That was me. If Heiwanders ignores the total system and focuses only on the spacecraft, it has indeed LOST kinetic energy. He tries to make up for this by arbitrarily changing the sign of the equation. He lies to himself about his own equation.
This is more evidence of ham-fisted fumbling rather than knowledgeable study. In orbital mechanics we commonly formulate specific energies such that they often come out negative, based on how we set up the frames of reference. It's no big deal to have a negative number. Because we know qualitatively how they relate to the geometry of the reference frame, we know algebraically what the numbers should look like, even if they happen to be negative.
People trying to fumble their way through a physics problem by "mathematizing" intuitively-derived concepts and properties often freak out when a number they associate with a real-world property comes out to be negative. "How can I have
negative energy? I can't have negative energy; I must have done something wrong." And so they arbitrarily change the arithmetic signs in their equation to make the values come out "right," showing that they really don't understand the formulation either way.
One of the things you learn very early as a professional engineer is to trust the numbers. Which is to say, understand
why the equations you use come up with the kinds of numbers they do. That means understanding the finer nature of the relationship between measured values and their reference frames, but also "letting go" and trusting the abstract nature of some of what you do.
That leads to the second major point of his fumble-around method, which we've belabored. The "system" as it applies to a momentum or energy computation -- anything where conservation is an expected property -- is an abstract concept. Anders consistently fails to do the problem right because he has an intuitive, concrete idea of what his "system" is: the physical, geometric boundaries of the spacecraft. That prevents him from considering that the expended propellants, now many kilometers away from the spacecraft, are still part of the system he defined at the outset.
People who use these formulations correctly, and are facile with them as part of their jobs, have no problem with such abstract, counter-intuitive definition. They have no problem considering the system as composed of the spacecraft dry mass separately from its propellant, and to properly account for them as coupled mass initially, but then physically distinct mass (or energy) later on. (In the real world, propellant slosh within the spacecraft is actually part of the energy-balance. That's how fine-grained some of these analyses can get.)
That's the facility that fakers and charlatans can't match. Not only do they fail to achieve appropriate abstraction in their own work, they cannot recognize it when it is presented to them. They are forever stuck in the layman's feeble practice of shoehorning their intuitive misconceptions into the formalisms. Smart students realize that it doesn't fit, and gradually adjust their thinking to embrace the formalism and the abstraction it expresses. Conspiracy theorists just blame someone else.
