Technical math problem: How to figure out varibales as they change? With/out cal

[Beamin]

In the 1960's how did they figure out how fast the rocket to the moon would go as it lost fuel gained speed? Or can this be done dynamically with one equation without a computer or even a calculator? For simplicity ignore wind resistance and say you only had to figure out how to just get it only into orbit: Since you have the constant of the thrust of the engines is the same all the way to orbit, but speed is changing, acceleration is changing as you go up. One way to figure out would be to break the problem down into sections: X amount of thrust and Y amount of weight for 10 seconds =Z, then you do for 20 seconds then 30 etc. To get an accurate measurement you would keep doing this calculation using smaller and smaller time intervals 10 sec 11 sec …

But how would you figure this out dynamically without having to keep doing the equation over and over with your accuracy limited by the number of times you do the equations. iE 120 seconds to orbit with a 1 sec accuracy would require doing the equation 120 times. Is it possible to do one "elegant" equation that's valid for anytime? OR does this have to be done using computers? Does this make sense? Its hard to explain. Since your weight is going down constantly your variables are constantly changing making the one you used 1 second at lift off not work 119 seconds after because the rocket is lighter. Related question: Did the Apollo computer have to figure this out so it knew exactly when to fire/stop the rockets or was a mainframe on the ground doing this and sending telemetry because calculating this as too complex for onboard computers?

[MasterT]

I think, it's called https://en.wikipedia.org/wiki/Differential_equation
There is also https://en.wikipedia.org/wiki/Integral_equation
People were able to solve complex non-linear equations centuries before first computer was invented.

[T3sl4co1l]

Exactly. You've discovered calculus, congratulations

Tim

[djacobow]

Let's cut the dude some slack. Most kids learn calculus, and maybe diff eqs, before or without the slightest internal motivation or curiosity about why these areas of math exist and why they should care.

There's something nice and honest about first being curious about how something works and then discovering that there are whole branches of math to help answer the question.

[rstofer]

And when the differential equations get out of hand, like controlling the rocket nozzles to keep the rocket vertical just after launch, they used analog computers.  In design and in flight.  My differential equations instructor was involved with the nozzle control of the Atlas missile.  It isn't easy keeping those monsters vertical until they get enough velocity to fly.

Here's a picture of one of Convair's analog computers.  They had a larger unit up at Kearney Mesa (Astronautics Division) when I was there in the early '70s.  There is an excellent description of actually programming the solution to a fairly simple second order ordinary differential equation.  The solution technique is due to Lord Kelvin.

http://www.rfcafe.com/references/radio-news/convair-analogue-computer-january-1954-radio-television-news.htm

https://www.google.com/search?q=convair+analog+computer

Here's a discussion re: gimballed thrust

https://spaceflightsystems.grc.nasa.gov/education/rocket/gimbaled.html

Lord Kelvin:
https://en.wikipedia.org/wiki/Differential_analyser

I am slightly into analog computing with 3 machines (2 Comdyna GP-6s and a DIY) and I'm looking forward to my grandson's differential equations course coming up in a couple of semesters.  I didn't really enjoy it in college but I'm definitely up for it today.   We have excellent tools for solving or modeling differential equations (MATLAB) and the Simulink add-in allows me to build an analog model just like an analog computer.  Very nice!  Much better than slide rules and the dreaded Spirule used in the Laplace Transforms (Control Systems) course.  I still don't understand that process.

Engineering is all about math!

[T3sl4co1l]

Let's cut the dude some slack.

Not sarcastic, mind.  Newton had a heck of a lot more work cut out for him, when making the same realization and having to create his own framework to use.  Fortunately it's a lot easier today!  (Though actually cranking through the analysis is not much easier... )

Tim

[rstofer]

Let's cut the dude some slack. Most kids learn calculus, and maybe diff eqs, before or without the slightest internal motivation or curiosity about why these areas of math exist and why they should care.

This is correct and regrettable at the same time.  As I help my grandson through math, I try to provide context.  "Why do we do this?", "Well let me show you an example!".

Don't understand 'e' and 1-e^-t/tau?  Let's toss in a charging voltage and I'll show you the voltage on a capacitor as it charges.

It's all about context!

[Beamin]

Let's cut the dude some slack. Most kids learn calculus, and maybe diff eqs, before or without the slightest internal motivation or curiosity about why these areas of math exist and why they should care.

There's something nice and honest about first being curious about how something works and then discovering that there are whole branches of math to help answer the question.

Yes unfortunately the seizure that left me disabled wiped clean ALOT of my memories from when I was younger. I know I took calculous in high school too. Worst injury is a brain injury I can still think just chunks of long term are gone.

[larsdenmark]

They are all good questions!

The rocket equation has been derived a couple of times, but Tsiolkovsky's is probably the most famous one since he wrote a book on it (that Oberth and van Braun based their work on):
https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

And no, you don't necessarily need a computer to calculate the burns of the rockets as long as you know where you are. There are examples where the burns had to be recalculated/corrected in flight and people did this using pencil and paper (and slide rules). But computers on Earth and on Apollo were used as well for normal situations.

Here is some background info from 1967