I wish people had more exposure to building mathematical models of things. I am fairly convinced that the only real exposure I was given was to models that we knew worked. So much so, that we didn't even execute many.
Specifically, parabolic motion is something you can obviously do by throwing something. You can, similarly, plot over a time variable where things are observed. You can then see that we can write an equation, or model, for this. For most of us, we jump straight to the model with some discussion of how it translates. But nothing stops you from observing.
With modern programming environments, you can easily jump people into simulating movement very rapidly and let people try different models there. We had turtle geometry years ago, but for most of us that was more mental execution than it was mechanical. Which is probably a great end goal, but no reason you can't also start with the easy computer simulations.
Something I really like is that the curve that a rope or thread makes when fixed in two points but not under tension, it's not a parabola. It really looks like one though, but it isn't. It's a catenary.
That's something you can verify by writing some simulation code, then drawing the curve, and then drawing the best matching parabola on top. It doesn't fit.
To model the issue mathematically you need some not-too-advanced calculus. On both the computer simulation and the mathematical model, you model the rope as being made of very small elements that are linked together (like a chain). In the simulation those elements are small, but finite. In the math you take the limit as the volume of the element tends to zero.
It's the same way of thinking but math gives some different tools, enabling you to solve the curve analytically
That's totally a tangent but I was reading a bit and it happens that, in practice, real cables bend in an inbetween curve between catenaries and parabolas
> Comparison of a catenary arch (black dotted curve) and a parabolic arch (red solid curve) with the same span and sag. The catenary represents the profile of a simple suspension bridge, or the cable of a suspended-deck suspension bridge on which its deck and hangers have negligible weight compared to its cable. The parabola represents the profile of the cable of a suspended-deck suspension bridge on which its cable and hangers have negligible weight compared to its deck. The profile of the cable of a real suspension bridge with the same span and sag lies between the two curves. The catenary and parabola equations are respectively, y = cosh x and y = x²( (cosh 1) − 1) + 1
> If the chain is carrying nothing other than its own weight, the resulting shape is a "catenary". If the chain is like a suspended cable carrying a deck below it, and its own weight is nothing compared to that of the deck, the resulting shape is a "parabola".
Which shows that sometimes your model (either using pure math or a simulation) is too simple to capture whatever is going on in the real world. (it gets further complicated when one considers elasticity etc)
Exactly! I think this scenario alone would be an amazing set of lessons for many grade schools.
Move this into modeling and then guessing stuff like bridge tensions, and you can easily show what many of the maths are good for.
We used to have this with the attempts at building tooth pick bridges and such. Which I still think is very illuminating. But, I think there are a lot of questions you can expose with models that were often only seen by the more advanced students. And again, I agree that getting people to mentally model these things is a good goal. Right now, people rarely ponder things on paper, it seems.
Specifically, parabolic motion is something you can obviously do by throwing something. You can, similarly, plot over a time variable where things are observed. You can then see that we can write an equation, or model, for this. For most of us, we jump straight to the model with some discussion of how it translates. But nothing stops you from observing.
With modern programming environments, you can easily jump people into simulating movement very rapidly and let people try different models there. We had turtle geometry years ago, but for most of us that was more mental execution than it was mechanical. Which is probably a great end goal, but no reason you can't also start with the easy computer simulations.