# How do i gain intuition to learn and remember physics?

Throughout my life, most of the things I've learned have come naturally, and seem to commit to memory without the need for much effort.

However now while I'm in the process of self-studying math and physics, i find that i constantly seem to forget everything i learn and can only make progress by making rigorous use of memorization techniques and disciplined note review.

However i had been hoping to just read physics materials and naturally absorb content and get an intuitive feel for it overtime, as note taking and the need for memorization techniques kind of suck the joy out of the process.

The problem is i feel like I'm just reading a bunch of random facts and equations and cant solidify or connect the information in some meaningful structure, so i keep forgetting it all.

I also forget because many of the equations seem to make no intuitive sense to me so i cant deduce them in practical situations.

i.e. I have no idea why centripetal acceleration is velocity times itself and then divided by the radius.

So i suppose my questions are:

1. Does anyone have an alternative perspective or mental framework they could suggest which would help me latch on to, intuitively understand, and categorize physics concepts more efficiently?
2. How can i get a better natural intuition/understanding for equations?

## Answers and Replies

Simon Bridge
Science Advisor
Homework Helper
Throughout my life, most of the things I've learned have come naturally, and seem to commit to memory without the need for much effort.

However now while I'm in the process of self-studying math and physics, i find that i constantly seem to forget everything i learn and can only make progress by making rigorous use of memorization techniques and disciplined note review.
If your learning style so far has concentrating on memorization, then you are discovering that this does not work any more.

However i had been hoping to just read physics materials and naturally absorb content and get an intuitive feel for it overtime, as note taking and the need for memorization techniques kind of suck the joy out of the process.
Just reading the aterials will not work either ... that is just another, more casual, memorization process.
The only way to learn physics is to practice it.

I also forget because many of the equations seem to make no intuitive sense to me so i cant deduce them in practical situations.
So you need some way to relate the physics you are being taught to some sort of core concepts?

There will always be some memorizing equations to do though - this is because the point of physics is to discover the few core concepts that are needed to derive everything else... such a discovery would be called the "grand unified theory of everything" and we don't have one yet.

But you will be able to settle for just some sort of linkage.

i.e. I have no idea why centripetal acceleration is velocity times itself and then divided by the radius.
It comes from Newton's Laws ... if an object travels in a straight line with speed v, then applying a constant force, perpendicular to it's motion, will change the object's direction by F=ma. Keep the force perpendicular to the velocity, and the object will go in a circle. So calculate the radius of the circle.

In this case, the core concepts you need are in Newton's Laws - you do just have to know those.

1. Does anyone have an alternative perspective or mental framework they could suggest which would help me latch on to, intuitively understand, and categorize physics concepts more efficiently?
Probably - but I don't know enough about you, or your current education, to make a good suggestion.

2. How can i get a better natural intuition/understanding for equations?
The core concept behind equations in physics is that maths is a language. In physcs, the maths is used to describe the situation - which gives you an equation.
The way to get used to that is to do lots of word problems at a basic level - like the kinematic ones.
Instead of doing consant-acceleration problems using the suvat equations, try doing them by making a velocity-time graph.

1 person
Nathanael
Homework Helper
In physics, no effort should go towards memorizing; but that is not to say that you can just look at an equation and understand it.
It will take effort, and more importantly, it will take patience. By patience, I mean the patience to spend several days on a single problem.

When you first learn a new equation, think about what it is supposed to represent, and then try to find a way to derive it yourself.
Deriving the equations yourself is the perfect solution to memorizing. If you ever forget an equation, you'll be able to derive it "from scratch" (and it will usually take no time at all, if you've done it before).

Preferablly, every once in a while, you should stop before your book shows you the equation (if you think you can) and see if you can guess it yourself. Have a little patience and try to figure it out for a while. More often than not, you won't be able to figure out the equation. But even if you don't figure it out, it is still a useful exercise. When you do finally read the equation, it will be more understandable and insightful.

It is absolutely great that you are self-studying, this gives you an excellent advantage, because you are not on a fixed schedule. You can learn things at your own rate. You're free to spend days thinking about a single idea (or a few ideas) because you don't have a test next week (or whatever; you just don't need to keep up with a class).

I have no idea why centripetal acceleration is velocity times itself and then divided by the radius.
This is a great example. I actually spent several weeks figuring out a derivation for this equation when I first learned it, because it bothered me very much that it made no sense! Of course, over those several weeks, I wasn't constantly thinking about it, I just kept it on the back of my mind.

It still doesn't make "intuitive sense" in the sense that I can verbally explain why it's v^2/r, but it makes "intuitive sense" in the sense that I understand why it has to be that equation (in other words, the derivation is what makes intuitive sense).

Centripetal acceleration is a good place to start, but have patience. To do it the way I did it, you'll need to know basic (very basic) triginonemtry and you'll need to know how to take the limit. The reason you need to know how to take a limit is because the direction of centripetal acceleration is constantly changing over time, so to find the formula, you will have to take the limit as the change in time approaches zero.
(The bigger the time interval, the bigger the angle, the bigger the inaccuracy.)

This is the way I thought about it:
An object is moving in a straight line with velocity v. Draw a circle of radius R such that the object is on the circumfrence. Now if no force acts on the object, it will continue in a straight line (tangent to the circle). So say there is a constant acceleration perpendicular to the direction of the velocity. (Ignore the fact that it changes direction, that will be taken care of in the limit.) What does the magnitude of that acceleration need to be so that the object (after a short time) will end up on the circle (instead of on the straight path that it was already going)?
Remember, the longer the time, the more inaccruate your formula will be. So write the formula, then take the limit as Δt→0

[[[EDIT: Hint: Or take the limit as Δθ→0 ]]]

If you don't know about triginometry or limits, then save this problem for later and study more math.
(Often, it is not the physics that is challenging, it is the math.)

And most importantly, don't worry about it. It will not happen over night, understanding physics is a skill that you develop. You will only be able to develop this skill (the skill of problem solving) if you enjoy thinking about problems just for the fun of it.

(Geez ... :uhh: ... how did this post get so long)

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1 person
Nathanael
Homework Helper
Instead of doing consant-acceleration problems using the suvat equations, try doing them by making a velocity-time graph.
I second this! There is absolutely NO reason to memorize those stupid "kinematics equations" if you understand what the graphs mean! It's all in the velocity-time graph (for those problems).

(At most, you will need to "memorize" the formula for the area of a triangle.)

P.S.
A note about memorizations:
You will end up memorizing many things by accident, simply because you use them so much. But this does not mean any effort should go in to memorizing! (Your effort should go into solving problems.)

Throughout my life, most of the things I've learned have come naturally, and seem to commit to memory without the need for much effort.

However now while I'm in the process of self-studying math and physics, i find that i constantly seem to forget everything i learn and can only make progress by making rigorous use of memorization techniques and disciplined note review.

However i had been hoping to just read physics materials and naturally absorb content and get an intuitive feel for it overtime, as note taking and the need for memorization techniques kind of suck the joy out of the process.

The problem is i feel like I'm just reading a bunch of random facts and equations and cant solidify or connect the information in some meaningful structure, so i keep forgetting it all.

I also forget because many of the equations seem to make no intuitive sense to me so i cant deduce them in practical situations.

i.e. I have no idea why centripetal acceleration is velocity times itself and then divided by the radius.

So i suppose my questions are:

1. Does anyone have an alternative perspective or mental framework they could suggest which would help me latch on to, intuitively understand, and categorize physics concepts more efficiently?
2. How can i get a better natural intuition/understanding for equations?
When you learn, don't try to memorize the formulas as numbers and letters. They describe something from the reality.
It is much easier to remember real examples and connect them to the formulas.
Some examples:

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/dopp.html

http://www.thenakedscientists.com/HTML/content/kitchenscience/exp/balloons-in-cars/

http://en.wikipedia.org/wiki/Kelvin_water_dropper

A.T.
Science Advisor
As others said, you learn it by using it, not by memorizing it. You should try this book: "Thinking Physics: Understandable Practical Reality" by Lewis Carroll Epstein.

However now while I'm in the process of self-studying math and physics, i find that i constantly seem to forget everything i learn and can only make progress by making rigorous use of memorization techniques and disciplined note review.
If you are self-studying at University level then you, as you are finding, need to make rigorous use of memorization techniques and disciplined note review! Of course, you need to do more than this, like solve all the problems in that 1000 page calculus text (and review them several times to make sure you can still solve them...)

However i had been hoping to just read physics materials and naturally absorb content and get an intuitive feel for it overtime, as note taking and the need for memorization techniques kind of suck the joy out of the process.
Who said that studying physics would be a joy? Think of Edison's 99% perspiration and 1% inspiration and you will get somewhere close to what it is like (I'm assuming the perspiration came from pain & suffering rather than a light jog...)

The problem is i feel like I'm just reading a bunch of random facts and equations and cant solidify or connect the information in some meaningful structure, so i keep forgetting it all.
Many physics textbooks are badly written, ask for suggestions for better ones. Even with the better ones, you'll still need note review, memorization, etc... That said, I don't think you can make much of University physics more palatable, or less disjointed - all those mathematical tricks come across like a bunch of random facts the way they are presented in mathematical techniques books. But to make them "hang together" you would have to prove all the results with rigour, and there isn't the time for that in a physics course.

I also forget because many of the equations seem to make no intuitive sense to me so i cant deduce them in practical situations.
Live with it :) Why do you need to deduce them in practical situations? If you have them, use them.

i.e. I have no idea why centripetal acceleration is velocity times itself and then divided by the radius.
Are you expecting to be able to intuit this instantly? Sorry, you can't do that. You can learn how to derive this result from Newton's laws, but that takes long pages of calculations rather than "instant intuition". And after you've done it, you are still in the same situation. You may feel happier having done the calculation. Myself, today, I'm happy just accepting it! I'm sure the other guys got the calculation correct. I might have derived this myself 30 years ago, when I did my physics degree, but now I'm happy to accept it and move on.

2. How can i get a better natural intuition/understanding for equations?
First, try and build up some "simplest case" pictures. With your example, try visualising a satellite orbiting the earth, at constant velocity. You know centripetal acceleration comes from a change in velocity (a vector), and the only change in velocity is in direction, not magnitude. Increase the velocity and obviously the direction changes faster, it accelerates more - and the same if you decrease r (you get a smaller tighter circle...) So this gives you a rough feel for the equation, if not the exact equation.

Second, do lots of problems and hope that increases your intuition.

Third, don't worry if you don't have an intuition - in the final analysis, it's the equations that count. You might end up studying the motion of 26 dimensional objects in 26 dimensional space. Good luck with visualising that! But you don't have to visualise it, because you can calculate it, and that is all that is required.

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
It still doesn't make "intuitive sense" in the sense that I can verbally explain why it's v^2/r, but it makes "intuitive sense" in the sense that I understand why it has to be that equation (in other words, the derivation is what makes intuitive sense).
Here are my 5 cents on the intuition behind the magnitude of centripetal acceleration:

In uniform circular motion, the velocity vector is rotating (without changing magnitude) with a constant angular velocity which must be the same as that of the orbit. The change in the vector is orthogonal to the vector itself and with magnitude equal to the change in the angle multiplied by the magnitude of the vector itself (in this case v). Thus, the change in the velocity vector per time unit (i.e., acceleration) is
$$a = \omega v = \frac{v^2}{r}$$
where we have used that ##\omega = v/r## for circular motion.

Here are my 5 cents on the intuition behind the magnitude of centripetal acceleration:
Sorry, I didn't get that! Maybe I'm too thick, or maybe the explanation wasn't good enough. Whatever, I don't feel bad about being thick, and you shouldn't feel bad about not producing a "good enough" explanation. Physics texts often don't provide "good enough" explanations.

In these situations, it's probably best to just accept the equation... and move on. This tactic is enouraged by Fields medallist Timothy Gowers in "Mathematics: A Very Short Introduction". He points to the fact that ambn= abm+n is an identity for which there is an intuitive explanation that many schoolkids just do not get! So does Gowers suggest they quit maths?! No he recommends they just accept the equation and move happily on without worrying their little heads too much. Intuition is hard, acceptance is easy...

sophiecentaur
Science Advisor
Gold Member
2020 Award
So many people seem to have the view that 'intuition just happens' and is somehow a bit magical. If you are honest, when you feel something intuitively, it's just that you are so familiar with all the steps to arrive at the 'proof' or derivation that you feel you can accept it. Use something that you have accepted, enough times, and it becomes intuitive. It's a big ask if you want to get intuition - just like that.
Pick something that you have an intuitive feel for and you can always track back in your history files and will find a pathway,starting with something you were told and just accepted. But without taking the formal path through the process your personal intuition can easily let you down when you want to apply it to real Physics. (Just read some of the posts on PF)

Rote learning has a really bad press these days but people of all levels of intelligence can easily learn the meaningless words in a song, the scores of past premier league football matches and the names of their favourite football team. There's nothing intuitive or logical about that stuff; they just had Motivation.
I love many of the 'facts' that I learned and now know and I'd bet that goes for all other PF members. That applies to the poems and speeches I had to learn at school, as well as the Science bits. I still remember lots of stuff that I was 'forced' to learn in English classes (and, no, the 'monks' didn't teach us in Latin).

jtbell
Mentor
i had been hoping to just read physics materials
What materials are you using? If you're trying to do everything using online video lectures and notes, you should use those in conjunction with a real textbook. Use the textbook for your "real studying" and use the lectures as a guide and supplement.

Do lots of exercises from the book. The concepts and equations that are important will eventually come back to you naturally through practice, because you'll use them over and over again, without your having to "memorize" them explicitly.

Dale
Mentor
2020 Award
2. How can i get a better natural intuition/understanding for equations?
Do homework-style problems, even if you are self-studying.

Chestermiller
Mentor
I want to strongly second what Dale Spam and others have said. You'll never get it if you just read through the book and don't do any problems. Applying what you have learned in solving problems is key. You can even dream up some problems of your own (and solve them) to solidify your understanding.

Regarding centripetal acceleration, it's only a matter of geometry, not physics. To understand the geometry, you need to understand that the direction of the velocity vector is changing. When the direction of the velocity vector changes, this translates into an acceleration perpendicular to the velocity. I'm sure your physics book shows how this plays out geometrically, and how the acceleration is quantified.

Chet

I want to strongly second what Dale Spam and others have said. You'll never get it if you just read through the book and don't do any problems...

Regarding centripetal acceleration, it's only a matter of geometry, not physics. To understand the geometry, you need to understand that the direction of the velocity vector is changing. When the direction of the velocity vector changes, this translates into an acceleration perpendicular to the velocity. I'm sure your physics book shows how this plays out geometrically, and how the acceleration is quantified.
Yes, but that's nothing to do with intuition. Dictionary definition:

intuition: "the ability to understand something instinctively, without the need for conscious reasoning."

All that geometry and quantification involves a heap of reasoning!

I think with equations like this we don't somehow gain intuition about them, we just become happy with using them, through familiarity. We see v2/r and we think "that old boy again", or, after dealing with a shed load of tricky equations we think, "another darned tricky equation, OK I'll accept that, it looks OK!"

We have an instinctual feel for very few equations, that's why Newton came along rather late in human history.

WannabeNewton
Science Advisor
Intuition just means understanding on a deeper physical level why an equation is what it is. Sometimes the derivations are what lead to the intuition, as in Chestermiller's case, and other times purely physical reasoning in retrospect leads to the intuition. Either way, simply getting used to the equations computationally without really getting a feel for what they are saying and why it is they are what they are is certainly a bad way to do physics.

If you want to develop intuition then you have to do hard problems. Purely computational problems will be useless to you in that regard because they tend to be straightforward if tedious, although getting adept at computations is also extremely important for physics. The problems that develop an intuition for a subject usually force you to think a lot to arrive at answers, perhaps in multiple different ways, using the minimal amount of computation and the maximum amount of physical reasoning e.g. "what symmetries do I use?", "what kind of approximations can I make?", "what coordinate system is best adapted to the calculation?", "how much can I guess using dimensional analysis?", "how can I use the conservation laws?", and "does my answer actually make sense physically?".

So to echo what others have said, buy introductory physics books that have conceptually difficult problems; there are a myriad of them. Also as Chestermiller said coming up with your own questions or problems can also be of great help. These problems could stem from just everyday observations or from discussions in texts/papers.

"In every scientific category, from evolution to astronomy to thermodynamics, students paused before agreeing that the earth revolves around the sun, or that pressure produces heat, or that air is composed of matter. Although we know these things are true, we have to push back against our instincts, which leads to a measurable delay.

What’s surprising about these results is that even after we internalize a scientific concept—the vast majority of adults now acknowledge the Copernican truth that the earth is not the center of the universe—that primal belief lingers in the mind. We never fully unlearn our mistaken intuitions about the world. We just learn to ignore them."

"The D.L.P.F.C. is located just behind the forehead and is one of the last brain areas to develop in young adults. It plays a crucial role in suppressing so-called unwanted representations, getting rid of those thoughts that aren’t helpful or useful. If you don’t want to think about the ice cream in the freezer, or need to focus on some tedious task, your D.L.P.F.C. is probably hard at work.

According to Dunbar, the reason the physics majors had to recruit the D.L.P.F.C. is because they were busy suppressing their intuitions, resisting the allure of Aristotle’s error. It would be so much more convenient if the laws of physics lined up with our naïve beliefs... science is full of awkward facts. And this is why believing in the right version of things takes work.

Of course, that extra mental labor isn’t always pleasant..."

http://www.newyorker.com/tech/frontal-cortex/why-we-dont-believe-in-science

2. How can i get a better natural intuition/understanding for equations?
Well, if you know me, this is the point where I recommend reading Visual Complex Analysis by Tristan Needham. I've been doing a lot of that lately, but there really is no better answer to this question that I know of, aside from maybe, the other book that I like to recommend by Hilbert and Cohn Vossen, Geometry and the imagination, or perhaps, Lines and Curves, a practical geometry handbook, being maybe the most elementary and having some relevance to basic physics, with its kinematic approach. The trick is that you have to make an argument that's susceptible to visual or conceptual reasoning. Not just any argument will do.

I agree 100% that memorization and mere acceptance of equations takes the joy out of it. Happily, it's not really necessary, if you know how to avoid it.

For example, for the acceleration of a rotating object, v^2/r, that is instantly obvious to me. It's not obvious in the sense that I could just look at v^2/r the first time I saw it. It's instantly obvious because over time, I learned to mentally take the derivative of a rotating vector based at the origin, getting a vector that's ahead by 90 degrees and multiplied by the angular velocity. So that's w^2 r, which is wv because I also have the picture that you multiply w by r to get arclength at my fingertips. Then, to get it all in terms of v, multiply by r over r, simplify again, and you have v^2/r. So, a little visualization and mental algebra makes it obvious. You just have to mentally rehearse this kind of thing in your mind enough and then eventually it becomes obvious after enough practice.

It's sometimes true that it's easier to accept and memorize, but to me, that takes all the fun out of it, and the whole purpose, to me, is to have fun. If it was all matter of doing ugly calculations and then memorizing the result and/or memorizing other people's ideas, I would just go do something else because it just wouldn't be interesting. I would go be an artist or a musician. Why bother with lifeless and boring physics, consisting of large messes of symbols? I wouldn't want to do that, and I don't see why anyone else would. Sure, it's good to be able to manipulate the laws of nature, but if you can't make them come alive, well, then, I pity anyone with the misfortune to pursue such a boring and lifeless pursuit. Luckily, that isn't what physics is about, as people like Feynman and many others have shown us. At most, that sort of thing is an annoyance that you might have to put up with some of the time in order to also do the fun part.

Also, one reason why it's easier to just accept things, rather than bothering to understand them is that a lot of times people don't present things in a way that's susceptible to conceptual understanding, so that if you want to do that, you have to do a lot of the work yourself.

Nathanael
Homework Helper
In these situations, it's probably best to just accept the equation... and move on. This tactic is enouraged by Fields medallist Timothy Gowers in "Mathematics: A Very Short Introduction". He points to the fact that ambn= abm+n is an identity for which there is an intuitive explanation that many schoolkids just do not get! So does Gowers suggest they quit maths?! No he recommends they just accept the equation and move happily on without worrying their little heads too much. Intuition is hard, acceptance is easy...
I don't recommend you quit physics if you don't understand a formula. Nor do I recommend stopping everything completely until you understand it. I'm not even against accepting an equation or an idea. I think it's a good point that you may need to accept that you'll just have to accept an equation (for the time being).
But, I also do not think the kids should spend the rest of their lives not knowing why ambn= abm+n.
(Besides, that equation isn't even true! Maybe their motive to learn why it's true can be so that they don't misremember it. But curiousity is a more enjoyable motive, anyway.)
Accept any and all equations you want to, get a feel for things. That's fine. But do not think that's the end of it.

If you just want to pass a class, then don't 'worry your little head about it.'

(But I don't think the OP just wants to pass a class, seeing as they're not in a class.)

Who said that studying physics would be a joy?
Who said it can't be?

It comes from Newton's Laws ... if an object travels in a straight line with speed v, then applying a constant force, perpendicular to it's motion, will change the object's direction by F=ma. Keep the force perpendicular to the velocity, and the object will go in a circle. So calculate the radius of the circle.

In this case, the core concepts you need are in Newton's Laws - you do just have to know those.
Though Newton's laws are essential to learn physics, they are not needed in order to understand why centripetal acceleration = v2/r. This is simple kinematics and calculus (and geometry).

Read books like "Mad About Physics" and "The Flying Circus of Physics."