How could a rocket/missle know it's exact speed [With TL;DR]?

Question

So i was reading my book and it seems that it makes a very odd statement, claiming that differentiation is used by rocket scientists to calculate the speed of the rocket.

Now that seems very weird to me. As far as i know, there is no way that there is a proper formula to calculate the speed at which the rocket is actually going (You can have a formula for an ideal environment, but that formula shouldn't face up to many random factors like air friction, difference in the performance of the engine e.t.c).

That leaves me to conclude that the rocket must be calculating it's velocity in some other manner. That's an interesting question in itself but not what i'm here to ask. Let's assume that the rocket has some means (Or most likely, several) of calculating it's velocity within a finite time range. For sake of convenience, let's say milliseconds, although i think that may be too exact?

Now the trouble is, how does the rocket know it's instantaneous velocity? If you were to give data to a human about the speed in the last minute, the last half minute, the last quarter minute, last second, last half a second ... last millisecond, the human can make a reasonable guess as to what number the data is tending to. But without an algorithm, how could a computer do the same?

TL;DR Is there any algorithm that when provided with the average change over a range of range of time, allows you to make an effective guess of the instantaneous rate of change? How does differentiation figure into the picture?

Answer 1

I strongly suggest you read this Wikipedia section about average and instantaneous velocities. Yes, differentiation is used to get the instantaneous velocity of a rocket. We usually assume an ideal environment. In truth, the real world does leave too many variables for consideration. However, in certain cases, variables like air friction make an extremely small difference, so an approximation using differentiation can be satisfactory.

How does the rocket know its instantaneous velocity? It doesn't. Rockets are inanimate objects. They don't know anything.

Your TL;DR question says "average change over a range of range of time". That doesn't make any sense. In any case, knowing the average change over some period $(0,t)$ does not let you know the instantaneous velocity at any point in $(0,t)$: the only thing that's certain is that by the intermediate value theorem, at some point the instantaneous velocity will be (momentarily) equal to the average velocity.

Answer 2

TLDR: curve fitting.

Given a set of discrete data which measure a continuous process, for example GPS coordinates at millisecond intervals, there are a number of methods to obtain extremely precise interpolations, for example a Taylor series.

Once an expression for the data has been defined, calculus can be applied to find the derivative of this curve.

The accuracy of this derivative is limited only by the accuracy of the original data and the algorithms used and the computer using them, and thus it can be made as accurate as any direct measurement of the speed (like say from external LIDAR) if given sufficiently precise and capable input and processing.