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this question may sound weird but I have notised some similarities between many physics formulas like $Ek = frac {1}{2} mv^2$ and $s = frac {1}{2} at^2$. Is it just a coincidence or are the actually some "versitile/universal" formulas, which i could use for almost anything (if i understood physics)? Or do I have to remember hundreds of formulas to be able to solve problems? (high school type of problems)

Following questions may be for a whole another topic, but i was thinking that maybe ..
Does it have something to do with derivatives and integrals? We have learned in maths how to count these but i have no idea how to use it in physics. And how do you know that the derivative of $s$ is $v$ and derivative od $v$ is $a$ etc? Do I just have to remember?
What about graphs? How do you understand these? I mean, how do I know that for eg $s$ is the area under the curve (integral) when i draw the graph of dependence of speed on time?

I know I have many questions but I just want to know how to see the bigger picture in physics, because in my school the teacher just tells us "use this formula for this, this one for this .." and I don't want to be like a machine that learns hundreds of formulas but doesn't understand anything.

So maybe if you have any advice on how to learn to understand, or basically any advice i would appreciate anything.

I completely disagree with the way I hear of many teachers approaching physics that basically boils down to "memorize the formulas" or "just find the right formula to use". In physics is is much better to know the principles and how the equations relate to these principles. Then you can start from the definitions and reason through to other things, thus removing the need for memorization.

With that being said, there is some "memorization" involved. As you say, there isn't a way to derive definitions. Those are things you have to know. Like how velocity is the rate of change of position, or how momentum is mass times velocity. You do have to know these to progress in physics.

However, once you know these, then you can start to see how other "formulas" are derived. For example, if you know that $a=frac{d^2x}{dt^2}$, then if you know your object has a constant acceleration starting at rest, then you can determine using calculus the equation you give of $x=frac12at^2$. No memorization is required. Through working problems you will find that the definitions and "basics" become part of how you think about more advanced topics and problems. Throughout my entire physics career I have never had to sit down and actively memorize anything, and when in classes I never used the supplied equation sheets. This is because if you truly understand the equations, what they mean, and where they come from, they are a lot easier to use.

As for your specific example, I think it is just a coincidence. Equations having the same mathematical form does not necessarily mean there is a physical relationship.

Finally, for your questions about graphs, learning how to use the graphs should come along for the ride in your studies. You might also pick up some things in your math classes

So overall, my advice would be:

1. Make sure you understand definitions


2. Make sure you understand where an equation comes from, what it means, and the context in which you are allowed to use it


3. Do practice problems. Just how athletes need to do drills to get better at their sport, build muscle memory etc., physics students need to do practice problems to develop understanding, build intuition, and learn how to think about physics problems.


4. In your practice problems, focus on problem solving rather than just memorizing formulas or methods. You have to understand why you are doing what you are doing. Two problems that may seem similar might need different techniques in order to be solved. Your first thought should never be "What formula do I use?" It should instead be "how am I going to think about and solve this problem?" At times this can be a subtle distinction, but I think it is necessary to be successful in learning physics.

Calculus didn't exist before Newton and Leibniz, but physics did.

Newton had his three laws.

Kepler had his three laws.

Galileo showed how to measure things, like how fast things fall.

Those formulas you're asking about are trying to simplify things, so go slowly, learn them one at a time, and what they're talking about.

Whatever you do, memorizing is the last thing you should do. The first thing is understand it.

"How to understand physics and formulas?"

Glib answer, but true: do the exercises, especially the ones you are afraid of......

Also, when you a see an equation, LOOK hard at it, and without writing anything down, try to see what it would mean if you set one variable to 0 or infinity, in your head see if you can change things around to make it easier to solve.

Draw a curve whenever you can, see where it cuts the x axis (the roots of the equation) and see by putting in values for x near the roots, which way is the curve going.

Look at the equation, is it odd or even, (you should look up what this means if you don't know), and how this will look in the curve.

Tricks and shortcuts are perfectly valid, but they take practice to learn.