid-fueled "stretched" version of the Redstone missile, took care of all of the altitude problem and a little bit of the horizontal velocity. Then, three separate solid-fueled stages took over the task of boosting Explorer to orbital speed.
The interesting part, to Hoagland, was that the three upper stages over-performed. The overspeed was 0.197 km/sec (roughly 650 ft/sec in old money). Hoagland suspected that this was more of an over-performance than could be accounted for by variability of normal factors like rocket fuel performance and high-altitude winds. So he sat down to do some calculation.
The "rocket equation" enables accurate calculation of the velocity contribution of a rocket stage (known as delta-V,) given knowledge of the power of the fuel (measured by what's called the specific impulse) and the mass of the rocket before and after the fuel has done its job. Hoagland's first mistake was that he did one single calculation grouping all three stages together, instead of figuring delta-V stage by stage then adding them up. You can't do that because your result will then ignore the very important point that rocket staging progressively drops off empty stages along the way. That's the whole point of it, really. His second mistake was that he failed to evaluate a logarithm which is at the heart of the equation. It's as if he didn't even notice it.
Well, obviously, he got entirely the wrong answer and it enabled him to state, quite incorrectly, that the velocity excess was more than could be accounted for by those conventional uncertainties I mentioned. He said it was 17%, but the real answer is 4.2%. What that means is that Hoagland's proposition that Wernher Von Braun had "an anti-gravity secret" is pathetically, irretrievably, wrong. Back in 2012, when he was still contributing to his Facebook, he actually acknowledged that his math was faulty but claimed that it didn't matter because the velocity excess was still more than could easily be explained. That, of course, is a lie. And what is truly reprehensible, for someone who has claimed to be a scientist, is that he has never amended the "Von Braun's Secret" web page. Today, it's still as wrong as when he first wrote it in August 2008.