How does one build up intuition in physics? How does one build up an intuitive gut feeling for physics that some people naturally have? Physics seems to be a hodgepodge of random facts.
Is that a sign to quit physics and take up something easier?
Thanks for all the answers. On a related note, how many years does it take to master physics? 1-2 years for each level multiplied by many levels gives? 
 A: Intuition is great, but it can also be a trap. I had extreme inutition about physics during highschool and undergraduate days. Probably at the one in a million level. It made much of physics and math almost effortlessly easy. But I basically trainwrecked in grad school. And the reason is that things reached a level of abstraction that was beyond what I could intuit. And all my classmates -none of whom half the intuition I had, had been doing it the hard way, learning symbolically, and mathematically for many years -probably out of necessity. And I suddenly realized I would need to back up several years and relearn things that way. And that was far too much backtrcking to contemplate at that stage.
So I suspect, that having one area of cognition that you are supergood in at is actually a setup for failure. Train all of the needed abilities with respect to physics. Not just the ones that you are most good at. Otherwise you might find yourself in a col-de-sac.
A: If physics seems to be a hodgepodge of random facts... well, it always seems that way when you're first learning. But physics is all about patterns and relationships, and if you keep working with it you should eventually get to a point where you see how different facts, formulas, and principles are related to each other. In my experience, for any given "piece" of physics this point comes 1-2 years after you first learn it. :-P
EDIT: In response to your related note, I don't know that anyone ever really "masters" physics. I'm not even sure exactly what that would mean, since physics as a whole is far too broad for any one person to ever understand in its entirety. But my impression is that, given the usual educational path (college then grad school), it takes anywhere from 3-10 years, depending on the topic and your level of preparation, to learn enough that you can start contributing to research in a single specific topic. Of course you never really stop learning.
Also, to clarify, the 1-2 year figure I mentioned isn't something you can multiply by a number to figure out how long it takes to learn physics ;-) And physics doesn't neatly split itself into levels (well, sort of, although the splitting you'll probably encounter is just a result of the requirements of our educational system). All I meant is that, from the first time you encounter any given physics concept (just to name a few: Newton's laws, the Euler-Lagrange equation, multipole expansion, free energy, quantum entanglement), you may not feel like that concept makes sense to you until 1-2 years later, on average. And that's perfectly normal.
A: Intuition is just pattern recognition. This comes with doing many problems that force you to think and keep your brain completely engaged with the material. I remember when I first started physics, I thought it was impossible to keep track of all of the types of problems. There were balls rolling down inclined planes and masses on pulleys and people pushing blocks up hills with friction. Then there was a certain point when a lightbulb went off and it made sense: these are all the same problem. All you do is figure out where all the forces are pointing and how strong they are, then add them up  to find the net force, and you've got the equation of motion in hand. Eventually, it becomes second nature, so I can glance at a problem in mechanics and pretty quickly have an intuition for what should happen.
The same thing happens in more advanced physics. You start to recognize classes of physical situations and types of symmetry. Let me give you a a more advanced example that was a similar lightbulb for me - it's a bit lengthy, and you don't need to understand every step. The important part is the conclusion. Try evaluating this integral:
$\int_{-\infty}^{\infty}e^{ip(x-x_0)/\hbar}e^{-|p|/p_0}dp$
This looks horrible, doesn't it? And yes, if you just charge in and try to integrate it, you'll wind up doing a long series of painful integrations by parts. But there are a bunch of things here that a practiced physics student will recognize that make it easy. 
Since $e^{ip(x-x_0)/\hbar}=\cos(p(x-x_0)/\hbar)+i\sin(p(x-x_0)/\hbar)$ (that's clever trick no. 1), this is actually an integral of a cosine plus a sine, multiplied by a decaying exponential, over $(-\infty,\infty)$. But sine is an odd function, so it evaluates to zero over $(-\infty,\infty)$ (clever trick no. 2)! That means the integral is actually
$\int_{-\infty}^{\infty}\cos(p(x-x_0)/\hbar)e^{-|p|/p_0}dp$ Now, we have an even function inside the integral, so its value over $(-\infty,\infty)$ is twice its value over $(0,\infty)$ (clever trick no. 2, reused). Furthermore, $\cos(p(x-x_0)/\hbar)=Re(e^{ip(x-x_0)/\hbar})$ (going backwards with clever trick no. 1). This makes the integral
$2Re(\int_{0}^{\infty}e^{ip(x-x_0)/\hbar}e^{-p/p_0}dp)$
$2Re(\int_{0}^{\infty}e^{(i(x-x_0)/\hbar - 1/p_0)p}dp)$
$2Re[\frac{1}{i(x-x_0)/\hbar + 1/p_0}]$
$\frac{2p_0}{1+(x-x_0)^2p_0^2/\hbar^2}$ (Maybe clever trick no. 3 if you aren't very used to complex numbers?)
Here's the kicker: If you do it enough, this sort of integral starts coming intuitively and quickly. The even-and-odd tricks become obvious; the conversion back and forth between exponential and trigonometric functions becomes totally natural. An integral like this flies by at the speed of thought, and I don't need to write out all those steps - it just feels sort of obvious at each point. That's just pattern recognition from doing it so many times - or, if you like, "intuition." 
If you want to build up your intuition, do many, many problems. As many as you can get your hands on. That's what it takes.
A: This question has similarities to at least this previous Stack question on learning physics. I would advise that you check the online links there.
An intuition for physics is built by learning the subject and finding out the major theories and their domain of application. So considering a question on Gravity (say working out the gravity on the surface of the Moon) what physics theories apply? As one learns more physics one learns about the limitations of older theories. Perhaps velocities have to be small, or masses below a certain limit: above that limit another related theory applies.
There are general principles from the major theories which derive much of one's analysis: energy and its conservation and transmission, radiation (of the various kinds). In applied examples a major test of intuition is to include the contributions of factors that are relevant and omit those which are irrelevant. So for example does the chemical composition of the Moon affect its surface gravity, are hollows and bumps on its surface likely to be relevant; could its nearness to the Earth affect the calculation? If we are considering a Pulsar (Neutron star) rather than the Moon do any of these factors change in significance?
Mathematically there are parallels between many theories which contain fields which decay at $1/r^2$; there are many phenomena describable by similar mathematical models like Harmonic Oscillators (from electronics to quantum physics);then there is the Wave Equation. This allows one to learn to expect similar behaviour in a unfamiliar sub-branch of the subject whenever similar phenomena appear. Indeed one may have already solved a given problem in other variables already.
Finally there is some evidence to support the hope that all of physics is united under a set of underlying equations and principles. They may not all have been discovered yet, but this suggests that understanding one set of concepts very well could teach us how everything works. We already know of the major branches of physics which are very comprehensive: Thermodynamics; Electromagnetism; Gravitation; Dynamics; Quantum Theory; then there are the subatomic theories.
A: Speaking of patterns, you may want to read Shive's Similarities in Physics.  It discusses common behaviour patterns such as filtering, dispersion, resonance, interference, exponential decay, noise, etc which can be observed across acoustics, optics, mechanics and electronics.  
There is a book by Pikovsky et al Synchronization: A Universal Concept in Nonlinear Sciences. is also interesting in this regard.
Also take a look at Terence Tao's article on Universality in large dimensions.
A: I think the intuitive sense (your 'intuitive gut feeling'mentioned above)that one develops for physics starts with a passion for the subject itself. In my case that's where it all started. As a child, I recalled asking my father why his airplane flies. Later, when I was able to study physics, I could recall some relational experiences such as scuba diving (gas laws), airplane flight (Bernoulli's principle), list goes on. In all instances, I recall thinking oh yes, this makes sense, and from yhere I could extrapolate to another scenerio for example; fluid flow in tunnels as related to Bernoulli's principle. It gives you a sense of being part and parcel of the whole physics experience which is extremely gratifying. As I progressed through my graduate curriculuum to a PhD, the main areas of physics; Newtonian mechanics, quantum, statistical mechanics and electromagnetic theory seemed to be intricately linked together to explain the structure of the universe along with processes occuring within. This is not a 'hodge-podge of random facts' as you mention. By studying physics and getting the feel and understanding of physics, one doesn't neccessarily have to memorize equations, they just come naturally.
