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What I am looking for is an example where real persons in the real world really need to mathematically model a situation before they need to calculate the average and instantaneous rate of change. I want to to use this example as an entry point to calculus for students. It's supposed to be a big question to pursue/problem to solve while learning about those three aspects. Every time we move to the next aspect we learn something new to solve this big problem.

I already asked this questions over at matheducators.stackexchange and received an answer I thought was worth investigating:

What about something like a rocket taking off into outer space?

  • The process needs to be modelled up front because of the cost and risk to life involved in a failed mission.
  • The average rate of change needs to be calculated in order to ensure that the rocket gains enough speed to reach escape velocity, otherwise the mission will fail.
  • The instantaneous rate(s) of change need to be calculated in order to ensure that the rocket materials and crew can cope with the stress of acceleration.

This answer seems somewhat convincing to me, although I am fully aware that I would need to use a simplified version of this example. Simplifying this example while keeping it plausible however raises some question:

  • How is the process of launching a rocket modeled up front (in a simplified sense)?
  • I know there have been engine tests for the Saturn V rocket. Did NASA – in simple words – use the data from these tests to project the velocity the rocket would reach in order to calculate the average and instantaneous rate of change?

Rough design-sketch of my explanation for the students so far

  • When a rocket is launched into space it has to be ensured, that its acceleration for any given moment doesn't drop under a certain value, since that would mean that the rocket's velocity gets to low for escaping the earth's gravitational pull. On the other hand the acceleration can't be to high, since the resulting g-forces could damage the rocket itself or injure the crew.
  • This can't be solved in a try-and-error attempt, because of the cost and risk to life involved in such an attempt
  • So what do NASA-Engineers do?
  • By testing the rocket engine, the velocity of the rocket for any given moment of its voyage into space can be projected.
  • [insert time-velocity data].
  • NASA-Engineers take these data and insert them into a coordinate system. If we compare the time-velocity-pairs we'll notice that the velocity-values can always be described by inserting the time value into a specific formula
  • [insert plausible function and add some explanation what a function and a graph is]
  • [I would then go on an explain how to find the average rate of change (aka average acceleration) and how to find the steepest point of that curve (aka highest acceleration)]

So the big question is:

Is this introduction anywhere near "the real thing"?

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  • $\begingroup$ Take a look at my answer regarding launching Space Shuttle SRBs and you will see some of the equations that must be integrated over time to simulate the launch. This cannot be done analytically since the equations are too non-linear, but gives you an idea of the calculus involved. $\endgroup$ – Brian Lynch Jan 9 '16 at 16:57
  • $\begingroup$ When it comes to testing and then simulating based on results, the idea is to understand what the performance of the rocket motor will be -- thrust, mass flow rate, and specific impulse being the important factors for trajectory. Testing allows you to characterize those features and then you can simulate with the same values to ensure the vehicle will meet requirements. $\endgroup$ – Brian Lynch Jan 9 '16 at 17:16
  • $\begingroup$ This article discusses an amateur simulation of the Saturn V ascent with very good results. The author doesn't share his code, but he describes the elements of the simulation in some detail: braeunig.us/apollo/saturnV.htm $\endgroup$ – Russell Borogove Jan 9 '16 at 18:17
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    $\begingroup$ We mathematically modeled everything about the shuttle (at least to the level the crew and MCC could see) for training purposes, in the Shuttle Mission Simulator. Just pointing out that preflight simulation doesn't have to be for analysis but can serve other functions like training. $\endgroup$ – Organic Marble Jan 9 '16 at 23:03
  • $\begingroup$ Grab Zipfel's book. $\endgroup$ – Deer Hunter Jan 10 '16 at 20:54

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