Using these kinds of relationships you can make models that accumulate over time: that is involve rates of change. The foundation idea is that accumulations (numerical integration) is rather simple—just like stacking up Cuisinaire rods, and so fundamentally based on addition. However differentation, and all that talk of rates of change and definitions such as
vt = a
can be rather off-putting.
You are working in this quadrant of the modeling space:
This is well trodden ground, and has been since Nuffield A level introduced paper and pencil methods for numerical integration, followed by the DMS system arriving on the BBC Micro in 1987. Modellus developed these ideas further, and was integrated into Advancing Physics. Stella, Coach and other
stocks and flows based modeling systems, based on systems dynamics thinking, follow a parallel path, and some are devotees.
In elementary physics there are rather few accumulating relationships, although some lie at the heart of difficulties with kinematics, which is all about such accumulations.
Using a triangle to suggest an accumulation—a visual reminder that one quantity changes another with time—you could write:
a tells v how to grow)
v tells x how to grow)
Time does not appear here, because it is a Newtonian universe, where the universal time is ticking away in the background, and we're dealing with processes (which necessarily evolve over time).
Add some more, such as
and you are essentially done. So the trick is to make it also appear to be that simple, and yet remain accessible.
In the imagined worlds rendered as computational models something happens on screen as the process unfolds, so there is both feedback and the chance to explore consequences. You can alter the rules and the initial conditions and see the effects, rather than having to conjure all of them up by a heroic act of the imagination .
So let's look at a completed model, rather than building one up by stages:
Accumulations can also be negative...
Some accumulations are so common that you might want something quicker. Here is one that accumulates all of the dynamical and kinematic quantities, incorporates a feedback link, and so models simple harmonic motion from first principles.
The secret is all in the code, and here there is not much. The trick is to isolate the physics and represent it in ways that offer an appropriate reasoning tool for the children. And a tool that they can use, not just look at.
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Even further back
If you prefer something more flexible, but still graphical, there was Variables and Relationships (VnR), developed as a downloadable Java app for the ASE science year 2000 series of CD-ROMs.
The idea of accumulations probably originated as