Was Isaac Newton's solution to the problem of solid of least resistance comfirmed in space programs?

Question

I was very surprised when i found out that Newton's solution to the problem of solid of minimal resistance was a body with somewhat strange nose cone - the end of the nose cone is a plane perpendicular to the body's axis. So my question is, was Newton's calculation confirmed and used somehow by space programs?

Answer 1

TL;DR

Newton got some things spectacularly right, and other things spectacularly wrong. In this case, he didn't remotely understand the real physics of the problem - and neither did anybody else until a few hundred years after his death.

You shouldn't believe everything in Newton's writings on alchemy either - nor his predictions of the date of the end of the world based on his analysis of the Bible.

Longer answer

The great importance of Newton's work attacking this type of problem in Principia was not that he got almost all the details wrong, but that he attempted to solve the problems at all using mathematical modelling.

In Newton's time, there was no clear understanding of the difference between energy and momentum. The greatest achievement of Principia was in pinning down the concept of "momentum", and the relationship between "force" and "rate of change of momentum", as given by Newton's second law. And the mathematical edifice that Newton built on that foundation was of course spectacularly right.

But with no real concept of energy, Newton's ventures into fluid mechanics were much less successful. I very much doubt whether anyone working in the space program would be naïve enough to consider use Newton's models - and most likely they would know nothing about them.

To show the sort of errors in Newton's fluid dynamics models, it might be instructive to start with a simpler problem which he also considered in Principia, namely to calculate the speed of sound in air. Newton attempts to solve this by comparing the vibrations of the air with the vibrations of a pendulum (which he could both analyze and measure experimentally), which was a good way to start. But he had no notion of the concept of the internal energy stored in the fluid via the motion of individual molecules, which is related to the temperature of the fluid. In fact, in Newtons' time temperature measurement had barely progressed beyond the subjective and qualitative human notions of "hot" and "cold."

Using modern terminology, he incorrectly assumed that all the internal energy in the fluid was accounted for by changes in its pressure and volume. This led him to the equation of state $PV = \text{constant}$, instead of the correct adiabatic relationship $PV^{\gamma} = \text{constant}$ where $\gamma \approx 1.4$ for air.

After deriving the properties of air from experiment, this error produced a calculated value for the speed of sound that was about 10% slower than what had been measured. (His measured speed of 339 m/s was probably accurate, but there is no record of the temperature, humidity, wind speed, etc, when the measurement was made).

To account for this discrepancy, Newton makes the wild assumption (Principia Vol 2 Section 8 proposition 50) that if the particles making up the air are rigid spheres of finite size, and separated from each other by 10 times their diameter, then (by analogy with a "Newton's cradle" device with gaps in between the balls) the speed of sound would be about 10% higher than his calculated result. (It is perhaps unkind to wonder if he had forgotten his grand declaration "Hypotheses non fingo" - "I don't make speculative assumptions!") He doesn't attempt any mathematical justification of that idea, nor does he explain how the air particles align themselves with the direction in which the sound is travelling!

He then goes on to consider the effect of water vapor on the speed of sound. He seems to consider the particles of water vapor to be simply inert bodies that occupy some proportion of the volume of the air, and therefore that water vapor increases the speed of sound further by adding more "balls" to his supposed "cradle". Of course this logic is actually backwards, but Newton had no idea that the molecular weight of water is about half that of air, and the presence of water thus reduces the density of the air while having little effect on its compressibility.

His analysis of the air resistance of a projectile (Principia Book 2, Section 7, Proposition 34, theorem 28) is equally far from the truth. He attempts to do this by considering the change of momentum of the projectile as it collides with stationary particles of air and knocks them aside. He assumes the collisions are perfectly elastic, and therefore (in modern terminology) no mechanical energy would be converted into heat. That means he is assuming the fluid is inviscid. But the correct mathematical analysis of inviscid irrotational flow gives the result that the drag force is zero for any shape of body! Newton made a second mistake, in ignoring what happens to the air behind the projectile.

In fact, Newton's model is not too unrealistic for a completely different scenario, namely piercing a hole through a (fairly loose) granular material like earth. Indeed, the shape of the nose of a typical "mole" tool used for trenchless tunneling is the simple pointed cone shape that Newton claims would give the least drag, as shown here: http://en.terra-eu.eu/underground-piercing-tools/2-16-produkte. But any similarity between that scenario and the analysis of a hypersonic space capsule is purely conincidental.