Question

What are good questions to ask yourself to see if you understand an relation well?

Answer 1

I change the form of relations, or equations: usually, putting it in the a less simplified and more form resembling the original "engineered" form helps understanding its intuition better.

But for some reason, even though i know that there are infinite algebraic forms to any one equation, I try to understand a relation in the form I see it. I try to do that because I think it gives a new perspective to the concept when it's related to other inputs/outputs.

That gets frustrating sometimes, I don't know what questions to ask, or what to investigate to understand a relation more without changing forms/ IOs\[V^2-V_0^2 = 2a(X-X_0)\]
Take this for example, I do understand the relationship. I can restate it in different forms, but I want to understand the mechanical procedure here, what it means.

I get frustrated asking myself seemingly complicated and simplistic questions like "Why are those squared? What does that mean in this context? why is it the difference between two squared quantities? Why is 2a multiplied by the distance covered??"

What do you guys think about this line of thought? is it bad????

Answer 2

I always want to know why a formula is a certain way and a lot of formulae can be proven in a trivial way. which is always a nice thing to have in mind. Of course it's not always easy to prove a formula so the next best thing is to know what the formula does and when to apply it.

Answer 3

actually it reminds me of a cellular neuroscience course I did cover electrophysical properties in cells and neurons, and there were a heap of equations relating to resistance, current, membrane voltage difference, ion flux etc.

The course did require some derivations involving the Nernst eqn etc. but that sort of helped understand it better when you play with the formulae, but obviously, not all students were keen on the mathematical rigour involved.

Answer 4

however I will specify it is not always helpful to try and consider an equation as it is. As you said there are infinite algebraic forms for a given equation. You just see the final result, the sculptor's work but never what it came from. So I think you can waste your time trying to demand an interpretation that is garnered from a more elementary form (especially if there are specific conditions that push the elementary form into the final form).

It goes with no surprise that if possible they will teach you the 'elementary' form and move towards the simplified form which you may employ in your calculations etc. as a simple example, we know that the derivative of any power of x with respect to x is nx^(n-1).
Does this help you understand why necessarily? Not really. There is a bit of work that leads to the result you could not interpret independently.

Answer 5

You can find that equation using two of the more standard equations, derived from calculus, which may help you understand it better

\(\large v = v_o + at\) <=== solve for t

\(\large x = x_o + v_ot + \frac{ 1 }{ 2 }at^2\) <=== substitute in here

Answer 6

^oh yes of course, I knew it was a familiar formula.
which adds to point how you need to line of work leading into the equation to truly understand it.

Answer 7

@aagent0smith ooooooh that's neat! I didn't know that. so it's it's x(t(v)) or x(v^-1) that's interesting!!!

@mww Yeah. yeah that's whyit's frustrating to see one equation and asking what a separate mechanical operation "means." or how it relates to other parts. The operations themselves are of course pretty simple to understand, but it's sort of like trying to make a straight connection between a messy rubik's cube and a solved rubik's cube. the operations you perform to solve it are simple, but many. but many of them have been performed, and it gets pretty complex to express a direct connection.

Yeah that's true. but I think for some equations at least, the alteration in form gives an interesting process, or relation between different variables, which can boost insight, like you said. I guess I should stop questioning the "meaning" of composite algebraic operations but instead try to seek meaning in the process of arriving to that conclusion by proving.

Good feedback you too, thanks!

Answer 8

$$\large v^2-v_0^2 = 2a(x-x_0)$$
This is one of the kinematic formulas that relate velocity , acceleration and displacement. Note that this equation is only valid when acceleration is constant. It's a useful equal because in a kinematic problem you may not be given the time elapsed.

It might be more instructive to see how its derived from more basic principles and definitions. The best way to get comfortable with a formula or equation is to plug in numbers.

Answer 9

Hmmm... plugging in numbers? I guess if you make different variables constant, you check for extreme case usages of the equation? those can show interesting properties

Answer 10

Or rather doing actual problems.
And yes if you make variables constant you can see how that affects the equation.

Answer 11

Yes! Makes sense! Thank you!